Annual Rainfall Extremes at Manilla NSW: V

V. Extremes marked by high kurtosis

Manilla annual rainfall kurtosis

This graph shows how the extreme values of annual rainfall at Manilla, NSW have varied, becoming rarer or more frequent with passing time.
The graph quantifies the occurrence of extreme values by the kurtosis of 21-year samples centred on successive years.

The main features of the pattern are:
* Two highly leptokurtic peaks, showing times with strong extremes in annual rainfall values. One is very early (1897) and one very late (1998).
* One broad mesokurtic peak, in 1938, showing a time with somewhat weaker extremes.
* Broad platykurtic troughs through the 1910’s, 1920’s, 1950’s, 1960’s and 1970’s, decades in which extremes were rare.
All these features were evident in the cruder attempts to recognise times of more and less occurrence of extremes in Parts I, II, III and IV of this series of posts. This graph is more precise, both in quantity and in timing.

However, kurtosis (the fourth moment of the distribution) does not distinguish extremes above normal from those below normal. It is known that some early dates at Manilla had extremes that were above normal, and some late dates had extremes that were below normal.

Use of skewness

Extremes above normal are distinguished from those below normal by the third moment of the distribution, that is, the skewness.
Manilla Annual rainfall history: SkewnessThe post “Moments of Manilla’s Yearly Rainfall History” shows graphs of the time sequence of each of the four moments, including the skewness (copied here) and the kurtosis ( the main graph, copied above). The skewness function, like the kurtosis function, relates to the most extreme values of the frequency distribution, but to a lesser extent (by the third power, not the fourth).

I have shown the combined effect of kurtosis and skewness on the occurrence of positive and negative extremes in this data set in the connected scatterplot below.

Manilla rain skew vs.kurt

The early and late times of strong extremes were times of strongly positive and strongly negative skewness respectively. As kurtosis fell rapidly from the initial peak (+0.9) in 1897 to slightly platykurtic (-0.4) in 1902, the skewness also fell rapidly, from +0.7 to +0.3.
Much later, in mirror image, values were almost the same in 1983 as in 1902, then kurtosis rapidly rose while skewness rapidly fell, until kurtosis reached +0.9 and skewness -0.3 by 1998.
Between 1902 and 1983, while kurtosis remained below -0.2, the pattern was complex. In the decades of strong platykurtosis (below -0.9) there were extremes of skewness: +0.7 in 1919 and -0.3 in 1968.
Note that the skewness range was as high in times of low kurtosis as in times of high kurtosis, and the same applies to kurtosis range in relation to skewness. Conversely, when either moment was near its mean, the range of the other was not high.


See also:
“Rainfall kurtosis matches HadCRUT4” and “Rainfall kurtosis vs. HadCRUT4 Scatterplots”.

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Australian climate Quasi-Biennial Oscillations.

Australian temperature and rainfall from 1950

[See the Note below: “2010 data re-posted.”]

The above graphs plot the time series of monthly data for the whole of Australia, smoothed with a Gaussian window of half-width 6 months.
The two independent series of (a) mean maximum monthly temperature anomaly in degrees celsius and (b) total monthly rainfall anomaly in mm are plotted on the same graphs, but the scale for temperature is inverted for easy comparison.
One degree on the left axis corresponds to 10 mm on the right axis, but the zero lines may differ.

There is an obvious sine-wave cycle with a wavelength of between one and three years. (This is the “quasi-biennial cycle” that A.B. Pittock identified in 1971.)
Most peaks and troughs on these independent time series almost coincide, and their relative heights and depths tend to agree. In fact. the correlation between values of temperature and rainfall is poor, but the shapes of the sinusoidal curves match extremely well.
Peaks and troughs on the rainfall curve tend to lead those on the (inverted) temperature curve by one, two, or three months. In these graphs, I have lagged rainfall values one month, to show that many of the peaks and troughs are aligned.

Taking the whole of Australia in the last 60 years, it is fairly clear that:
* points of lowest maximum temperature have generally lagged about one month behind points of highest rainfall;
* points of highest maximum temperature have generally lagged about one month behind points of lowest rainfall.

Given this lag effect, times of lowest rainfall cannot be caused by times of highest temperature, but it is possible that times of highest temperature may be caused by times of lowest rainfall.
I find it plausible that temperature swings would closely follow rainfall swings (but in the opposite sense) due to lack of cooling by evapotranspiration in times of drought and effective cooling by evapotranspiration in times of deluge.

