Improved loss estimation for Victoria

There is now a “Jurisdictional Specifics” page for Victorian on the ARR data hub which is based on a project undertaken by HARC which I was involved in. Our work looked at whether adopting losses from the ARR Data Hub would result in “good” flood estimates from hydrologic modelling. We defined good estimates as when the design flood peaks from modelling are close to the equivalent peaks from flood frequency analysis of gauged data.

In short, the Data Hub losses tend to be too large, which means modelled flood estimates are biassed low. We recommend this is addressed by increasing the preburst rainfall – using the 75th percentile value rather than the median. A caveat is that his recommendation only applies for continuing loss region 3 (see the figure below). There wasn’t enough information to make recommendations for the other loss regions.

Loss regions for Victoria

There is more information available from:

Of course, the Data Hub isn’t the only source of loss estimates. A key outcome of the project was to develop an ordered list of approaches for estimating losses. Using losses from the Data Hub should only be contemplated when there is no better alternative.

Preferred approaches to loss estimates are, in order:

  1. Reconciliation with at-site flood frequency quantiles: initial and continuing losses are varied within their expected range to achieve a reasonable level of agreement between estimates derived from rainfall-based modelling and flood frequency analysis.
  2. Reconciliation using within-catchment transposed flood quantiles: streamflow observations are commonly available at gauging stations upstream or downstream of the site of interest, and flood quantiles derived from these sites can be transposed to the site of interest and used for reconciliation as described in approach 1.
  3. Event-based calibration: continuing losses obtained from calibration of historical events provide some indication of typical design values, noting that past historical events are biased towards wet catchment conditions; initial losses from historical events are highly variable and information from a small sample of events are of low utility (and therefore some form of reconciliation with other sources of information is recommended).
  4. Reconciliation using nearby catchment transposed flood quantiles: regional flood quantiles derived using RFFE and other procedures (Section 3, Book 3, ARR2019) can be used for reconciliation as described in approach 1.
  5. Transposition of losses: initial and continuing loss estimates validated on nearby catchments which are considered to be hydrologically similar.
  6. Regional losses (ARR Data Hub): unmodified initial and continuing loss estimates obtained from the Data Hub losses can be adopted in data poor areas, noting that in loss region 3 these should be combined with 75th percentilepre-burst values.

A similar list has also been developed for NSW which, unfortunately, differs from the this one developed for Victoria. The top recommendation from the NSW list is to use calibration losses from the study catchment. Calibration losses are the initial and continue loss that is required to make the output of a hydrologic model match a historical flood peak when historical rainfall is used as the input. This is number 3 on the Victorian list.

Usually, we calibrate to large floods. On average, large floods are likely to have small losses – that is one of the factors that caused the large flood. Therefore, calibration losses may not provide a representative sample of the loss from a catchment; they may be biassed low. Using losses that are biassed low means modelled flood estimates may be too high. Where possible, it would be checking losses through reconciliation with flood frequency analysis of gauged data, using tools such as the RFFE and comparing losses on nearby catchments (Items 2, 4 and 5 on the Victorian list).

A review of temporal patterns from Australian Rainfall and Runoff

Paper presented to the Hydrology and Water Resources Symposium 2021

Abstract

The recommendation in Australian Rainfall and Runoff (ARR) is that an ensemble of 10 rainfall temporal patterns is used for modelling. Patterns depend on region, catchment area and duration and are provided on the ARR Data Hub.  Most available patterns are consistent and appropriate but there are a few which are physically unrealistic and appear to be erroneous.  A method was developed to identify problem patterns and all available areal and point patterns across Australia were checked.  

Nineteen problem areal temporal patterns were identified in 3 regions.  Areal patterns listed on the Data Hub for one region are sometimes borrowed from neighbouring regions and it was found that all the problem patterns originated in the Murray Basin in a small area near Jerangle, 100 km south of Canberra.  The problem seems to be that rainfall accumulated over 24 hours has been allocated to a shorter time step in the pluviograph record.  

