Annual Rainfall Extremes at Manilla NSW: II

II. Platykurtic, Bimodal Annual Rainfall

Histogram annual rainfall frequency Manilla NSW

Manilla’s 135 years of rainfall readings yield the graph above. There are several features to notice.

A ragged pattern

Despite having as many as 135 annual rainfall values, the graph is still ragged. Some of the 20 mm “bins” near the middle have less than 2% of the observations, while others have over 5%. The pattern has not yet become smooth.

It is not near a normal distribution

Rainfall is thought of as a random process, likely to match a curve of normal distribution. On the first two graphs I have drawn the curve of normal distribution that best fits the data.

Smoothed annual rainfall frequency Manilla NSW

In this second graph, I have smoothed out the ragged shape of the plotted data, using a 9-point Gaussian smoothing. You can see more clearly where the actual curve (black) and the normal curve (magenta) differ. The dotted red line shows the differences directly:

The peak is low;
The shoulders, each side of the “peak”, are high;
Both of the tails are thin.

These three features describe a platykurtic curve: one with low kurtosis. This fact makes the highest and lowest annual rainfalls at Manilla less extreme than would be expected in a normal distribution.

Another departure from normality is that the curve is skewed: the tail on the left is shorter than the one on the right. That is a positive skew, but it is small. (By contrast, most of the rainfall distributions for individual months at Manilla have large positive skew. In them, the peak is well below the mean, and a tail extends to rare high values.)

In summary, four of the leading features of the shape of Manilla’s annual rainfall distribution are:

Mean or average: 652 mm per year.
Standard Deviation (measuring spread or scatter): 156 mm.
Skewness: 0.268 (slightly positive).
Kurtosis: -0.427 (strongly platykurtic).


A platykurtic curve matches the Manilla annual rainfall frequency curve to some extent.

The sum of two Gaussian curves gives a much better match.


Fitting a platykurtic near-normal curve

Much of the poor fit of a normal curve to the data is due to the data having a platykurtic distribution. Being platykurtic produces a reduced peak, high shoulders, and thin tails, as was noted.

Smoothed rainfall frequency and a platykurtic curve

In the third graph, I have drawn (in magenta) a new model distribution that is platykurtic. It is a transform of the normal distribution with a weighted sinusoidal correction. The new curve fits much better up both flanks of the data curve. It cannot be made to fit in the peak area between 500 mm and 820 mm.

Fitting a bimodal model made of two normal curves

The shoulders of the smoothed rainfall distribution curve (black) are not simply high; they are higher than the  zone in the middle where the peak would normally be. There is a major mode (peaking at 5.1%) on the left, a minor mode (3.9%) on the right, and an antimode (3.7%) between them.

Smoothed rainfall frequency and a bimodal curve

Continue reading

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.

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

Continue reading

When is the First Frost?

This year (2017) the first frost at Manilla came on the 11th of May, close to the middle date for it: the 13th of May. In just half of the years, the first frost comes between ANZAC Day (the 25th of April) and the 19th of May.

Graphical record of first frost dates
(See the notes below: “Observing Frosts in Manilla.”)

The date of first frost from year to year

The graph shows the dates of first frosts in the last nineteen years. One feature stands out: from a very early date of the 4th of April in 2008, the dates got later each year to a very late date of the 6th of June in 2014. Otherwise, the dates simply jumped around.

Graphical log of frostsThe date of first frost hardly relates at all to the number of frosts in a season. This graph, copied from an earlier post, shows the mismatches. The earliest first frost, in 2008, was in a year with a normal number of frosts. In the least frosty year, 2013, the first frost did not come late.

The central date and the spread

To find the central value and the spread of a climate item like this calls for readings for a number of years called a “Normal Period”. (See note below on Climate Normals.) I chose the first eleven years of my readings (1999 to 2009) as my Normal Period. For this period I found these five order statistics:

Lowest (earliest) value: 4th April;
First Quartile value: 25th April (ANZAC Day);
Median (middle) value: 13th May;
Third Quartile value: 19th May;
Highest (latest) value: 24th May.

These five values divide the dates of first frost into four equal groups. For example, the first frost comes before ANZAC Day in one year out of four. This could confirm what Manilla gardeners know already!

Is the first frost getting later?

