Annual Rainfall Extremes at Manilla NSW: IV

IV. Some distributions had heavy tails

Graph of history of heavy tails in Manilla annual rainfall

This graph is based on applying a 21-year sampling window to each year in the Manilla rainfall record, then adding smoothing. (See “Note about Sampling” below.)

“Heavy tails”

In the previous postI plotted only the most extreme high and low values of annual rainfall in each sampling window. Now, I choose two rainfall amounts (very high and very low) to define where the “Tails” of the frequency distribution begin. These Tails are the parts that I will call “extreme”. I count the number of values that qualify as extreme by being within the tails.
In this post, I recognise heavy tails, when before I recognised long tails.


Back to the prelude “Manilla’s Yearly Rainfall History”.
Back to Extremes Part I.
Back to Extremes Part II.
Back to Extremes Part III.


Making the graph

The long-term Normal Distribution

The graph relies on the long-term Normal Distribution curve (“L-T Norm. Dist.” in the legend of the graph). That is, the curve that I fitted earlier to the 134-year record of annual rainfall values at Manilla NSW.
Histogram annual rainfall frequency Manilla NSWThe graph is copied here.

I defined as “Extreme Values” those either below the 5th percentile or above the 95th percentile of the fitted Normal Distribution. That is to say, those that were more than 1.645 times the Standard Deviation (SD = 156 mm) below or above the Mean (M = 652 mm). When expressed in millimetres of annual rainfall, that is less than 395 mm or more than 909 mm.
These ‘Tails’ of the Normal Distribution each totalled 5% of the modeled population, making 10% when added together.

The data

For each year’s 21-year sample, I counted those rainfall values that were lower than 395 mm (for the Low Tail) and those higher than 909 mm (for the High Tail). I added the two to give a count for Both Tails. To get a percentage value, I divided by 21.
I then found the ratio of this value to that of the fitted long-term Normal Distribution by dividing by 5% for each tail, and by 10% for both tails together. Ratios above 1.0 are Heavy Tails, and ratios below 1.0 are Light Tails.
That ratio, when smoothed, is plotted on the main graph at the head of the page.

Results

The resulting pattern of heavier and lighter tails, shown above, is similar to that found by using more and less extreme values, shown in the graph copied here.

Graph of history of extremes of annual rainfallAs before, there were less extremes in the 1900’s, 1910’s, 1920’s and 1930’s.
As before, there were more extremes in the 1940’s and 1950’s.
In the 1890’s, the “Tails” graph did not confirm the more extreme values that had been found earlier.

The 1990’s discrepancy

Extremes had been near normal through the last five decades in the earlier graph. By contrast, the “Tails” graph shows extremes in the most recent decade, the 1990’s, that were just as high as those in the 1950’s. Those two episodes differ, however: in the 1950’s only the high tail was heavy; in the 1990’s, only the low tail was heavy.
(For the 1990’s heavy low tail, see the Note below.)

The inadequacy of the data

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Annual Rainfall Extremes at Manilla NSW: III

III. When extreme values were more or less extreme

Graph of history of extremes of annual rainfall

This graph is based on applying a 21-year sampling window to each year in the Manilla rainfall record, then adding smoothing. (See “Note about Sampling” below.)


Back to the prelude “Manilla’s Yearly Rainfall History”.
Back to Extremes Part I.
Back to Extremes Part II.
Forward to Extremes Part IV.


Making the graph

For each year, I have identified the highest and lowest values of annual rainfall in its 21-year sample. I already know the long-term mean annual rainfall at Manilla: 652 mm. From those values I have plotted the height of the maximum value above the mean (red) and the depth of the lowest value below the mean (green). Both may be called “Extreme Values”.
The difference between the maximum value and the minimum value in each 21-year sample is the Total Range. That also is a measure of Extreme Value, which I graphed in an earlier post.
The Total Range is equal to the sum of the two Extreme Values that are plotted. To make it easy to compare the three measures, I have divided the Total Range by two. I have  plotted that value in blue.

