Category Archives: Hydrology

ARR point temporal patterns

**Table 1: Least uniform point temporal patterns**
EventID	Durtation (min)	Duration (hr)	Time step (min)	Region	AEP	NUI	Source Region
1208	8640	144	180	Rangelands (West)	frequent	419	Rangelands (West)
1182	10080	168	180	Rangelands (West)	intermediate	416	Rangelands (West)
1973	10080	168	180	Rangelands	frequent	375	Rangelands
1183	8640	144	180	Rangelands	intermediate	357	Rangelands
1181	8640	144	180	Rangelands (West)	intermediate	357	Rangelands (West)
2870	10080	168	180	Central Slopes	frequent	342	Central Slopes
1083	10080	168	180	Rangelands	intermediate	341	Rangelands
1079	10080	168	180	Rangelands (West)	intermediate	341	Rangelands (West)

Large peaks 24 hours apart

Analysis of areal patterns showed here were some that had large peaks 24 hours apart which could have been caused by accumulated rainfall in pluviograph records that had not been processed correctly. The same analysis was undertaken for point patterns. The four patterns with largest peaks 24 hours apart are shown in Figure 3. None of these appears to be particularly unusual.

Figure 3: Point patterns were large peaks are separated by 24 hours

In summary, point patterns do not appear to have the problems that were identify for areal patterns. Modellers still need to check for embedded bursts which depend on local values of design rainfall.

Code to reproduce the figures and analysis is available as a gist.

ARR areal temporal patterns – II

Check ARR temporal patterns before you use them

2 Replies

Recently, I had a task to troubleshoot some odd results from a hydrologic model. High flood discharges were being produced from very long duration events which was not expected. The issue came down to a few unusual temporal patterns.

The recommendation in Australian Rainfall and Runoff is that an ensemble of 10 temporal patterns are used for modelling. These depend on the region, catchment area and duration and are available from the ARR Data Hub. The temporal pattern regions are shown in Figure 1.

There are two types of patterns; point and areal. Areal patterns are appropriate when the catchment area exceeds 75 km². This blog only considers areal patterns.

Figure 1: Regions used for Temporal Patterns (ARR Book 2, Figure 2.5.7)

An example of an unusual pattern is as follows. The 10 areal temporal patterns for the southern slopes (mainland) for 120 hours duration for 200 km² catchment area are shown in Figure 2. Pattern 7 (event ID 6884) stands out because of the small number of large values. The largest peak in the pattern implies that 48.82% of the 120 hour rainfall occurs in a single 180 min (3 hour) increment. This is likely to cause an intense embedded burst that may produce misleading results from modelling.

Figure 2: Ensemble areal temporal patterns for Southern Slopes (mainland) for 120 hours duration, and catchment area of 200 km²

Embedded bursts

“Embedded bursts” occur when a period of rainfall within a temporal pattern has an annual exceedance probability rarer than the burst as whole. Embedded bursts can cause issues with modelling and modellers should consider if they should be removed or smoothed (see Book 2, Section 5.2.1 and 5.9.4; Scorah et al., 2016). We can check if pattern 7 is likely to lead to such a problem.

The areal temporal patterns in Figure 2 are for the Southern Slopes region which includes Melbourne (Figure 1), so let’s use the Melbourne CBD as an example (lat = -37.8139, lon = 144.9633). The 1 in 100, 120 hour rainfall for Melbourne is 194 mm (from the BoM). Applying pattern 7 means there would be a 3 hour period when 48.82% of this rainfall occurs i.e. 94.7 mm in 3 hours. Looking at the IFD data for Melbourne shows that an event of 94.7 mm in 3 hours has an AEP of between 1 in 500 and 1 in 1000 years. So the 1 in 100, 120 hour storm contains a very rare 3 hour embedded burst. If we run a model using this pattern, the results are likely to depend on this one 3 hour period of rain rather than being indicative of the long 120 hour storm what was of interest.