[Note: 2010 data re-posted.
This material appeared originally in a “Weatherzone” forum “Observations of climate variation”, Post #810237 of 27 December 2009.
Since the graphs as posted are lost, due to the action of the “Photobucket” image-hosting web-site, I am re-posting the graphs from my records.]


Data source.
The data was sourced at the following web-page.
http://reg.bom.gov.au/silo/products/cli_chg/
That web-page no longer exists.


[Note posted to “weatherzone”.

I never did find Barrie Pittock’s 1971 article in which he referred (I believe) to a quasi-biennial oscillation in Australian surface climate.
However, here is a detailed discussion of that particular topic:

“Historical El Nino/Southern Oscillation variability in the Australasian region” by Neville Nicholls, Chapter 7 (p151-173) in “El Nino: Historical and Paleoclimatic Aspects of the Southern Oscillation”, Henry F.Diaz and Vera Markgraf (eds.), Cambridge U P, 1992, 476pp.

On p.158, Nicholls has a section headed “Biennial cycle” that refers to papers written in the 1970’s, 1980’s, and 1990’s. He says:

“The biennial cycle is observed over the equatorial Pacific and Indian Oceans and is tightly phase-locked with the annual cycle. It varies in amplitude from cycle to cycle and sometimes changes phase. It is not strictly a 2-year cycle so it may be characterised better as a quasi-biennial cycle…..Rainfall over much of Australia displays a quasi-biennial cycle (e.g.Kidson 1925).”]

Rainfall kurtosis vs. HadCRUT4 Scatterplots

These scatterplots and Connected Scatterplots support a relationship between the kurtosis of annual rainfall at Manilla NSW and the de-trended smoothed HadCRUT4 series of global temperatures.

Scatterplot rainfall kurtosis vs. HadCRUT all data

The raw data, as observed

The first scatterplot compares (y-axis) all the calculated unsmoothed values of kurtosis of annual rainfall at Manilla, NSW with (x-axis) the unsmoothed values of the HadCRUT4 series of global near-surface temperature at those dates.
[I have plotted rainfall values lagged by five years on all of the scatterplots shown. This lagging makes little difference to the first two scatterplots.]

On this first graph, the fitted linear trend barely supports a positive relation of kurtosis to temperature. The slope is low (1.05) and the R-squared only 0.16. There is an aberrant cloud of points in the top left corner.

Scatterplot rainfall kurtosis vs. HadCRUT detrended (all data)

The raw data, HadCRUT4 de-trended

This graph takes a first step towards a better model for the relationship: the secular trend of the temperature series (that is, the global warming) is removed. For comparison, I have not re-scaled the x-axis.
Although still very weak, the relation is much enhanced. The slope (2.35) is twice as steep and the R-squared (0.24) increased by 50%.

Connected Scatterplot rainfall kurtosis vs. HadCRUT all data

Smoothed data, HadCRUT4 de-trended

This third graph uses smoothed data. The HadCRUT4 series is  “decadally-smoothed” (as published) with a 21-point binomial filter to remove high frequency noise. The rainfall data, already damped by its 21-year sampling window, has been further smoothed with a 9-point Gaussian filter.
This graph is a Connected Scatterplot, that shows the trajectory of the rainfall-temperature relation with the passing of time.

Line chart rainfall kurtosis vs. HadCRUT (detrended)Smoothing both data sets has given a much closer relation. The R-squared value is almost doubled again, to 0.43, and the slope is increased to 3.70. The date labels show that the relation before 1910 was different from that at later dates. (This had also been clear in the Dual axis line chart, copied here, from the post “Rainfall Kurtosis Matches HadCRUT4”.)

Connected Scatterplot rainfall kurtosis vs. HadCRUT from 1908

Smoothed data, HadCRUT4 de-trended, from 1908 to 2002

In this final graph, I have discarded the first eleven years. The linear regression based on smoothed values from 1908 to 2002 has a steep slope of 5.21 and a respectable R-squared value of 0.84.

I had prepared similar graphs for lag values of rainfall kurtosis from zero up to nine. The lag value of five years tends to maximise the slope and the R-squared values.
Choice of a five-year lag tends to form hair-pin loops in the trace, while shorter lags give wider clockwise loops and longer lags give wider anti-clockwise loops.
The lag value of five years implies that the Manilla annual rainfall kurtosis value for a given year matches the de-trended HadCRUT value that occurs five years later.

