Introduction
The atmospheric water holding capacity and thus potential
precipitation intensity depends exponentially on air temperature according to
the Clausius–Clapeyron (CC) relationship. As empirically documented by
several studies, high precipitation quantiles rise with temperature,
increasingly so with shorter duration, such as hourly or shorter. This
CC scaling describes a log-linear dependence of precipitation intensity on
temperature (P–T relationship) that roughly follows or exceeds the CC rate of
7 % K-1 for water vapor. Similarly well documented is a breakdown or even
reversal of that relation for temperatures beyond some thresholds, usually
somewhere between 15 to 20 ∘C, as indicated in
Fig. . This drop was also observed by
, ,
, and . More
details about the methods used in each referenced article can be found in
Tables and .
P–T relationships (99 % quantile, hourly intensities) digitized from several figures in the literature on a logarithmic scale.
Red dashed lines indicate CC scaling by the August–Roche–Magnus approximation (7 % at 0 ∘C, 6 % at 20 ∘C),
see and .
Across regions and studies, P rises with T but then decreases.
(a) , (mm day-1), and .
(b) , , and (converted from mm day-1).
The last two articles use temperature bins of varying width with a semi-constant number of observations per bin.
More details on study region and temperature variables can be found in Table .
P–T analysis methods used in the cited literature.
Article
Bin width
Min nbin
Quantile + estimation method
2 ∘C
unknown
mean amount
2 ∘C
unknown
mean amount
2 ∘C
unknown
75+90+99+99.9 %, emp. + GPD top 5 and 10 %
2 ∘C
300
99 %, GPD top 20 %; mm day-1 (for each month)
variable
median 233
99 %, empirical order stats1
2 ∘C (overlap: 1 ∘C steps)
2001
90+99+99.9 %, empirical + GPD top 4 %2
variable, avg. 2 ∘C
150
99 %, unknown, presumably empirical
5 ∘C (overlap: 3 ∘C steps)
300
99 %, unknown, presumably empirical
1 ∘C
unknown
95+99 %, unknown, presumably empirical
2 ∘C (overlap: 1 ∘C steps)
100
90+95+99 %, emp.: Cunnane unbiased estimator1
–
–
as in
1 Personal communication per email. 2 As in climexp.knmi.nl (p).
Regions and temperatures used in the literature cited in Fig .
Article
Region
Temperature variable
SW Germany
Refers to
Western Europe
Surface temperature
Germany
Presumably air temperature
Hongkong + NL
Dew-point temperature
Australia
Surface temperature
Japan
Presumably air temperature
Several explanations for this phenomenon have been proposed, such as an
increase in the proportion of rainfall stemming from convective events as
opposed to large-scale stratiform precipitation
. Other explanations include a slower increase
in moisture availability than in moisture storage capacity according to the
CC relationship or fully saturated conditions
lasting less than event duration .
There may be several different mechanisms in process at different timescales
and locations . The decrease in precipitation
intensity at high temperatures coincides with a decrease in the number of
observations. The aim of this study is to examine whether this drop could
(partly) be a sample size artifact. For this purpose, we contrast two
different approaches to estimate very high precipitation quantiles, namely
empirical quantiles (which are based on plotting positions), and parametric
quantiles (which are derived from fitting the generalized Pareto distribution (GPD) to
the data). We compare both estimation methods with regard to their sample
size dependency and their effect on the shape of P–T relationships, using
both observed hydrometeorological and synthetic data.
Data and methods
Climate data
We analyzed publicly available time series of precipitation, temperature, and
relative humidity from 142 stations across Germany from the German Weather
Service . The stations are selected based on the length of
available hourly time series. All selected datasets contain at least 15 years
of observations, mostly 20 years. The R code for data selection, download, and
analysis is available at https://github.com/brry/prectemp.
To analyze only the nonzero precipitation records that are actually of interest for this article, values below 0.5 mm h-1 are omitted.
This cutoff is in line with the cited literature and is suitable because measurements of very low rainfall intensities have a high relative uncertainty.
The values are then logarithmized to enable a comparison of rates of precipitation change across temperatures.
Because of the very skewed nature of rainfall values, this also allows for better distribution fits.
Temperature binning
Throughout this paper, event dew-point temperature is used as an integrated
measure of air temperature and water vapor saturation (or moisture supply).
It is defined as the average dew-point temperature of the 5 h preceding
each rainfall hour, similar to the procedure by
. Dew-point temperature is calculated with the
Magnus formula based on observed relative humidity and air temperature at 2 m height see.
Following the analysis method of and
, we partition the hourly precipitation depths
according to the event dew-point temperature. We use moving temperature bins
with a fixed width of 2 K. Bin midpoints increase in 0.1∘ steps.
Empirical quantiles
Empirical quantiles are estimated by a monotonic mapping of the ordered sample to sample-size-specific probabilities called plotting positions.