Where did this pattern come from?

The creation of design areal temporal patterns is explained in ARR Book2, Chapter 5.6. They are scaled from real storms but if there were insufficient patterns for a particular region then patterns may be donated from neighbouring regions. For example, the Southern Slopes region can take patterns from the Murray Basin (see ARR Book 2, Table 2.5.10); pattern 7 is an example of this. From the data hub, it is possible to download a file that contains information about the storms that were used to create the patterns. For the southern slopes region, this file is Areal_SSmainland_AllStats.csv. Looking at pattern 7 (Event ID 6884) shows this pattern came from Jerangle, in the Murray Basin, 100 km south of Canberra (lat = -35.8875, lon = 149.3375). IFD information for this location shows that the 1 in 100, 120 hour storm is 336 mm. Pattern 7 means that 48.82% of this rain (164 mm) falls in 3 hours. For the Jerangle region this is much larger than a 1 in 2000 year event (the 1 in 2000 year, 3 hour rainfall is 96.7 mm). This suggests that pattern 7 is based on a very unusual storm so may not be appropriate for modelling “ordinary risks” such as those associated with 1% events. Another possibility is that there is an error in the data.

Searching for other odd patterns

Finding a single strange pattern naturally leads the problem of searching for others. Whether a rare embedded burst is present depends on both the temporal pattern and the rainfall characteristics of a location but it was not practical to obtain and process pattern and IFD data for the whole of Australia. Instead a non-uniformity index (NUI) was developed which is a measure of the difference between an ARR temporal pattern and a uniform pattern (which has the same rainfall in each increment). The details of the NUI are provided below. A high value of the NUI won’t necessarily indicate an issue with embedded bursts but will identify patterns that require checking. The NUI was calculated for all 10,692 areal temporal patterns on the data hub. The top 17 least-uniform patterns are listed in Table 1.

The first 3 entries are actually the same pattern that originated in the Murray Basin and which has been used in the neighbouring regions of Central Slopes and Southern Slopes (mainland). This next two patterns are the same and also originated in the Murray Basin and there are similar examples throughout the table. Of the top 17 least-uniform patterns, 15 are from the Murray Basin and all originated in the Jerangle area in 5 separate but nearby locations as mapped on (Figure 3). The pattern listed in row 14 originated near Charleville, Qld about 600 km west of Brisbane and the final pattern in the table, from the Monsoonal North is based on a storm from Rum Jungle, 100 km south of Darwin, NT.

**Table 1: Least uniform areal temporal patterns**
No.	Event ID	Duration (hour)	Region	NUI	Source region	Area	Lat	Lon
1	4207	120	Central Slopes	481	Murray Basin	200	-35.89	149.34
2	5989	120	Murray Basin	481	Murray Basin	200	-35.89	149.34
3	6884	120	Southern Slopes (mainland)	481	Murray Basin	200	-35.89	149.34
4	4119	96	Central Slopes	385	Murray Basin	200	-35.89	149.34
5	5901	96	Murray Basin	385	Murray Basin	200	-35.89	149.34
6	5644	36	Murray Basin	378	Murray Basin	500	-35.91	149.46
7	3950	48	Central Slopes	347	Murray Basin	500	-35.96	149.51
8	5731	48	Murray Basin	347	Murray Basin	500	-35.96	149.51
9	4116	96	Central Slopes	335	Murray Basin	200	-35.94	149.49
10	5898	96	Murray Basin	335	Murray Basin	200	-35.94	149.49
11	6797	96	Southern Slopes (mainland)	335	Murray Basin	200	-35.94	149.49
12	4029	72	Central Slopes	307	Murray Basin	200	-35.91	149.41
13	5810	72	Murray Basin	307	Murray Basin	200	-35.91	149.41
14	4170	96	Central Slopes	252	Central Slopes	10000	-26.39	147.71
15	4043	72	Central Slopes	229	Murray Basin	500	-35.96	149.51
16	5822	72	Murray Basin	229	Murray Basin	500	-35.96	149.51
17	5004	96	Monsoonal North	210	Monsoonal North	100	-13.06	130.96