[Back to the main post on this topic: “Rainfall kurtosis matches HadCRUT4”.]

Rainfall kurtosis matches HadCRUT4

Line chart rainfall kurtosis vs. HadCRUT (detrended)

The kurtosis of annual rainfall at Manilla NSW forms a time-series that matches the time-series of global surface temperature when de-trended.

Features of the data

Data sources, noted on the graph, are specified below. The best match is achieved by decadal smoothing, by scaling 1.0 units of kurtosis to 0.16 degrees of temperature, and by lagging the rainfall data five years.

Closeness of the match

Although both variables have irregular traces, their patterns are almost the same. They begin and end very high, have a broad peak near 1943, and are low in the 1910’s, 1920’s, 1950’s, 1960’s and 1970’s.
The match is very close for ninety years from 1915 to 2005, except for one decade (at 1972). In all this time, both the values and the slopes (as scaled) agree. [See the Note below “1991-1992”.]

Before 1915, the patterns do not match well, but they remain similar. Both traces descend rapidly together from 1903 to 1910. The initial peak in the rainfall trace at 1903 (actually 1898) is similar in height (as scaled) to a peak of the de-trended temperature trace just off the graph at 1879.

Discovering the pattern match

I was seeking a robust measure of the occurrence of extreme values in annual rainfall at Manilla, NSW. As kurtosis is just such a measure, I calculated it. I then plotted out the time-series, as shown here. It reminded me of the well-established time-series of smoothed HadCRUT4 global near-surface temperature. In particular, I recalled a locally-dominant peak near 1940.

Line chart rainfall kurtosis vs. HadCRUT
Simply reconciling the vertical scales of the two time-series gave me the second graph.
While not matching in details, the two curves remain very close from 1940 to 1995. Matching over the whole rainfall record is prevented by a difference in trend. While the rainfall kurtosis has no trend, the HadCRUT4 curve has a secular trend rising at half a degree per century (known as “global warming”).
To extend and improve the match, I subtracted the linear trend from the global temperature curve, and lagged the rainfall points by five years. The first graph is the closely-matching result.

What it means

As evidence of extreme behaviour in climate

It is said that more extremes in climate will occur as the world becomes warmer. The evidence is not strong. Most data sets are overwhelmed by noise, and “extreme” is seldom defined with rigor.
In the present case, I believe that the definition of “extreme” that I use is sound: that is, the kurtosis of a frequency-distribution. Only the instability of kurtosis when based on small samples is an issue.

My rainfall data set that displays more and less extreme behaviour is not general but local. It can merely suggest that data elsewhere may reveal functional relationships.

Connected Scatterplot rainfall kurtosis vs. HadCRUT from 1908A very strong and persistent empirical relationship is shown by the graphical logs above. In another post, “Rainfall Kurtosis vs. HadCRUT4  Scatterplots”, I show scatterplots like this in support of it.

De-trended global temperature

This strong link between local annual rainfall kurtosis and global climate has a surprising feature. Although this extreme behaviour seems to relate to global temperature, it does not relate to global warming! It relates to a temperature trace from which the global warming trend has been removed. Times of high kurtosis, denoting enhanced extremes, correspond to times when the global temperature was highest above trend. Such times occurred not only in the twenty-first century, but equally in the nineteenth century. There was another (widely-known), lower peak in de-trended global temperature near 1940: at that time also kurtosis was above normal.

Should global temperature remain static for a time, it would be falling rapidly below its rising trend. According to this data set, that should bring reduced extreme behaviour in annual rainfall at Manilla.


Data Sources

(i) Global temperature time-series

From the three available century-long time series of global near-surface temperature I have chosen to use HadCRUT4, published by the British Met Office Hadley Centre. The link is here.

I selected from the section: “HadCRUT4 time series: ensemble medians and uncertainties”.
From this, I downloaded two files:
(i) “Global (NH+SH)/2, annual”;
(ii) “Global (NH+SH)/2, decadally smoothed”.
[The “Decadally smoothed” data supplied is annual data smoothed with a 21-point binomial filter.]
From each data file, I used only the first column: the year date, and the second column: the median value.

I established the secular trend of global warming using the linear trend function in Charts for “Excel”. I found the linear trend of the whole HadCRUT4 annual series data (1850 to 2016) to be:

y = 0.005x – 0.52.

I then subtracted the annual value at the trend line from the decadally smoothed HadCRUT4 value to get the de-trended smoothed value shown on the first graph.

