Relations Among Rainfall Moments

Six graphs of rainfall moment relations

Twelve-monthly values of rainfall since 1883 at Manilla NSW yield the four moments of their frequency distributions: mean, variance, skewness, and kurtosis. I plotted the history of each moment (when smoothed) in an earlier post.
Here, I compare the moments in pairs. Connected scatterplots reveal the trajectory of each relationship with time.
Some linear and cyclic trends persist through decades, but none persists through the whole record.
The first image is an index to the suite of six graphs of pair-wise relationships that I present below.

Rainfall variance vs. mean

Trajectory of Variance versus Mean

Rainfall variability (the variance) did not relate to annual rainfall amount (the mean) in a simple way. However, contrary to folklore, the driest time, around 1942, was a time when rainfall was not variable, but steady.
Annual rainfall was steady again at later dates: around 1973, around 1990, and since 2000. The latest date (2005) and the year 1937 shared the least rainfall variance.
By contrast, annual rainfall was extremely variable about 1957, in a decade when mean rainfall fell from very high values to normal values.

Rainfall skewness vs. mean

Trajectory of Skewness versus Mean

Skewness related negatively but weakly to rainfall amount (Mean). For the regression, R-squared was only 0.22.
However, there were episodes when this relation was strong: 1902-1943; 1963-1969; 1974-1986, a total 60 years in 108 years.
Such relationships imply that falling rainfall brought positive skewness and rising rainfall brought negative skewness. (On a histogram, the mode would move the opposite way to the tails.)
Both the first four and the last four years of record moved contrary to this: skewness decreased with decreasing rainfall in each case. (On a histogram, the mode would fall more slowly than the tails.)

Rainfall kurtosis vs. mean

Trajectory of Kurtosis versus Mean

Kurtosis was generally low: excess kurtosis was below -0.5 most of the time.
It was high in only two episodes, that of recent times (since 1985) when mean rainfall was near the long-term mean, and the time of extremely low rainfall around 1941 (from 1930 to 1946).
In the 1941 episode, kurtosis rose as rainfall fell, becoming leptokurtic briefly, then fell as rainfall rose.

[This completes the set of three graphs relating the other moments to the Mean.]

Rainfall skewness vs. variance

Trajectory of Skewness versus Variance

Skewness showed little relation to variance.
At times of low variance, there was a wide range of skewness values.
The single episode of very high variance began (in 1950) with skewness just slightly positive. Skewness became strongly positive (right skew) while very high variance persisted until 1959, then fell to strongly negative (left skew) as variance became low after 1966. By 1969, skewness (as smoothed) had reached its lowest point (-0.125).
In the first thirty years from 1896, variance remained close to its mean value, but skewness traversed a wide range.

Rainfall kurtosis vs. variance

Trajectory of Kurtosis versus Variance

Kurtosis fell with rising variance. Although the R-squared of the regression line was only 0.32, the plot consists largely of several trends that were almost parallel to the regression.
This relation would tend to keep the total spread constant through time. Variance is a measure that is dominated by values close to the mean, while Kurtosis is dominated by extreme values that are remote from the mean. Frequency distributions at the lower right of the graph, such as 1953, have very high variance (a wide central spread) and very low kurtosis (few extreme values). Frequency distributions at the upper left, including all recent years, have very low variance (a narrow central spread) and very high kurtosis (many extreme values).

Rainfall kurtosis vs. skewness

Trajectory of Kurtosis versus Skewness

Kurtosis related to skewness as a linear trend only briefly, between 1930 and 1950. In that period, kurtosis and skewness rose together to a maximum in 1941 and fell together after that.
For 55 years. from 1950 to 2005, there is a cyclic pattern of two-and-a-half anti-clockwise loops. In cycles of about 20 years, kurtosis lags skewness by 6 years (+/-5 years). Negative skewness is followed in sequence by platykurtosis, positive skewness, and leptokurtosis.
Cyclic results of this kind were published in 1957 by Robert L. Folk and William C. Ward, on the frequency distributions of sediment grain size: “Brazos River bar: a study in the significance of grain size parameters”.
Fig. 18 of that study revealed a helical trajectory involving mean, standard deviation and skewness. In that case, kurtosis was less clearly related to the other moments.

[My previous work, using annual data, seemed to show that kurtosis began and ended very high, while skewness began very high and ended very low. These augmented data do not show that pattern.]

Implications for rainfall at Manilla NSW

These graphs are empirical, revealing changes in Manilla’s rainfall regime decade by decade. They say little about climatic process.
The immediate cause would seem to be changes in the latitude and intensity of the Sub-Tropical High-pressure Ridge. This must be implicated in the two kinds of bimodality in Manilla’s rainfall: a bimodal frequency histogram of annual rainfalls, and a bimodal distribution across the seasons of the year.

These data will have to be reconciled with two other data sets presented earlier in this blog: (a) the length and severity of droughts, and their relative frequency; (b) the variation in time of summer rain versus winter rain.
Using the connected scatterplots of this post, and the time charts of each of the four moments, I can select significant dates for further study. For such dates, I propose to prepare rainfall histograms, as well as graphs of seasonal rainfall.

Implications for Climate Change

These graphs are for a very simple set of climate data: the quasi-annual changes in rainfall at a single station through 134 years. Yet the patterns are extremely complex, with trends and cycles that are transitory and that persist only to a limited degree.
They plainly illustrate one side of what Judith Curry calls the “Fundamental disagreement about climate change”.
On one side of the disagreement, Judith Curry has “External forcing: CO2 as climate ‘control knob'”. On the other side she has “Climate chaos: no simple cause and effect”.

These graphs support the second view, not the first.

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