This problem means that these patterns are not likely to be suitable for modelling and may lead to unrealistically large flows from long-duration events.  A work around is to exclude these patterns and if necessary, replace them, or use ensembles with fewer than 10 patterns.

Analysis also identified the least uniform point and areal patterns.  These are not necessarily in error but do have the potential to contain “embedded bursts” which occur when a period of rainfall within a temporal pattern has an annual exceedance probability rarer than the burst as a whole.  Embedded bursts can cause issues with modelling and modellers need to consider if they should be removed or smoothed.

Pattern 7 is an example of a ‘problem’ pattern

RegionDurationAreaEvent ID
Central Slopes485003950
 4810003951
 722004029
 725004043
 7210004049
 962004116
 962004119
 1202004207
Murray Basin365005644
 485005731 
 485005732 
 722005810 
 725005822 
 7210005829 
 962005901 
 962005898
 1202005989
Southern Slopes (mainland)962006797
 1202006884
Table 1. List of the 19 problem patterns

Paper

Presentation

Citation:

Ladson, A. R. (2021) Review of temporal patterns from Australian Rainfall and Runoff 2019. 39th Hydrology and Water Resources Symposium. Engineers, Australia.

Can flood control works change a state boundary?

In 1961, Minnesota agreed to ceed land to North Dakota because of a change in course of the Red River of the North as a result of flood control works. Two parcels of land had become detached from Minnesota and attached to North Dakota because of a project undertaken by the US Army Corps of Engineers which cutoff bends of the river.

The State legislatures agreed that State boundaries should be redraw and sought and were granted approval from the US House and Senate. The Minnesota-North Dakota Boundary Line Compact was approved by the US House and Senate on August 25, 1961 as Public Law 87-162. Details are available here.

Both areas are near Fargo North Dakota. The screenshots show the areas and a description of the land is provided in the captions. The maps are sourced from https://www.randymajors.org/township-range-on-google-maps.

NE 1/4 of Section 29, Township 140 North, Range 48 West of the 5th Principal Meridian, Clay County Minnesota (9.78 Acres)
NE 1/4 Section 7, Township 139 North, Range 48 West of the 5th Principal Meridian (12.76 Acres)

Errors in variables regression

The usual assumption in regression is that that all the error is in the y values; the dependent variable; the x values (the independent variable) are known without error.

For a recent project, I had to undertake some analysis where this assumption wasn’t reasonable. There were errors in both variables. In this case, we were looking at the relationship between turbidity and clarity. Turbidity was measured by a meter; clarity was measured as the minimum depth of water that just obscured a target.

As example of the issues, let’s consider a simple case; 10 made up values.

Figure 1: 10 values with random errors around a 1:1 line

For ordinary least squares regression, we calculate a line that minimises the sum of the squared distances between all the points and the line. Only the y distances are considered.

Figure 2: Ordinary least squares finds a line that minimises the sum of the squared distances in the y direction

Next consider the case where the y values are known perfectly all all the error is in the x values. This will give a different regression line.

Figure 3: A different line is obtained if the squared distances are minimised considering only the x-values

We can also consider errors in both x and y values.  Different approaches to this problem are referred to as Errors-in-variables regression, Deming regression or total least squares.  Figure 4 shows the special case of orthogonal regression where the line is calculated by minimising the squared perpendicular distances. This occurs were the variance of the x and y values are equal.  Deming regression can also handle cases where the ratio of the variances is not equal to 1.

The three lines are compared in Figure 5.

Figure 4: Orthogonal regression minimises the sum of the squared perpendicular distances
Figure 5: Comparing the three lines

R code to produce the figures is available as a gist. There is also an example of orthogonal regression in excel available at this link.

On prebust depths and ratios

Preburst rainfall is the rain that falls before a ‘burst’.  Burst rainfall is what you get from IFD information on the Bureau of Meteorology website.  In hydrologic modelling, it is often important to add preburst to burst rainfall to create a ‘design’ storm.