(ii) Kurtosis of Manilla annual rainfall

The rainfall data is that for Manilla Post Office, Station 055031 of the Australian Bureau of Meteorology. Station 055031 functioned without gaps from 1883 to March 2015. Since then, the official record is fragmentary.
I found kurtosis values for annual rainfall by using the (excess) kurtosis function in “Excel”. I used sub-populations of 21 successive years, referred to the median year. I found values for the years 1893 to 2006. I smoothed these values with a 9-point gaussian filter (yielding similar smoothing to that of HadCRUT4). Smoothing reduced the plottable years to those from 1897 to 2002.

Manilla yearly rainfall history: four momentsI posted a line graph of this kurtosis data earlier, in “Moments of Manilla’s Yearly Rainfall History”.


Note: 1991-1992

The most striking match in the graph is that both traces pause at 1991-1992 within a two-decade-long steady rapid rise. That pause in the global temperature series has been attributed with little doubt to the injection into the atmosphere of seventeen million tonnes of sulphur dioxide by the eruption of Mount Pinatubo in the Philippines. That eruption cannot have affected the rainfall kurtosis five years earlier.

Moments of Manilla’s Yearly Rainfall History

Manilla Annual rainfall history: Four moments

Comparing all four moments of the frequency-distributions

Annual rainfall for Manilla, NSW, has varied widely from decade to decade, but it is not only the mean amounts that have varied. Three others measures have varied, all in different ways.

I based the graph on 21-year sub-populations of the 134-year record, centred on consecutive years. I analysed each sub-population as a frequency-distribution, to give values of the four moments: mean (drawn in black), variance (drawn in red), skewness (drawn in blue) and kurtosis (drawn in magenta).

[For more about the moments of frequency-distributions, see the recent post: “Kurtosis, Fat Tails, and Extremes”. See also the Note below: “Instability in the third and fourth moments.”]

Each trace of a moment measure seems to have a pattern: they are not like random “noise”. Yet each trace is quite unlike the others.

The latest values are on the right. They show that the annual rainfall is now remarkable in all four respects. First, the mean rainfall (black) closely matches the long-term mean, which has seldom happened before. By contrast, the other three moments are now near historical extremes: variance (red) is very low, skewness (blue) very negative, and kurtosis (magenta) very positive.

To my knowledge, such a result has not been observed or predicted, or even suspected, anywhere.

Manilla Annual rainfall history: Mean

The mean yearly rainfall (the first moment)

As I have shown before, the mean annual rainfall was low in the first half of the 20th century, and high in the 1890’s, 1960’s and 1970’s. Rainfall crashed in 1900 and again in 1980.

Manilla Annual rainfall history: Variance

Yearly rainfall variance (the second moment)

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Kurtosis, Fat Tails, and Extremes

sketch demonstrating kurtosis

PLATYKURTIC left; LEPTOKURTIC right

Why must I explain “kurtosis”?

Manilla 21-year rainfall mediansThe annual rainfall at Manilla, NSW has changed dramatically decade by decade since the record began in 1883. One way that it has changed is in the amount of rain each year, as shown in this graph that I posted earlier.

Another way, unrelated to the amount of rain, is in its kurtosis. Higher kurtosis brings more rainfall values that are extreme; lower kurtosis brings fewer. We would do well to learn more about rainfall kurtosis.

[A comment on the meaning of kurtosis by Peter Westfall is posted below.]

Describing Frequency Distributions

The Normal Distribution

Many things vary in a way that seems random: pure chance causes values to spread above and below the average.
If the values are counted into “bins” of equal width, the pattern is called a frequency-distribution. Randomness makes this pattern form the unique bell-shaped curve of Normal Distribution.

Histogram of annual rainfall frequency at Manilla NSWThe values of annual total rainfall measured each year at Manilla have a frequency-distribution that is rather like that. This graph compares the actual distribution with a curve of Normal Distribution.

Moments of a Normal Distribution: (i) Mean, and (ii) Variance

The shape of any frequency-distribution is described in a simple way by a set of four numbers called moments. A Normal Distribution is described by just the first two of them.
The first moment is the Mean (or average), which says where the middle line of the values is. For Manilla annual rainfall, the Mean is 652 mm.
The second moment is the Variance, which is also the square of the Standard Deviation. This second moment says how widely spread or scattered the values are. For Manilla annual rainfall, the Standard Deviation is 156 mm.