Talk of global warming leads us to expect the date of first frost to get later. By how much?
Dates on the graph after 2009 seem to be later in the season than during the Normal Period. As shown, a linear trend line fitted to the data points slopes steeply down towards later dates in later years. A curved trend line (a parabola) slopes down even more steeply. However, with so few data points, these trend lines are wild guesses, not to be relied on for forecasting future frosts.
Data for NSW from 1910 shows that daily minimum temperatures have been rising at 0.11° per decade.  (That is much faster than the rate for daily maximum temperatures, which is 0.07° per decade.) To work out how this might affect the date of first frost in Manilla, one needs to know that the daily minimum temperature in this season gets lower each day by 0.15°. One day of seasonal cooling will more than cover a decade of climate warming. The effect of global warming is to make the date of first frost only one day later in fourteen years. If the middle date of first frost was the 13th of May in the Normal Period, centred on 2004, the forecast middle date of first frost next year (2018) would be the 14th of May. This is shown by the flattest of the three trend lines on the graph.

Looking ahead, it seems unlikely that the date of first frost will get later by as much as a week within a lifetime.


Notes

1. Observing Frost in Manilla

Continue reading

Global Warming Bent-Line Regression

HadCRUT global near-surface temperatures

HadCRUtemp2lineThis graph, posted with permission, shows a bent line fitted to the HadCRUT annual data series for global near-surface temperature. Professor Thayer Watkins of San Jose State University Department of Economics posted it on his blog about 2009.

HadCRUTsmoothWithout knowing of this work, I constructed the second graph. I used data from the same HadCRUT source, but a data set smoothed by the authors.

In April 2013 I posted it to a forum thread in”weatherzone”.

Next, I added to that graph a logarithmic plot of global carbon emissions, similarly fitted with a series of straight trend lines.

Log from 1850 of world surface air temperature and carbon emissionsThis I included in posts to several forums: in a post to “weatherzone”, in a post to the Alternative Technology Association forum, and finally in a post to this blog.

Both Professor Watkins and I have fitted bent lines to the data. I fitted the lines by eye (for which I was accused of “cherry-picking”). Professor Watkins used an explicit process of Bent-Line Regression, minimising the deviations by the method of least-squares. Like me, he initially chose by eye the dates of the change points where the straight lines meet. But he then adjusted them so as to minimise the least squares deviations.
[See notes below on the method of Bent-Line Regression.]

The trend lines and change points are practically the same in the Thayer Watkins and the “Surly Bond” graphs:
1. (Up to Down) TW: 1881; SB: 1879.
2. (Down to Up) TW: 1911; SB: 1909.
3. (Up to Down) TW: 1940; SB: 1943.
4. (Down to Up) TW: 1970; SB: 1975.
As I said at the time, once straight trend lines are chosen, the dates of change points to fit this data series closely do not allow of much variation.

Relation to the IPO (or PDO) of the Pacific

Not by coincidence, Watkins and I both went on to relate the multi-decadal oscillations of Pacific Ocean temperatures to the global near-surface average temperatures.

My approach

I merely plotted my chosen global temperature change points on to the Pacific graphs (I chose to cite the IPO (Inter-decadal Pacific Oscillation)). In two posts I noted (i) the way the change points in the HadCRUT global temperature series were close to those in the IPO, and (ii) the way the IPO seemed able to explain why the trend in global warming was “bent” in 1943 and 1975 but, in that case, could only sharpen the bends of 1910 and 1880.

Professor Watkins’ approach

AGT_PDO7Professor Watkins did a separate Bent Line Regression Analysis on the Pacific graphs (He chose to cite the earlier-developed PDO (Pacific inter-Decadal Oscillation)). His analysis “A Major Source of the Near-Sixty Year Cycle in Average Global Temperatures is the Pacific (Multi)Decadal Oscillation” is here.

He admits the match is poor, with various lags and a different period. He concludes:
“Thus while the Pacific (Multi)Decadal Oscillation appears to be involved in the cycles of the average global temperature there have to be other factors also involved.”

The significance of the IPO

Continue reading

HadCRUT Global Temperature Smoothing

Graph of recent HadCRUT4

As a long-term instrumental record of global temperature, the HadCRUT4 series may be the best we have. [See Ole Humlum’s blog in the notes below.]
I like to use the published smoothed annual series of HadCRUT4.  I find that this smoothing gets rid of the “noise” that makes graphs about global warming needlessly hard to read. I used the smoothed HadCRUT series to point out the curious inverse relation between the rate of warming and the rate of growth of carbon emissions in this post from 2014.  I will refer again to that post in discussing the use of bent-line regression to describe global warming.