Two discordant results

All three measures agree well except at two dates: 1897 and 1980. On both occasions the Minimum Value (green) was not extreme at all, being only about 200 mm below the long-term mean. The Maximum Value in 1980 was rather extreme (about 330 mm above the mean). The Maximum Value in 1897 was the most extreme value that appears on this graph: 475 mm above the mean!
The pattern of this graph is dominated by this single feature. It is due to just one data item: the annual rainfall reading of 1129 mm in the year 1890, which was the highest ever.

The pattern

For extreme annual rainfalls at Manilla, this graph suggests the following:

They were more extreme than usual at the end of the 19th century and in the 1940’s.
They were less extreme than usual from the 1900’s through to the 1930’s.
They have been no more or less extreme than one should expect through all of the last five decades.

Comment

This graph depends on very simple statistics: the maximum, the minimum and the mean. Such a sparse data set is subject to the effect of chance. Also, although this is not obvious, this graph assumes that other features of the distribution of annual rainfall have not been changing, which is not true.

I have more to say on this topic.


Note about Sampling

I chose a 21-year sampling window to be wide enough to contain enough points for analysis, without losing time-resolution, or losing too many years at each end of the record from 1883 to 2016.
The first mid-year of a sampling window was 1893 and the last, 2006.
To remove jumps in the trace on the graph, I then applied a nine-point Gaussian smoothing function. That further reduced the years that could be plotted to those from 1897 to 2002.

Annual Rainfall Extremes at Manilla NSW: I

I. Better graphs of Manilla’s annual rainfall and its scatter

Manilla 21-year rainfall medians

Background

The first two graphs  are new versions of graphs in an earlier post, published also as an article in “The Manilla Express” (28/2/17) and in the “North West Magazine” (20/3/17).

In that article, I said:
“This Manilla rainfall record is one counter-example to the snow-balling catalogue of reported extreme climatic events.”
My claim was not well supported. While the two graphs showed that recent annual rainfalls have been normal, with little scatter, they do not show whether there were any extreme events.

However, Manilla’s annual rainfall record can be analysed to show extreme events. This post considers the Total Range within a 21-year sampling window as a measure of extremes. Using that measure, extremes were at their highest in the 19th century, before anthropogenic global warming began.

A following post discusses kurtosis as another measure, with a different result.


Back to the prelude “Manilla’s Yearly Rainfall History”.
Forward to Extremes Part II.
Forward to Extremes Part III.
Forward to Extremes Part IV.


The two improved graphs

The re-drawn graphs of historical records in this post use a 21-year sampling window, as before. They now have an improved smoothing procedure: a 9-point Gaussian curve. (The weights are stated below.)

1. Yearly Rainfall Totals

The first graph (above) represents the normal rainfall as it changes. The earlier version showed the arithmetic mean. The new version uses the median value (the middle, or 50th percentile value) instead.
The new version is less “jumpy” due to better smoothing. The median varies much more than the mean does. All the same, most features of the shape are unchanged: very low annual rainfall from 1915 to 1950; very high rainfall from 1955 to 1982; normal rainfall since 1983. There are some shape changes: rainfall before 1900 does not plot so high; from 1911 to 1913 there is a respite from drought; the highest rainfall by far now appears from 1970 to 1980.

As before, one can say:
“Present rainfall will seem low to those who remember the 1970’s, but the 1970’s were wet times and now is normal. Few alive now will remember that Manilla’s rainfall really was much lower in the 1930’s.”

In addition, this new version makes the pattern of growth and sudden collapse obvious. Collapses amounting to 100 mm came within a few years after both 1900 and 1978. Growth in the 58 years from 1920 to 1978 came at the phenomenal and unsustainable rate of 33 mm per decade. By the 1970’s, elderly residents of Manilla would have seen rainfall increase decade by decade throughout their lives.
(I noted this pattern of growth and collapse in an earlier post about Manilla’s summer rainfall.)

Manilla 21-year rainfall Inter-quartile Range

2. Yearly Near-Mean Rainfall Scatters

The plot on this second graph is changed only by better smoothing. However, the titles are changed. I realised that the Inter-quartile Range is not a good general indicator of spread or, in this case, of reliability of rainfall (as I had assumed). Inter-quartile Range measures the scatter of values that are close the middle: just the middle 50%. My new title refers to “near-mean” scatter. Any values that could be called “extreme” fall very far beyond the Inter-quartile Range.