This can be done in a variety of ways as reviewed by .
Common to all is the fact that the portion to the right of the sample maximum is left unresolved (no extrapolations) and receives the same probability as the maximum.
Quantiles representing return periods larger than the sample length are consequently mapped to that maximum.
They are therefore underestimated – a fact apparently too trivial to have warranted any publication.
The empirical quantiles used in this article are computed based on the k-1/3n+1/3 plotting positions n= sample size, k = 1,..., n; see.
Parametric quantiles
The parametric quantile estimates are obtained in a peak-over-threshold
approach, where the generalized Pareto distribution is fitted to the
top 10 % of the sample. Quantiles are calculated from the fitted GPD.
We use the method of L moments to fit the GPD parameters. They are analogous
to the conventional statistical moments (mean, variance, skewness, and
kurtosis) but “robust [and] suitable for analysis of rare events of
non-normal data. L moments are consistent and often have smaller sampling
variances than maximum likelihood in small to moderate sample sizes.
L moments are especially useful in the context of quantile functions”
.
To obtain the quantile from the fitted distribution, the given probabilities
must be scaled with the conditional probability of the truncation. For
example, if the 99 % quantile (Q0.99) is to be computed from the top 10 %
of the data, Q0.90 of the truncated sample must be used. We refer to Q0.99 as
the “censored 99 % quantile”. Because five values are required to obtain
L moments, the minimum sample size at 90 % truncation is 50 (45 values are
discarded).
Selecting a suitable fitting method is of great importance in the context of
sample size bias. For example, unlike moment-based procedures, maximum
likelihood estimation (MLE) can still show an underestimation bias at small sample
sizes, as shown in the Supplement. This happens in small samples
(n<200) for distributions with bounded parameters (and the optimum of the
likelihood function lying on the boundary). We refer to the Supplement for a comparison of the different methods.
The GPD quantile computation formula used in the source code of lmomco is
x(F)=ξ+ακ1-(1-F)κif κ≠0ξ-α×log(1-F)if κ=0,withξ=location,α=scale,κ=shape.
Sample size dependency
In Sect. , we pointed out that empirical methods
inherently underestimate high quantiles in small samples. In order to
quantify the potential effect in the context of P–T relationships, we set up
the following experiment: to investigate the dependency of both quantile
estimation methods on sample size, we draw random samples from a defined
population. This should optimally be a large set of values following a
distribution observed in nature. We therefore use a pooled dataset with all
the precipitation values observed at any of the 142 stations. From this
population, we draw random samples of several sizes and compute empirical and
parametric quantiles from each sample. For each sample size, this is done
1000 times, resulting in a corresponding quantile distribution depending on
sample size.
Synthetic P–T relationship
We apply the results of the previous Sect. – that is, the
potential small-sample effects of empirical and parametric quantile
estimates – to P–T scaling relationships and analyze the drop at high
temperatures. To study that effect, we designed an experiment with synthetic
data. Here, precipitation values are generated in a way that exhibits a
stable temperature scaling over all temperature ranges. The CC-scaling rate
is constant, and the increase in high rainfall quantiles per degree Kelvin
remains the same over all temperatures. When sampling from such synthetic data,
any drop in the P–T relationship must be a statistical artifact. For this
purpose, we define a “temperature-dependent GPD” with parameters that depend
on temperature. To achieve a realistic temperature scaling, we base the
parameters on the linear regression of the fitted parameters at several dew-point
temperatures.
From that synthetic GPD, 1000 random samples are generated for each temperature bin.
The sample size corresponds to the average number of precipitation observations at the climate stations in each bin.
From these sets of random samples, the empirical and parametric 99.9 % quantiles are calculated.
Results
Sample size dependency
The dependence on sample size, as revealed by 1000 random draws per sample size from the pooled precipitation data, is shown in Fig. .
The 99.9 % quantile of this population (n=1.16 million) is 19.5 mm h-1.
It is strongly and consistently underestimated by the empirical estimator with shrinking sample size.
For a sample size of 50, the median estimate is only 7 mm h-1.
Realistic estimates are obtained only for samples larger than about 700, around which the estimates converge to the (true) population value.
The parametric estimators do not exhibit this bias – only their variance increases with smaller samples (the uncertainty range is wider).
This is a typical example of the well-known bias–variance tradeoff in estimation theory.
Median of the empirical and parametric 99.9 % quantile estimates depending on the size of
samples drawn from all the precipitation intensity values along with their uncertainty bands.
The horizontal dashed line marks the empirical quantile of the complete dataset (n=1.16 million).
For n>500, we used a step size of 10 (instead of 1) for the sample size, so the curve appears smoother there.
P–T relationship: empirical vs. parametric quantiles
The procedure of obtaining parametric (using the GPD) and empirical quantiles
was applied per temperature bin to the datasets of each of the 142 stations.