You can look up preburst rainfall on the ARR data hub.   An example is shown below which is for the data hub default location in central Sydney (Longitude = 151.206E, Latitude = -33.87N). These are median preburst values. Data is also available for the 10th, 25th, 75th, and 90th percentiles.

Each cell provides the prebust depth in mm and the preburst ratio (in brackets) for a specified burst AEP and duration.  For example, the top-left cell represents the 50% AEP, 60 min, preburst depth which is 12.0 mm and ratio which is 0.372.  

The ratio is the preburst depth/burst depth.

It’s possible to look up the burst depth, for a given AEP and duration, at the in the IFD website.  

For this location, the 50% AEP, 60 min burst depth is 32.3 mm.  Mutliplying the burst depth by the preburst ratio gives

32.3 x 0.372 = 12.02 mm which is the same as the preburst depth given in the table. 

Sometimes the numbers don’t quite work.  For example the 2160 min (36 hour) burst depth is 357 mm.  With a burst ratio of 0.032, the preburst depth works out to 11.4 mm, compared to 11.6 mm in the table.  A small difference possibly cause by a rounding error or a slightly different IFD values.  The preburst work was based on the interim 2013 IFD data which has since been superseded by the 2016 release.  

Sampling distribution of the 1% flood

A sampling distribution is the probability distribution of a statistic calculated from a sample. In this case, the statistic I’m interested in is the number of floods that may occur in a sample of a certain number of years.

In engineering hydrology, and land use planning, we are often interested in the 1% (1 in 100) AEP flood. If the period of interest is 100 years, say the design life of a house, how many 1% floods should I expect?

We can calculate this using the binomial probability distribution. The general formula is:

\binom{n}{r}p^r(1-p)^{(n-r)}

Where, n is the number of years (the sample size), p is the probability of a ‘success’ (a flood in this case), r is the number of successes.

So the probability of one 1% flood occuring in 100-years is:

\binom{100}{1}0.01^1(1 - 0.01)^{(100-1)} = 0.37

To determine the sampling distribution we need to calculate the probability of getting 0 floods, 1 flood, 2 floods etc. This is shown in Figure 1. The average, or expected number of 1% floods in 100-years is 1, but it’s likely the actual number will be between zero and 4. The probability of getting 5 or more is low, 0.34%

Figure 1: Probability of different number of 1% AEP floods occurring in 100-years

What if our estimate of the 1% flood is not accurate?

Usually the magnitude of the 1% flood is estimated using flood frequency analysis applied to a reasonably short gauge record. Figure 2 shows a flood frequency analysis of 46 years of flood data for the Tyers River at Browns (see this post for details). The best estimate of the 1% AEP discharge is 245 cumec. However, it is quite plausible that this discharge has a true probability of exceedance of 2% rather than 1%.

Figure 2: Flood frequency estimates for the Tyers River at Browns

Consider the scenario where 245 cumec is used for the design flood, and land use planning is based on water levels for this event, however the true exceedance probability is 2% rather than 1%.

The distribution of 2% AEP floods in 100-years is shown in Figure 3. The probability of getting at least 1 flood has increased from 0.63 to 0.87. The probably of getting 5 or more floods has increased by a factor of 15 and is now, 5%. Overall, the risk of flooding has doubled, but the probability of a possibly catastrophic outcome, multiple episodes of flooding, has increased substantially.

Going the other way, if our design flood has a lower AEP value, say 0.5% then the probability of at least one flood drops to 0.39.

Figure 3: Sampling distribution of 2% AEP floods in 100 years

This adds support to taking overall risk into account when selecting design flood levels, rather than just adopting a 1% value and ignoring uncertainty. A risk-based design approach is recommended in Australian Rainfall and Runoff (Book 1, Chapter 5).

This gist has the code to create the figures.