Moments of other (non-normal) distributions: (iii) Skewness, and (iv) Kurtosis

The third moment, Skewness, describes how a frequency-distribution may have one tail longer than the other. When the tail on the right is longer, that is called right-skewness, and the skewness value is positive in that case. For the actual frequency-distribution of Manilla annual rainfall, the Skewness is slightly positive: +0.268. (That is mainly due to one extremely high rainfall value: 1192 mm in 1890.)
Kurtosis is the fourth moment of the distribution. It describes how the distribution differs from Normal by being higher or lower in its peak or its tails, as compared to its shoulders.
As it was defined at first, a Normal Distribution had the kurtosis value of 3, but I (and Excel) use the convention “excess kurtosis” from which 3 has been subtracted. Then the excess kurtosis value for a Normal Distribution is zero, while the kurtosis of other, non-normal distributions is either positive or negative.

Smoothed rainfall frequency and a platykurtic curveManilla’s total frequency distribution of annual rainfall has a Kurtosis of -0.427. As shown here (copied from an earlier post), I fitted a curve with suitably negative kurtosis to Manilla’s (smoothed) annual rainfall distribution.

Platykurtic, Mesokurtic, and Leptokurtic distributions

Karl Pearson invented the terms: platykurtic for (excess) kurtosis well below zero, mesokurtic for kurtosis near zero, and leptokurtic for kurtosis well above zero.
The sketch by W S Gosset at the top of this page shows the typical shapes of platykurtic and leptokurtic curves.
(See the Note below: ‘The sketch by “Student”‘.)

In the two graphs that follow, I show how a curve of Normal Distribution can be modified to be leptokurtic or platykurtic while remaining near-normal in shape. (See the note “Constructing the kurtosis adjuster”)
In both of these graphs, I have drawn the curve of Normal Distribution in grey, with call-outs to locate the mean point and the two “shoulder” points that are one Standard Deviation each side of the mean.

A leptokurtic curve

A leptokurtic curve

By adding the “adjuster curve” (red) to the Normal curve, I get the classical leptokurtic shape (green) as was sketched by Gosset. It has a higher peak, lowered shoulders, and fat tails. The shape is like that of a volcanic cone: the peak is narrow, and the upper slopes steep. The slopes get gentler as they get lower, but not as gentle as on the Normal Curve.

A platykurtic curve

A platykurtic curve

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A drought has begun

A year ago, I showed that Manilla was far from being in a drought. That is not true now. There are severe shortages of rain.

Rainfall status at Manilla, September 2016 and September 2017.

The first graph has rainfall totals up the left margin. They are not expressed in millimetres but as percentile values, Along the bottom margin is the number of months included in calculating each rainfall total.

On the graph, I have compared the rainfall situation today, September 2017, plotted in red with that of September 2016, plotted in grey. Much has changed.

Take, for example, the 12-month (one-year) rainfall total. Rainfall totals for 12 month periods are directly above the value “12” at the bottom of the graph, near the label “Number of Months included”. In data for the month of September 2016 (grey), the 12-month total (actually 802 mm) had been at the 80th percentile, which was very high. In up-to-date data for the month of September 2017 (red), the 12-month total (actually 484 mm) is at the 17th percentile, which is very low.
Although rainfall totals for  periods longer than 12 months have not fallen so much, nearly all of them have fallen. Three that have not are those for 30 months, 36 months and 42 months. They were already low, due to including in them some months of low rainfall several years ago, in 2013 and 2014.

So far, real shortages have occurred mainly within the last 12 months. Beyond that, the two-year rainfall total of 1285 mm, for example, is still near normal, plotting at the 48th percentile.

The second graph shows in detail how shortages that are serious or severe have developed during the last six months. These were the monthly rainfall amounts, with the normal amounts in brackets:

April: 24.0 mm (39.3);
May: 55.6 mm (40.3);
June: 62.8 mm (44.3);
July: 13.2 mm (41.4);
August: 13.8 mm (39.5);
September: 5.5 mm (41.2).

As a result, the current situation is as shown below. There are already severe rainfall shortages, at the 2nd or 3rd percentile, in the two-month and three-month totals to date. There are also serious shortages, at the 8th and 9th percentiles, in the four-month and six-month totals to date.

Drought status at Manilla in September 2017

I will update these graphs each month to show how the situation changes.

[Serious rainfall shortages did not occur in any following months up to January 2018.]