The empirical precipitation quantiles per bin are presented in the left panel
of Fig. . The shape of the P–T relationships is consistent
with the behavior of P–T relationships shown in Fig. of the
introductory section. The empirical quantile estimates start decreasing
between 15 and 20 ∘C. Some stations show the empirical quantile drop
more distinctly than others. The figure also shows the average across
stations, where the drop becomes particularly clear. Compared to the red line
depicting the CC scaling of 7 to 6 % K-1, the precipitation increase follows a
super-CC scaling with a rise that is steeper than the CC rate. This is in
accordance with previous findings, e.g., by .
The parametric estimates are displayed in the right panel.
At temperature ranges where empirical quantiles decrease, parametric quantiles keep increasing.
This difference is less pronounced for smaller quantiles (see Supplement Sect. S4).
The 99.9 % precipitation intensity per temperature bin with empirical and parametric quantile estimate
(a and b respectively).
Each line represents one of the 142 stations, with the black line as the average across stations.
The red line denotes CC scaling as in Fig. .
The green line in (b) repeats the average from (a) for comparison.
(a) Parameters of a temperature-dependent GPD: ξ (location),
α
(scale),
and κ (shape). The orange lines show a linear regression as per Sect. .
(b) Corresponding 99.9 % distribution quantile (orange) and median of the 99.9 % quantile
estimates generated from samples in 1000 random draws along with their variance bands.
Synthetic P–T relationship
The synthetic P–T relationship that continuously rises with temperature (see
Sect. ) is defined with the parameters shown in the
left panels of Fig. , where each dot represents one of the
stations. The right panel shows the median of the 99.9 % quantile estimates
from random samples with the original sample sizes. Even though the
distribution continues to increase with temperature, empirical quantiles from
random samples stagnate or drop around 18 ∘C where sample size
decreases quickly. Parametric quantiles obtained by distribution fitting do
not drop and follow the theoretical quantile from the distribution function.
Discussion and conclusions
Precipitation quantile estimates rise with temperature until they reach a turning point,
beyond which they decrease. For this drop in the CC-scaling relation towards
higher temperatures, a number of explanations have been suggested. In this
study we offer the alternative view that the drop can be understood, at least
in some cases, as a statistical artifact of small samples. At higher
temperatures, fewer precipitation observations are available because (1) wet
events are less frequent at high temperatures and (2) precipitation events
at higher temperatures are generally convective in nature and very localized
in space; they are thus often missed by the observing network, resulting in
smaller sample sizes compared to large-scale precipitation at lower
temperatures. A rather simple argument shows that empirical quantile
estimators have an underestimation bias for return periods exceeding the
sample size, and we verified this behavior in a set of Monte Carlo
experiments. It turned out that the underestimation of high quantiles, such
as those relevant for the upper portion of the CC-scaling relationship, can
be substantial. We have shown that when empirical estimators are
appropriately replaced by parametric ones, the high-temperature drop in
CC scaling disappears. The method of parametric estimation is crucial,
nevertheless, as similar small-sample biases are known, e.g., from using MLE
estimators (see above and more examples in the Supplement). The most robust
estimates were obtained from moment-based methods. Past CC-scaling studies
that have relied on empirical or ML-based quantile estimators are likely
affected by the small-sample artifacts for high temperatures that we have
described here. For those, we find it necessary to revisit the corresponding
estimation step using other, e.g., moment-based, procedures. This may be
especially interesting for quantiles beyond the 99.9 % level.
To exclude potential physical effects related to precipitation as much as
possible, we have repeated the analysis with synthetic data and obtained
essentially the same results. Furthermore, we have used dew-point temperature
instead of air temperature in order to rule out that the drop in the
P–T relationship is caused by a lack of moisture supply. It should be noted,
though, that the use of dew-point temperatures only accounts for moisture that
is already stored in the local atmosphere. It does not account for
large-scale moisture convergence which becomes more important with longer
precipitation duration intervals. This is evidence that the drop in empirical
quantile estimates is precipitation independent; it is less a physical
phenomenon but rather a statistical artifact caused by small samples, and it
can largely be overcome by employing parametric estimators. Still,
alternative physical explanations considering physical processes should not
lightly be discarded. Some were summarized briefly in
Sect. . It might also, for example, be hypothesized that
near-surface temperature is not an adequate proxy for air temperature at the
height where precipitation-forming patterns unfold on very warm days.
Parametric quantiles from fitted distributions provide a means to retrieve less biased estimates of extreme quantiles.
The price to be paid is the larger uncertainty of those estimates.
This should be quantified by confidence intervals or application to several datasets to avoid singular non-representative results.
The parametric method requires significantly fewer data points in a sample than empirical quantiles need to converge to the actual (unknown) value.
In the combination of small sample sizes and very high quantiles, the use of parametric quantiles is recommended.