Smooth interpolation of ARF curves

As discussed in the previous post, the areal reduction factors, developed for Australia must be interpolated for durations between 12 and 24 hours. The long-duration equations apply for 24 hours or more. The short duration equations apply for 12 hours or less. In between, the recommendation is to interpolate, linearly (Figure 1) (Podger et al., 2015). This post looks at an alternative to linear interpolation.

Figure 1: Areal reduction factor as a function of duration. ARFs are linearly interpolated between 12 and 24 hours

One approach would be to interpolate in such a way that that there is a smooth curve between the short and long duration values. This can be achieved by matching the slopes, as well as the values, at the end points.

That means there are 4 conditions that must be satisfied. The curve must pass through the two end points and match the slopes (derivatives) at the two end points. This can be achieved with a cubic polynomial.

We have the equations for short and long duration ARFs, and with some help from an on-line differentiator, we can find the derivatives. Its also possible to numerically differentiate the ARF equations.

As in the earlier posts, D is duration (min), P is AEP, A is area in km2.

Short duration:

\mathrm{ARF}= \min[1,1-0.287(A^{0.265}-0.439\log_{10}D)D^{-0.36} \\  +0.00226A^{0.226}D^{0.125}(0.3+\log_{10}P) \\  +0.0141A^{0.213}10^{-0.021\frac{(D-180)^2}{1440}}(0.3+\log_{10}P)]

Derivative with respect to duration:

\frac{0.10332 (A^{0.265} - 0.190655 \log(D))}{(D^{1.36})}\\ +  \frac{0.0002825 A^{0.226}( \frac{\log(P)}{\log(10)} + 0.3)}{D^{0.875}}\\ -  9.46938\times10^{-7}10^{(-0.0000145833 (D - 180)^2)} A^{0.213} (D - 180) (\frac{ \log(P)}{\log(10)} + 0.3) \\ +  \frac{0.0547181}{D^{1.36}}

For the long duration equation, the constants, a, b, c, d, e, f, g, h, i, depend on region. See this post for a map of the Australian regions, and the values of the constants.

Long duration:

\mathrm{ARF}= \min[1,1-a(A^b-c\log_{10}D)D^{-d}\\  +eA^fD^g(0.3+\log_{10}P) \\  +h10^{\frac{iAD}{1440}}(0.3+\log_{10}P)]

Derivative with respect to duration:

a d D^{(-d - 1)} \frac{(A^b - (c  \log(D))}{\log(10))} + \\      \frac{a c D^{(-d - 1)}}{\log(10)} +      e g A^f D^{(g - 1)} (\frac{\log(P)}{\log(10)} + 0.3) + \\     (1/9)  2^{((A  D  i)/1440 - 5)} 5^{((A  D i)/1440 - 1)} A h i  \log(10) (\frac{\log(P)}{\log(10)} + 0.3)

For the example shown in Figure 1, we have:

AEP = 0.005, Area = 1000 km2 Region = ‘Tasmania’.

For a duration of 24 hours (1440 min), long duration ARF = 0.8771, slope = 2.019e-05

For a duration of 12 hours (720 min), short duration ARF = 0.8171, slope = 6.554e-05

A cubic polynomial can be expressed as:

ax^3 + bx^2 + cx + d

So we need to solve for the four unknowns, using the 4 conditions. In this case the x value is duration.

We can set this up as a matrix equation

\begin{bmatrix} x_1^3 &x_1^2  &x_1  & 1 \\  x_2^3&x_2^2  &x_2  & 1 \\  3x_1^2 &2x_1  & 1  & 0 \\  3x_2^2& 2x_2 &1  &0  \end{bmatrix}\begin{bmatrix} a\\  b\\  c\\  d \end{bmatrix}=\begin{bmatrix} f(x_1)\\  f(x_2)\\  f'(x_1)\\  f'(x_2) \end{bmatrix}

Where, x1 is 720 and x2 = 1440, f(x1) = ARF at 720 min, f(x2) = ARF at 1440 min. f'(x1) = derivative of the short duration ARF equation at 720 min f'(x2) is derivative of the long duration ARF equation at 1440 min.