Figure 3: Map showing the origins of 15 of the 17 patterns in Table 1. This area is about 100 km south of Canberra, ACT

The patterns listed in Table 1 are plotted as follows. Murray Basin pattern 5989 (row 2) is shown in Figure 2 as pattern 7 (southern slopes eventID 6884). The next 4 Murray Basin patterns from Table 1 (rows 5, 6, 8 and 10) are shown in Figure 4 below. All have similar characteristics with one increment producing 40% to 60% of the rain for the whole pattern. Figure 5 shows the next lot of patterns (rows 13, 14, 16 and 17 in Table 1). By the time we get down to row 17 (Monsoonal North eventID 5004) the pattern appears closer to expectations.

Figure 4: Non-uniform patterns specified in Table 1 rows 5, 6, 8 and 10

Figure 5: Non-uniform patterns specified in Table 1 rows 13, 14, 16 and 17

At least to me, its not clear why a range of highly non-uniform patterns should all originate in the same area of NSW (Figure 3), but it does suggest the original data should be checked. In the interim, modellers could consider excluding these patterns and running ensemble or Monte Carlo analyses without them.

I haven’t looked at point temporal patterns but it would seem appropriate to check that they are reasonable and suitable for modelling.

Analysis is available as a gist.

References

Scorah, M., P. Hill, S. Lang and R. Nathan (2016). Addressing embedded bursts in design storms for flood hydrology 37th Hydrology & Water Resources Symposium. Tasmania, Engineers Australia. (link)

Appendix: NUI non-uniformity index

A non-uniformity index was used to rank patterns and find the least uniform. These were likely to contain rare embedded bursts. The approach was to use the chi-squared probability calculated using a quantile based on the difference between a particular temporal pattern and a uniform pattern. The number of degrees of freedom was the number of increments in the pattern minus one.

The quantile was calculated as:

$Q = \sum \frac{(P_p - P_u)^2}{P_u}$

Where

$P_p$ is the proportion of rainfall in a increment in the pattern
$P_u$ is the proportion of rainfall in an increment in a uniform pattern
The sum is over all the increments in the pattern
Degrees of freedom (df) is the number of increments less one

From the value of $Q$ , and df, find the probability $p$ . I used the pchisq function in R do this.
This probability very close to one for the highly non-uniform patterns so was transformed to be in a range that is easy to work with with maximum values about 500 for the most extreme patterns listed in Table 1.

$\mathrm{NUI} = -log(-log(p))$

Fitting a probability model to POT data

1 Reply

Australian Rainfall and Runoff 2016 has an example of fitting a probability model to POT (peak over threshold) data (Section 2.8.11 in Book 3). It took me a long time to understand this example. To these similarly challenged, I hope this blog will help make it easier.

Calculations are available as an excel spreadsheet; R code is available as a gist. The R code reproduces all the results and figures from this blog.

The case study is the Styx River at Jeogla with the POT series providing 47 peaks above the threshold of 74 m³/s for a period of record of 47 years. The data are as follows:

878 541 521 513 436 411 405 315 309 301 300 294 283 258 255 255
238 235 221 220 206 196 194 190186 164 150 149 134 129 129 126
119 118 117 117 111 108 105 98 92.2 92.2 91.7 88.6 85.2 79.9 74

Our goal is to fit a probability model to these data and estimate flood quantiles.

You can download the data as a csv file here. This is the same data set that was used in the 1987 edition of ARR (Book IV, Section 2.9.5). I’ve sourced the date of each flood from ARR1987 and included it in the file. Interestingly the earliest flood discharge is from 1943 and the last from 1985, a record of 43 years not 47. This is because the floods between 1939 and 1943 were all less than the 74 m³/s threshold (from 1939 to 1942 they were: 58.0, 13.0, 22.1, and 26.1 m³/s).