Two more measures of scatter

An alternative measure of scatter in data is the Standard Deviation. In normally distributed data, the Standard Deviation extends 34% each side of the median (and mean). The “Standard Deviation Range” then extends from the 16th percentile to the 84th percentile. It includes a much larger proportion (68%) of a population than the Inter-quartile Range (50%) does. However, it also says nothing about extremes, which will lie far out in the residual 32% “tails” of the data.

The broadest measure of scatter is the Total Range from the lowest to the highest value. This measure does include any extreme values that exist in the data.
In the present case, each calculation uses a sample that includes only 21 points. The lowest data point is close to the 5th percentile and the highest data point is close to the 95th percentile of a similar continuous curve.

All three measures of scatter graphed

Manilla 21-year rainfall Total Range, Standard Deviation Range and Inter-quartile Range

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Manilla’s Yearly Rainfall History

Lately, Manilla’s rainfall is normal, and more reliable
than it ever was.

Manilla yearly rainfall record, 21-yr smoothed

Yearly rainfall totals

The first graph helps to make sense of the history of Manilla’s rainfall, using the totals for each year. The actual figures make little sense, jumping up or down from one year to the next. The figures here have been calmed down. First, I replaced each yearly figure by an average of twenty-one years, ten years before and ten years after the date. Then I smoothed that figure some more.
The pattern is plain. There were periods in the past when there was much more or less rain than usual.
In four decades the rainfall was some 30 mm higher than normal: the 1890’s, 1950’s, 1960’s and 1970’s. In four other decades, the rainfall was some 30 mm lower than normal: the 1900’s, 1910’s, 1920’s and 1930’s.
Rainfall here collapsed about 1900. The collapse was was widespread, as was recognised half a century ago.

Using the average line drawn across the graph (at 652 mm), you can see that rainfall was below average from 1902 to 1951: almost exactly the first half of the twentieth century. After 1951, rainfall was above average for the 44 years to 1995. Since then, the annual rainfall (as plotted) has been remarkably close to the 132-year average.
Present rainfall will seem low to those who remember the 1970’s, but the 1970’s were wet times and now is normal. Few alive now will remember that Manilla’s rainfall really was much lower in the 1930’s.

The pattern in Manilla’s history of annual rainfalls is better shown in a graph in a later post. The new graph, using 21-year median values, has a clear pattern of collapse, growth and collapse.

Manilla yearly rainfall scatters.

Yearly rainfall scatter

The second graph also groups the data twenty-one years at a time. It shows the scatter of yearly rainfalls in each group. More scatter or spread means the rainfall was less reliable. Comparing the graphs, times of high scatter (very unreliable rainfall) were not times of low rainfall, as one might think. Annual rainfall scatter and rainfall amount were not related.
Times of very unreliable rainfall came in 1919 (dry), 1949 (normal) and 1958 (wet). Times of reliable rainfall came in 1908 and 1936 (both dry). However, by far the most reliable rainfall came since 1992, extending to 2004 and likely up to this year.

Global warming

It has been argued that human-induced climate change will cause climatic extremes to happen more often in future. Already, when any extreme climate event is reported, someone will say that climate change has caused it.

The present steady rise in global temperature began about 1975. Does this Manilla rainfall record show more extreme events since that date? Definitely not! Quite the contrary. Continue reading

More Droughts After Heavier Rains III.

Graphical log of errors when droughts are predicted from rains

Droughts and flooding rains at Manilla NSW were related in a way that is remarkable and unexpected.

Part III. Predicting drought from heavy rain

[Back to Part II: Scatter-plots]

The graph above is derived from the first graph in this series (copied here) by using the blue regression trend-line from the scatter plot of selected data (also copied here). (For data details, sLog of 1-year droughts and 5-year lagged heavy rainfallsee Note 1, below.)