We can then solve for a, b, c and d and draw in the cubic polynomial (Figure 2). The polynomial interpolation doesn’t make a lot of difference but does mean a ‘kink’ in ARF values is avoided.

Figure 2: Polynomial interpolation between 12 hours (720 min) and 24 hours (1440 min)

Code is available as a gist

References

Podger, S., Green, J., Stensmyr, P. and Babister, M. (2015).  Combining long and short duration areal reduction factors. Hydrology Water Resources Symposium (link to paper at Informit).

ARR2019 – Areal Reduction Factors: interpolating between short and long duration ARFs

There are two equations provided for areal reduction factors (ARFs) in Australian Rainfall and Runoff 2019. One for long durations, greater than 1440 minutes (24 hours), and one for short durations, less than 720 minutes (12 hours).

To estimate ARFs between 12 hours and 24 hours we need to interpolate.  The recommendation in Australian Rainfall and Runoff 2019 is to do the required interpolation linearly (Podger et al., 2015)  Complicating matters, the short duration ARF equations are only suitable for catchment areas less than 1000 km2 while long duration ARFs are available for catchments out to 30,000 km2. The situation is summarised on Figure 1.

Surprisingly, ARR2019, recommends that interpolation can be used for catchment areas all the way out to 30,000 km2.  That is, we can use the short duration equations as end points for interpolation for catchment areas larger than 1000 km2.   This involves applying the short duration equations well beyond the range of the data that was used in their derivation (Stensmyr and Babister, 2015).

interp

Figure 1: Application of short and long duration ARF equations as a function of catchment area and duration

Let’s start with the equations:

Short duration:

 \mathrm{ARF}= \min[1,1-0.287(A^{0.265}-0.439\log_{10}D)D^{-0.36} \\ +0.00226A^{0.226}D^{0.125}(0.3+\log_{10}P) \\ +0.0141A^{0.213}10^{-0.021\frac{(D-180)^2}{1440}}(0.3+\log_{10}P)] (equation 1)

Long duration:

\mathrm{ARF}= \min[1,1-a(A^b-c\log_{10}D)D^{-d}\\ +eA^fD^g(0.3+\log_{10}P) \\ +h10^{\frac{iAD}{1440}}(0.3+\log_{10}P)] (equation 2)

where, A is area, D is duration and P is AEP.

The constants a, b, c, d, e, f, g, h, i depend on the region (Figure 2).  The values of these constants is shown below in a screen shot of Table 2.4.2 from ARR Book 2, Chapter 4.3.1.

2059

Figure 2: ARF regions for Australia (Copy of ARR Book 2, Figure 2.4.1)

ARR_Table_2-4-2

Here, we are interested in the case where area is larger than 1000 km2 and duration falls between 12 and 24 hours.  Figure 2 shows an example for a catchment area of 30,000 km2.  Between 12 and 24 hours, the ARFs are linearly interpolated between the long duration value at 24 hours and the short duration value calculated at 12 hours (solid line in Figure 2).  The short duration 12 hour value is subject to considerable uncertainty because the prediction equation was developed using data from catchments smaller than 1000 km2 , much smaller than the 30,000 km2 catchment area used here.

My first though was, if we require ARF estimates for durations between 12 and 24 hours, it would be better to just extend the long duration equations (the dashed line in Figure 2).   This would result in more conservative ARFs in the interpolation region.  And it is not obvious why extending the long duration equations is any worse than extrapolating short duration values and then interpolating from these values.  However, I now realise there are problems with this approach.

ARF_interp

Figure 2: Interpolating between long and short duration ARFs

Extrapolating the long duration ARFs does not work for catchment areas close to 1000 km2  This can be seen on Figure 3.  For a catchment area of 1000 km2 , both short and long duration ARFs can be calculated and the extrapolation between 12 and 24 hours is reasonable because the end points will be accurate.  Both equations are being used where there is data to support their derivation.   This is the red line on Figure 3.