Start by calculating the the plotting positions for these data points. ARR2016 recommends using the inverse of the Cunnane plotting position formula to calculate recurrence intervals (see ARR 2016 equation 3.2.79, in Book 3, Chapter 2.7.1).

$T = \frac{n+0.2}{R-0.4}\qquad \qquad \mathrm{(1)}$

Where:

$T$ average interval between exceedances of the given discharge (years)
$R$ is the rank of flows in the POT series (largest flow is rank 1)
$n$ is the number of years of data.

Note that $n$ is the number of years of record, not the number of peaks.

Also, its important to recognise that equation 1 works differently for the POT series compared to the annual series. Let’s look at an example that shows this difference. Consider the case where we were analysing 94 peaks from a POT series from 47 years of data. In that case, the smallest peak would have a rank of 94. The recurrence interval would be:

$T = \frac{47+0.2}{94-0.4} = 0.504$

Which is about half a year, or 6 months, which seems reasonable (there were 94 peaks in 47 years so there are 2 peaks per year on average and they will be about 6 months apart on average).

Interpreting the Cunnane plotting position as annual exceedance probability would give a result of:

$P = \frac{94-0.4} {47+0.2}= 1.98$

That is, the probability is greater than 1 which doesn’t make sense. So we can’t use a plotting position formula to calculate the AEP of data from a peak-over-threshold series. Instead, the inverse of equation 1 can be interpreted as EY, the expected number of exceedances per year.

The ARR example fits the exponential distribution as the probability model. This has two parameters, $q_*$ and $\beta$ , which can be determined by calculating the first and second L moments of the POT series data.

L1 is just the mean of the pot series (266.36 m³/s). To calculate L2 we use the formulas developed by QJ Wang (Wang, 1996)

$L2 = \frac{1}{2} \frac{1}{^nC_2} \sum\limits^n_{1=1}\left( ^{i-1}C_1-^{n-i}C_1\right)q_i\qquad \qquad \mathrm{(2)}$

Where:

$q_i$ are the flows (the POT series) in order from smallest to largest
$n$ is the number of flows

L2 = 79.12

I’ve calculated L2 using this formula as shown in the companion spreadsheet but it does require a fudge. Excel doesn’t automatically recognise that ⁰C_n is equal to zero. This is a standard mathematical convention that Excel doesn’t seem to implement.

The parameters of the exponential distribution can be expressed in terms of the L moments (See ARR2016, Table 3.2.3.) We are using the formulas for the Generalized Pareto (GP) but setting $\kappa$ to zero which results in the exponential distribution. The GP distribution may be useful in real world applications.

For the exponential distribution:

$\beta = 2 \times L2 = 158.24$
$q_* = L1- \beta = 68.12$

The example in ARR doesn’t include confidence intervals for these parameter estimates, but these can be calculated using bootstrapping.

The 95% confidence intervals are, for beta (37.4, 89.4) and $q_*$ (120.6, 244.7). The correlation between the parameters is about -0.5. Uncertainty in parameters is shown graphically in Figure 1.

Figure 1: Uncertainty in parameters. Ellipse shows 95% confidence limit

The exponential distribution relates the exceedance probability to the size of a flow. In this case, the probability that a peak flow, $q$ , exceeds some value $w$ in any POT event is:

$P(q>w) = \mathrm{e}^{\left( -\frac{w-q_*}{\beta} \right)}\qquad \qquad \mathrm{(3)}$
$P(q>w)= \mathrm{e}^{\left( -\frac{w-68.12}{158.24} \right)} \qquad \qquad \mathrm{(4)}$

The ARR example cross-references these formulas to Table 3.2.3, but the reference should be to Table 3.2.1. The final formula in Table 3.2.1 shows the cumulative distribution function for the exponential distribution. This is for the non-exceedance probability. In hydrology we usually use exceedance probabilities (the probability that a flood is greater than a certain value), therefore the required equation is one minus the equation shown.