The Met Office Hadley Centre published the smoothing procedure that they used for the time series of smoothed annual average temperature in the HadCRUT3 data set. The smoothing function used is a 21-point binomial filter. The weights are specified in the link above.
The authors discuss the fudge that they use to plot smoothed values up to the current year, even though a validly smoothed value for that year would require ten years of data from future years. Their method is to continue the series by repeating the final value. They had added to the uncertainty by using a final value from just part of a year.
They relate how this procedure had caused consternation when the smoothed graph published in March 2008 showed a curve towards cooling, due to the final value used being very cool.
They show the effect by displaying the graph for that date.
They maintain that the unacceptable smoothed curve (because it shows cooling, not warming) is due mainly to using a final value from an incomplete year, saying:
“The way that we calculate the smoothed series has not changed except that we no longer use data for the current year in the calculation.”
That web-page is annotated:
“Last updated: 08/04/2008 Expires: 08/04/2009”
However, this appears to be the current procedure, used with the HadCRUT4 data set.

For my own interest, I plotted the values from 1990 to 2016 of the annual series of HadCRUT4, averaged over northern and southern hemispheres. [Data sources below.]

On my graph (above), all points 1990 to 2016 are as sourced. I have plotted raw values 2017 to 2026 (uncoloured) as I believe they are used in the smoothing procedure. I have also left uncoloured the smoothed data points from 2007 to 2016, to indicate that their values are not fully supported by data.

I agree with Ole Humlum that it is very good of the Met Office to come clean on the logical shortcomings of their procedure for smoothing, but it would be even better if they ceased plotting smoothed points when the smoothing depends on data points for future years.
In my monthly series of parametric plots of smoothed monthly values of climate anomaly variables, I have faced the same problem. I smooth using a 13-point Gaussian curve. My solution is to plot “fully-smoothed” data points (in colour) up to six months ago. That gives a consistent mapping up to that date. The fifth month before now (plotted uncoloured) is smoothed with an 11-point Gaussian and so on, up to the latest month with a necessarily unsmoothed value.


Notes

1.
Ole Humlum’s blog “Climate4you”

[See: Index\Global Temperature\Recent global air temperature change, an overview\]

2.
HadCRUT4 data
Source of raw annual values:

Source of smoothed annual values:

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

House in a cold October

This October has been very cold. That has kept indoor temperature
in this solar-passive house almost too cool for comfort. I wore warmer clothes and opened windows to admit warm air.

Indoor/outdoor temperature scatter-plot.

The climate this October

The graph shows (on the x-axis) how cold this October [in red] was: the coldest of the new century.
Here on the North-west Slopes of NSW, October warms and cools more from year to year than other months. It is the month most affected by climate cycles such as the El Nino-Southern Oscillation (ENSO). As shown, October warmed by one degree each year from 2011 to 2015, then cooled by nearly six degrees from 2015 to 2016.
ENSO followed almost the same pattern, but October 2012 was warmer than October 2013.
For five months, world temperature has also been down: much lower than it was in the record-breaking months of February and March 2016. (HadCRUT4 Global monthly near-surface data set (Column 2 in the linked table.))

Indoor climate this October

As shown on this graph beginning 2005, the indoor mean temperature in October months has varied with outdoor mean temperature. This coldest October outdoors (15.9 degrees) was also the coldest indoors (20.8 degrees). (But see Note below.)
October is the final month that I keep the house in its winter warming regimen. In 2014 and 2015 it had been almost ideally warm, but in 2016 it was just above the comfort minimum. Since this figure is just an average, there were times when the house was too cool for comfort, especially in the mornings.

Successive unfavourable months this year

As in other seasons, I intend the indoor climate to be comfortable through each spring season.
As I posted in “Hard Winter for Solar-passive” this very cloudy winter had reduced solar gain, making heaters needed much more than usual. However, the mean indoor temperature at winter’s end (August) was normal, although the heat bank was 0.7 degrees cooler than normal.
In September months, the warmth indoors still depends on solar gain through the north windows. This time,the sky continued very cloudy, and the daytime temperature was a record low value. As a result, the indoor temperature was 0.9 degrees down and the heat bank 0.7 degrees down.
By October, there is no solar gain through the north windows: warmth is gained from the surroundings in daytime by conduction, convection and radiation and retained by closed curtains at night. This time, both day and night temperatures were three degrees below normal, reducing daily heat gain and increasing nightly loss. As a result, the indoor temperature was 1.2 degrees down and the heat bank 0.9 degrees down.

What I did

Continue reading