Now consider the case for a catchment of 1100 km2. The Short duration ARFs can no longer be calculated (because the limit of 1000 km2  is exceeded).   So, to calculate the ARF between 12 and 24 hours we could, either

1) Extend the long duration equation (dashed green line in Figure 3), or

2) Interpolate between the long duration value at 24 hours and the short duration value at 12 hours (the green solid line in Figure 3).

In this case, only approach 2 will work.  If we adopt approach 1, the ARFs for a larger 1100 kmcatchment will exceed those for the smaller 1000 km2 catchment (the dashed green line exceeds the solid red line).  This is not physically realistic.  As area increases, ARF is expected to decrease.

ARF_interp_3

Figure 3: Interpolating ARF values between 12 and 24 hours for a 30,000 km2 catchment

The upshot is, if you are working on catchments with areas only a bit larger than 1000 km2 it would be best to do the ARF calculations using the recommended method, to avoid discontinuities in ARF values.

If you are working on catchments much larger than 1000 km2 , and need to calculate an ARF between 12 and 24 hours, it may be worth checking the ARF values that result from extending the long-duration equations.   They will likely to be conservative, in that they will produce higher flood flows than the ARR recommended approach.  It is also in this region, i.e. where catchments area are much larger than 1000 km2,  that there is not strong justification for using the short duration equations as end points for the interpolation.  The support for these short duration equations reduces as catchment areas increase beyond the 1000 km2  limit that was used in their derivation.

Code to produce graphs is available as a Gist.

 

Reference

Stensmyr, P. and Babister, M. (2015) Short duration areal reduction factors.  Australian Rainfall and Runoff Revision Project 2: Spatial Patterns of Rainfall.  Stage 3 Report.  September 2015) (link).

Podger, S., Green, J., Stensmyr, P. and Babister, M. (2015).  Combining long and short duration areal reduction factors. Hydrology Water Resources Symposium (link to paper at Informit).

Comments on the report: “Impact of lower inflows on state shares under the Murray-Darling Basin Agreement”

First, this report is worth reading, its short and clearly written.  It’s produced by the Interim Inspector General of Murray-Darling Basin Water Resources, which is Mick Keelty AO, the former commissioner of the Australian Federal Police

The big story is that inflows to the Murray have decreased, a lot.

Inflows upstream of Albury

“…half of the driest years on record have occurred in the past 25 years”.

This sounds bad but its difficult to know what it means, however there is a telling graph of NSW inflows (Figure 1).  The last 20 years or so do not look some random perturbation of what has happened before.  This century looks very dry.  Counting the red bars suggests that 11 of the 12 driest years have occurred in the last 20 years.  And there is nothing in the historical record like the period from the mid 1990s to 2010/11 with year after year of low flows.

NSW_inflows

Figure 1: NSW inflows

Total inflows to the Murray

The report says: “More than half of the driest 10% of years in the historical record have occurred in the past two decades”.

So how unlikely is that?

Here is one interpretation.  The historical record is 126 years, from 1895 to 2020.  There would be, say, 13 years in “the driest 10% of years”, so “more than half” would be 7.

If we assume each year as independent, admittedly a strong assumption, then we can work out the probability of this occurring using the hypergeometric distribution*.

We need to calculate the probability of 7 or more “successes” from 13 items that could be labelled a success from a total population of 126.

The formula in excel is:  1-HYPGEOM.DIST(6,20,13,126, TRUE) = 0.000854

In R it’s 1-phyper(6, 13, 113, 20)

An alternative approach is to do a simulation.  Arrange the numbers 1 to 126 in random order and see how many of the numbers in the range 1 to 13 (representing the driest 10% of years) occur in the final 20 elements of that random series. Do this lots of times and count the number of occasions when this is 7 or more.

My answer is 0.000847 which is close to the theoretical answer so we are on the right track.