We can convert the POT probabilities to expected number of exceedances per year, $EY$ by multiplying by the average number of flood peaks above the threshold per year. The example in ARR states that the threshold is $q_*$ i.e. 68.12 m³/s but the threshold used to create the POT data was 74 m³/s. I’m not sure, but I think we should use the 74 m³/s threshold. There is no information available on how many peaks there are above 68.12 m³/s. It is likely the ambiguity has arise because of a typo in the example i.e. the threshold should not have been related to $q_*$ .

Sticking with the 74 m³/s threshold, there are there are 47 values in POT series and 47 years of data, so the average number of peaks per year is 1. Therefore, the EY can be expressed as follow:

$EY(w) = \nu P(q>w) =\nu \mathrm{e}^{ \left( -\frac{w-q_*}{\beta} \right)}\qquad \qquad \mathrm{(5)}$

Where latex $\nu$ is the average number of POT floods per year, which is 1 in this case. In real applications, $\nu$ is likely to be greater than 1. The discussion in ARR2016 mentions the flow threshold is typically selected so that there are 2 to 3 times more peaks in the POT series compared to the annual series (Book 3, Chapter 2.2.2.2). Although in Chapter 2.3.4 it is mentioned that there are circumstances where the POT and annual series should have the same number of peaks.

$w$ can also be related to $EY(w)$ as follows:

$w = q_* - \beta \log \left( \frac{EY(w)}{\nu} \right)\qquad \qquad \mathrm{(6)}$

We can also relate $EY$ to annual exceedance probability by the formula:

$AEP = \frac{\mathrm{e}^{EY}-1}{\mathrm{e}^{EY}}\qquad \qquad \mathrm{(7)}$

Therefore, a flood magnitude, $w$ can be related to an AEP, and an AEP to a flood magnitude, $w$ , through the following pair of equations (ARR2016 Equation 3.2.11).

$AEP = 1-\exp \left( -\exp \left(- \frac{w-q_*}{\beta} \right) \right)\qquad \qquad \mathrm{(8)}$

$w = q_* - \beta \left( \log \left( -\log \left( 1-AEP \right) \right) \right)\qquad \qquad \mathrm{(9)}$

These equations can be used to estimate quantiles at standard EY values. Confidence intervals (95%) have been estimated using bootstrapping (Table 1).

**Table 1: Flood quantiles and 95% confidence limits at standard EY values**
EY	AEP	AEP (1 in X)	ARI	Q	Lwr	Upr
1.00	0.63	1.6	1.0	68	39	90
0.69	0.50	2.0	1.4	127	106	150
0.50	0.39	2.5	2.0	178	151	215
0.22	0.20	5.1	4.5	308	253	404
0.20	0.18	5.5	5.0	323	267	427
0.11	0.10	9.6	9.1	417	335	565
0.05	0.05	20.5	20.0	542	434	754
0.02	0.02	50.5	50.0	687	548	965
0.01	0.01	100.5	100.0	797	627	1167

Now plot the data and the fit (Figure 2). The x values are the EY calculated from the inverse of equation 1; the y values are the flows. The fit is based on equation 6. Confidence intervals for quantiles were calculated using bootstrapping.

Figure 2: Partial series for the Styx River at Jeogla showing fit and confidence intervals

Personally I prefer the graph to be increasing to the right which is the way flood frequency curves are usually plotted, its also the way a similar example was presented in the 1987 edition of Australia Rainfall and Runoff. This just involves using the ARI as the ordinate (Figure 3).

Figure 3: Partial series for the Styx River at Jeogla showing fit and confidence intervals (same as Figure 2 except increasing to the right)

To see more, check out the excel spreadsheet and R code gist.

I’d also like to thank Matt Scorah for his help in clarifying the process of fitting POT data.