The equation of the trend line, y = 0.030x is used AS IF to use the daily rainfall excesses to predict the drought frequency five years later. The graph shows the “error” of this “prediction”. (In Note 2, below, I concede that this data set could not support such prediction.)
As expected from the previous graphs, the “prediction” is accurate at most data points to 1975. It is correct to the nearest percentage whole number at nine of the eighteen points. From 1940 to 1955, droughts are uniformly more frequent than predicted. After 1975, the error curve swings wildly up and down.

Could droughts have been predicted from heavy rainfalls?

Scatter-plot 1890 to 1975

By about 1915, it is conceivable that this relationship could have been discovered, either by analysis of such data, or by modelling of the climate system. Then, the data for the next 20 years, up to 1935, would seem to confirm it. Data from 1940 to 1955 would cause doubts, but data from 1960 to 1975 would restore confidence. Then the utter failure of the model in the following four decades would have led to its abandonment, at least for the time being.

Climate shifts of 1975

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More Droughts After Heavier Rains II.

Scatter-plot 1890 to 1975

Droughts and flooding rains at Manilla NSW were related in a way that is remarkable and unexpected.

Part II. Scatter-plots

[Back to Part I: Graphical logs]

I have made scatter plots to see how much correlation there is between the two data sets: the frequency % of severe 12-month drought and the total decadal daily rainfall excesses over 50 mm, when lagged five years. (For data details, see Note 1, below.)

A. The first 70% of the data

The first scatter-plot includes only the first 70% of the data, from 1890 to 1975, which showed matching patterns on the graphical log copied below. I have broken the data points into two groups: the aberrant group 1940 to 1955 (red) and the fourteen best-matched points (blue). The trend line that best fits those fourteen points is y = 0.028x + 0.407, with R-squared = 0.898. However, I have been able to fit the trend line y = 0.030x, that shows y proportional to x, without making R-squared worse than 0.892.
Similarly, the four decades centred on 1940, 1945, 1950 and 1955, had y = 0.050x, with R-squared equal to 0.902.

Expressed in words: for fourteen of the first eighteen data points, the frequency % of severe 12-month droughts remained close to 0.03 times the decade total of daily rainfall (>50 mm/day) measured five years earlier. For the other group of four adjacent points, the number was not 0.03, but 0.05.

B. All the data

Scatter-plot 1890 to 2010

The second scatter plot shows data for all 25 (five-year overlapped) decades. There is a “shot-gun” pattern, as expected. Continue reading

More Droughts After Heavier Rains I.

Log of 1-year droughts and 5-year lagged heavy rainfalls

Droughts and flooding rains at Manilla NSW were related in a way that is remarkable and unexpected.

Part 1. Graphical logs

As the first graph shows, for most of the 130-year record year-long droughts came in direct proportion to very heavy daily rainfall five years earlier. (For data details, see Note 1, below.)
The match between these two variables is astonishing. Both are based on rainfall readings, but they are scarcely related. Excessive daily rainfalls are transient extreme weather events; 12-month droughts are an aspect of climate.

Mackellar’s “Droughts and flooding rains”

Dorothea Mackellar’s famous line * is more apt for this graph than for other graphs where I use “flooding rains” to mean periods unlike drought. (See Note 2. below.) The rains and droughts that I plot here both bring hardship. Severe droughts lasting one year are among the worst of droughts: long enough to use up reserves, and not so long as to be eased by periods of rain. The daily rainfall events plotted are the ones that cause damaging floods.

Features of the graphical log

Log of 1-year droughts and heavy rainfalls

This second graph shows the data at the actual dates. Although the data points for the decade excess of heavy daily rainfall and those for frequency % of 12-month droughts have a matching pattern for much of the record, the pattern is offset. Heavy rainfall points come five years earlier than corresponding drought points. Notice that the heavy rainfalls do not (except in 1980) come squarely in gaps between droughts.
Lagging the rainfall points by five years (as in the first graph) makes some matches almost exact. Such matches occur at all data points from 1890 to 1975, except those from 1940 to 1955, where drought frequencies are relatively higher. Both variables show a two-decade-long, slow decline from 1905 to 1925. At the chosen scales, the amplitude of corresponding rises and falls are usually similar as well.
After 1975, daily rainfall oscillates through a wide amplitude with a twenty-year period, while the frequency % of drought varies Continue reading