Here’s the code

Num_in_last_20 = function()  
{ 
  Ranks = sample(1:126, 126, replace = FALSE) 
  # sort the numbers 1 to 125 in random order 
  Last20 = Ranks[(length(Ranks) - 19): length(Ranks)] 
  # Get the last 20 elements 
  sum(Last20 %in% 1:13) # Number of ranks that are 13 or lower 
}  

# Repeat lots of times to determine the 
# probability there are 7 or more ranks less than 13 
  
mean(replicate(1000000, Num_in_last_20() >= 7))

These probabilities are likely underestimates, because the years are not independent, but they do suggest we should consider if there is a downward trend rather than just a random fluctuation.

Darling inflows

“Median inflows into the Menindee Lakes have reduced by about 80% in the last 20 years relative to the recorded period prior. Eight of the 13 driest years on record occurred in this period, most yielding zero or close to zero inflows.”

This is a similar situation to the total inflows to the Murray where 7 of the 13 driest years had occurred in the last 20 years.

So what is causing the low flows since 2000?  The report says there are numerous factors:

  • low rainfall
  • high temperatures
  • catchment modification (including farm dams)
  • increasing development
  • floodplain harvesting
  • changes in extraction rules in water sharing plans
  • non-compliance.

Interestingly, there was virtually no change in inflows from the Snowy River.

Recommendation 1 of the report is:

“The MDBA should undertake further analysis of the causes of reduced inflows from the northern Basin and the extent to which this is affecting State water shares.”

It will be interesting to see what comes from that.

Pie chart and an alternative

There is lots of discussion on the internet about how pie charts are not a great way to show data.  People have difficulty judging angles and ordering pie slices from large to small (see the discussion in Wikipedia).  This report doesn’t shy away from a complicated pie chart.  The alternative column graph is shown below.  Do you think you could order the issues correctly just looking at the pie slices?

Code for this blog is available as a gist.

Issues_piechart

issues_bar

* Probability was calculated with some help from cross-validated.

ARR2019 – Areal Reduction Factors: wobbles in short duration ARFs

In the previous post we saw that short duration areal reduction factors (ARFs) looked a little wobbly for rare events (AEPs of 1% or less).  Here I look at this in more detail.

A plot of areal reduction factor against duration, for an AEP of 0.5% and for various areas, shows the issue (Figure 1). The long duration ARFs for durations greater than 1440 mins (24 hours), look to be well behaved.  But the short duration ARFs show a rapid and variable change in slope for durations between about 100 min and 720 min.  Also shown on Figure 1 is the gap between the short and long duration ARFs, between 720 min and 1440 min.  ARFs in this range are linearly interpolated. They don’t look linear on Figure 1 but that is because of the log scale on the x-axis.  I’ll write more about the interpolation between short and long duration ARFs in a later post.  Here we focus on the wobbles in the short duration ARFs.

ARF-duration-smallAEP

Figure 1: Areal reduction factor as a function of duration for various catchment areas and for an AEP of 0.5%

The equations that describe ARF for short and long durations, are functions of:

  • D, duration in min
  • A, area in km2
  • P, annual exceedance probability (AEP)

Short duration:

 \mathrm{ARF}= \min[1,1-0.287(A^{0.265}-0.439\log_{10}D)D^{-0.36} \\ +0.00226A^{0.226}D^{0.125}(0.3+\log_{10}P) \\ +0.0141A^{0.213}10^{-0.021\frac{(D-180)^2}{1440}}(0.3+\log_{10}P)] (equation 1)

Long duration:

\mathrm{ARF}= \min[1,1-a(A^b-c\log_{10}D)D^{-d}\\ +eA^fD^g(0.3+\log_{10}P) \\ +h10^{\frac{iAD}{1440}}(0.3+\log_{10}P)] (equation 2)

The constants in the long duration equation depend on the region (there are 10 regions across Australia; a region map is shown as figure 1 in this post).  R functions to calculate the ARFs are here.

Note that these equations are of the same form, part from the final part, with the short duration equation containing a term with (D-180)2 .