References

Wang, Q. J. (1996). “Direct sample estimates of L moments.” Water Resources Research 32(12): 3617-3619.

Time of concentration: comparison of formulas

References

Beran, M. (1979) The Bransby Williams formula an evaluation. Procedings of the Institution of Civil Engineers. 66(2):293-299 (https://doi.org/10.1680/iicep.1979.2355). Also see the discussion (https://doi.org/10.1680/iicep.1980.2512).

Bransby Williams, G. (1922). Flood discharge and the dimensions of spillways in India. The Engineer (London) 121: 321-322. (link)

French, R., Pilgrim, D.H., Laurenson, E.M., 1974. Experimental examination of the rational method for small rural catchments. Civ. Eng. Trans. Inst. Engrs. Aust. CE16, 95–102. CE16 95-102.

McDermott, G. E. and D. H. Pilgrim (1982). Design flood estimation for small catchments in New South Wales, Department of National Development and Energy and Australian Water Resources Council. Research project No. 778/104. Technical paper No. 73. Australian Government Publishing Service.

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

Time of concentration: some data

5 Replies

I’ve previously written about time of concentration formulas including the approaches developed by Bransby Williams and Pilgrim and McDermott. Pilgrim and McDermott also collected data on time of concentration from Australian catchments. This blog explores the data and makes it available as a CSV file.

“Time of concentration” is often assumed to be the time for runoff to flow from the furthest reaches of a catchment to the outlet. Pilgrim and McDermott take a different tack, and use the “characteristic response time of a catchment” which is the minimum time of rise of the flood hydrograph. This is the time taken for flow to increase from an initial low value to the hydrograph peak (French et al., 1974). They determine this characteristic response time from measurements at gauging stations and argue that it is a better approach because it is empirically based (McDermott and Pilgrim, 1982).

Pilgrim and McDermott also measured catchment characteristics: stream length, average slope, equal area slope and catchment area. They explored the data and ultimately adopted the formula that relates time of concentration to catchment area.

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

where A is measured in km².

Characteristic response times were collected for 95 stream gauging sites from Queensland, New South Wales, Victoria and Tasmania. Those sites where location information is available (62 of 95) are shown Figure 1. There are lots of sites in the Snowy Mountains and around Sydney and a scattering of sites elsewhere.

Figure 1: Locations of gauges (red dots) used to derive the Pilgrim McDermott formula for the time of concentration

Let’s have a look at the data. Figure 2 shows the relationship between response time and catchment area. The regression fit to the logged data is the same as that reported by Pilgrim and McDermott which is a good start. The fit seems reasonable and the Nash-Sutcliffe coefficient, and the adjusted r-squared, is 0.85.

However, there is substantial uncertainty. Also plotted on Figure 2 are the 95% prediction intervals which show that a large range of response times are possible for any given catchment size. For example, the fitted value for a 100 km² catchment is 4.4 hours but the upper and lower prediction limits are 1.8 hours and 10.5 hours.

Figure 2: Catchment response time as a function of area. Figure shows the fitted equation (equation 1) and the 95% prediction intervals

The relationship between catchment response time and the other catchment characteristics is interesting (Figure 3 and Figure 4). There is a strong association with stream length and catchment slope (either equal area or average slope). Steep, short streams respond quickly so have shorter characteristic times.

Pilgrim and McDermott considered including stream length and slope in the equation to predict time of concentration but found they were highly correlated with area which we can see in figures 5 and 6. These figures show that large catchments have longer stream lengths, which is expected. Also, large catchments have lower slopes. The upshot is that the Pilgrim McDermott formula is taking account of the effect of stream length and slope because of their correlation with area. Including these variables in the formula for time of concentration only made, at best, a marginal improvement in accuracy. The adjusted r-squared increased from 0.848 to 0.854 with the inclusion of slope. There was no advantage from including stream length. An equation based solely on catchment area is also simpler to use.