We can get a better appreciation of the change in slope of the short duration equation by looking at the derivative.

Differentiating equation 1 with respect to duration gives:

\frac{0.10332 (A^{0.265} - 0.190655 \log(D))}{(D^{1.36})}\\ + \frac{0.0002825 A^{0.226}( \frac{\log(P)}{\log(10)} + 0.3)}{D^{0.875}}\\ - 9.46938\times10^{-7}10^{(-0.0000145833 (D - 180)^2)} A^{0.213} (D - 180) (\frac{ \log(P)}{\log(10)} + 0.3) \\ + \frac{0.0547181}{D^{1.36}}

This is plotted on Figure 2 (lower panel) which confirms the rapid change in slope that occurs around a duration of 180 min.

ARF-derivative

Figure 2: Short duration areal reduction factor and derivative; note log scale on the derivative

Another way to investigate the issue is the break equation up into parts and see how each part varies with duration.

Part 1:

-0.287(A^{0.265}-0.439\log_{10}D)D^{-0.36}

Part 2:

+0.00226A^{0.226}D^{0.125}(0.3+\log_{10}P)

Part 3:

+0.0141A^{0.213}10^{-0.021\frac{(D-180)^2}{1440}}(0.3+\log_{10}P)]

As can be seen in Figure 3, it is the contribution of part 3, which has the (D-180)2 term, that is causing the change in slope at around 180 minutes. This component,

10^{-0.021\frac{(D-180)^2}{1440}}

will equal 1 when D is 180 and decreases for durations greater than, or less than, 180.

ARF-short_parts

Figure 3: three parts of equation 1, plotted separately with the sum which is the ARF estimate. AEP= 0.5%, area = 1000 km2

There was earlier work on short duration areal reduction factors which produced an estimation equation that did not include AEP, it was just a function of area and duration (Jordan et al., 2013):

\mathrm{ARF}=\min(1,[1+a(A^b+c\log_{10}D)D^d \\ +eA^fD^g(0.3+\log_{10}P)])

If we plot the current short duration estimates (equation 1) with AEP on the x-axis (Figure 4), it is apparent that for most areas and durations, AEP has little influence on ARF (the lines in Figure 4 are mostly flat).  In fact, the largest influence occurs for a duration of 183 mins and a catchment area of 1000 km2.  This will be the contribution of part 3 above, with a small amount from part 2.

Apparently, the effect described by these equations is real and shows up in measured ARF data particularly for Sydney and Brisbane (Podger et al., 2015).  This suggests that for rare (intense) storms of about 180 min duration, the rainfall intensity decreases with area in a way that is more rapid than we might otherwise expect.   This explains the dip in the ARF curve that is shown in Figure 1. I’m sure there is a good meteorological reason for this, but I’m not sure what it is.  Perhaps these types of storms move faster than others.

Code to create graphs is available as a gist.

ARF-AEP_facet

Figure 4: Short duration areal reduction factor as a function of AEP.

References

Jordan, P., Weinmann, E., Hill, P. and Wiesenfeld, C. (2013) Collation and review of areal reduction factors from applications of the CRC-FORGE method in Australia.  Australian Rainfall and Runoff Region Project 2: spatial patterns of rainfall.  Engineers Australia (link to report)

Podger, S., Green, J., Stensmyr, P., & Babister, M. (2015). Combining long and short duration areal reduction factors. The Art and Science of Water – 36th Hydrology and Water Resources Symposium, HWRS 2015, 210–218. (link to paper)

Stensmyr, P.,  Babister, M. and Retallick, M. (2014) Short duration areal reduction factors.  Australian Rainfall and Runoff Revision Project 2: spatial patterns of rainfall.  Stage 2 report. Engineers Australia. (link to report)

Stensmyr, P. and Babister, M. (2015) Short duration areal reduction factors.  Australian Rainfall and Runoff Revision Project 2: spatial patterns of rainfall.  Stage 3 report. Engineers Australia. (link to report)