Figure 3: Catchment response time as a function of slope

Figure 4: Catchment response time as a function of stream length

area_stream_length

Figure 5: Stream length as a function of area

Figure 6: Relationship between catchment area and catchment slope

The data set used by Pilgrim and McDermott is available in a csv file. Code to reproduce the figures and analysis is available as a gist.

References

French, R., Pilgrim, D.H., Laurenson, E.M., 1974. Experimental examination of the rational method for small rural catchments. Civ. Eng. Trans. Inst. Engrs. Aust. CE16, 95–102. CE16 95-102.

Grimaldi, S. Petroselli, A., Tauro, F. and Profiri, M. (2012) Time of concentration: a paradox in modern hydrology. Hydrological Sciences Journal 57(2):217-228. https://www.tandfonline.com/doi/full/10.1080/02626667.2011.644244?scroll=top&needAccess=true

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

Victoria won the hydrologic games, and they may not have cheated

3 Replies

As usual, the hydrologic games were held during the HWRS2018 conference dinner, and this time Victoria won. Competitors threw a dart to select an initial loss and rainfall temporal pattern which were used as input to a RORB model to simulate a 1% AEP peak flow. The team with the highest average peak was the winner.

Figure 1: Results of the hydrologic games (Source: A post by Ben Tate on LinkedIn)

Before the games started, I thought one of the small teams would win. There were only two international competitors and six from Tasmania, but likely over 100 from Victoria. Extreme results are more likely from small samples so I expected small teams to have both the highest and lowest scores with the Victorian team to be in the middle, near the true mean.

There is a nice discussion of the effect of sample size on variation in the wonderful book Thinking Fast and Slow by Daniel Kahneman. See the start of Chapter 10 “The law of small numbers”. Also, in Howard Wainer’s article The Most Dangerous Equation. Wainer argues that misunderstanding of this effect has caused confusion and wasted effort. For example, the small-school movement was based on the idea that students performed better in smaller compared to larger schools. In the US, grants from the Gates foundation were used to facilitate the conversion of large schools to several smaller schools. However, evidence presented by Wainer shows the small-school effect is likely the result of the extra variation when only a small number of students are assessed. Comparisons of many schools showed the smallest schools achieving both the highest and lowest scores.

Back to the hydrologic games. How likely is it that the mean of a large sample, the Victorians, would be higher than all the means from a number of smaller samples? I don’t know how many people participated in the hydrologic games, so let’s guess. Ben says there were “over 150 players”; I’ve divided these up as shown in Table 1.

Table 1: Teams for the hydrologic games

Now, to simulate the game we draw random numbers from the normal distribution. The Victorian score will be the mean of 100 random numbers, the NSW score will be mean of 15 random numbers and so on. Victoria wins if its mean exceeds the mean of all the other teams.

Using this approach, the probably of Victoria winning is estimated to be 0.00830 (1 in 120), based on the mean of 100 sets of 100,000 simulations. So its less likely than seeing a 1% AEP flood. If a hydrologist attends 20 hydrologic games during their career, and this game was played every time, the probability that they would see Victoria win at least once is about 15%.

The upshot is that when Victoria won, and provided the games were fair, we witnessed a rare, but not impossible event.

Calculations are available as a gist.

tonyladson

Hydrology, Natural Resources and R

Category Archives: Hydrology

ARR point temporal patterns

Large peaks 24 hours apart

ARR areal temporal patterns – II

Check ARR temporal patterns before you use them

Embedded bursts

Where did this pattern come from?

Searching for other odd patterns

References

Appendix: NUI non-uniformity index

Fitting a probability model to POT data

References

Time of concentration: comparison of formulas

References

Time of concentration: some data

References

Victoria won the hydrologic games, and they may not have cheated

	tonyladson on Check ARR temporal patterns be…
	LP on Check ARR temporal patterns be…
	tonyladson on Time of concentration: some…