-Supplementary material
S1: Review of notable past studies on historical precipitation trends in the United States.
Study
Study area
Data source
Time Period
Temporal
resolution
Annual and
seasonal
Spatial
resolution
National
Variables
Karl et al.
1995
CONUS1
NCDC2 data
1900-1993
Karl & Knight
1998
CONUS
Daily data for
182 stations
from USHCN3
1910-1996
Annual and
seasonal
Frequency and intensity
of total pr and pr divided
into ten quantiles
Least squares
linear regression
63 Canadian
stations and
1295 U.S.
stations from
various sources
For US, 188
stations from
USHCN3, same
as Karl and
Knight 1998
NCDC2 Daily
data: a few
hundred in early
twentieth
century to close
to 5000 in late
20th century
GHCN4 daily
data
1931–1996
for US
1951–1993
for Cananda
Annual and
seasonal
Calculated at
each station,
Summarized by
9 NCDC regions
At each station
Kunkel et al
1999
CONUS and
Canada
Extreme pr based on
recurrence interval (1and 5-yr) and duration (1,
3, and 7 day).
Non-parametric
test and the
Kendall slope
estimator
Groisman et
al 1999
8 major
regions
around the
world
1910-1996
Daily
precipitation
distribution
At each station
Mean and heavy
precipitation (defined as
above 50.8mm for US)
1900-1999
Focus more
on 19501999
Annual and
seasonal
Pooled into 9
climate regions
Mean, extreme pr (90%,
95%, and annual
maximum)
Statistical model
based on gamma
distribution: the
scale and shape
parameters
Least squares
linear regression
Groisman et
al 2001
CONUS
Groisman et
al 2005
Global
1893–2002
Annual and
seasonal
Summarized to
1 degree by 1
degree cells
Total and frequency of
wet (> 1mm), heavy
(>50.8mm), very heavy (>
101.6mm) pr.
Upper 5%, 1%, and 0.1%)
pr.
McRoberts
and NielsenGammon
(2011)
CONUS
A new
homogeneous
climate division
monthly
precipitation
dataset
1895–2009
Annual
344 climate
divisions
Mean annual
precipitation
Groisman et
al. 2012
Central US
NCDC2 COOP
stations and
hourly pr.
stations
Comparison
of two
periods:
1948–1978
1979-2009
Total amount, frequency,
heavy pr (upper 10%),
days with pr greater than
50.4mm
Trend detection
method
Least squares
linear regression
Least squares
regression and
Nonparametric
Spearman rank
order correlation,
autocorrelation
tested
Least squares
linear regression
Key finds

Since 1970 pr remained above the 20th century
mean, averaging about 5% higher than in the
previous 70 years.

Increase is mainly due to increases during the
second half of each year, particularly during the
autumn.

Since 1970 there have been about 2% more days
per year with pr. than earlier in the century
Increases of total precipitation are strongly affected by
increases in both frequency and intensity of heavy and
extreme precipitation events.

An increase in the number of 7-day, 1-yr events
over the period of 1931–96.

A composite index for US exhibits an average
upward trend of 3% per decade, statistically
significant at the 1% level
Scale parameters vary but shape parameters are
stable. The increase in the probability of heavy
precipitation is four times the increase in mean
precipitation.

Pr. increased significantly over most of CONUS in
all seasons except winter.

The trends range from 7% to 15% per century
during the summer and transition seasons.

The increase in heavy and extreme events is
amplified by a factor of 2–3 compared to the
change in the mean.
For US over the study period (1893–2002) the
frequency of very heavy pr. has increased by 20%
(statistically significant at the 0.01 level). All of the
increase has occurred during the last third of the
century.



Annual
Analyzed at
stations,
summarized
into three
climate regions
Intense pr. (> 12.7 mm)
moderately heavy (12.7–
25.4 mm), heavy (25.4–
76.2 mm), very heavy
(>76.2 mm), and extreme
(> 154.9 mm) pr.
Direct comparison
of the two
periods.


Linear precipitation trend is positive across most
of the United States.
Trends exceed 10% per century across the
southern plains and the Corn Belt.
Changes in gauge technology and station
location may be responsible for an artificial trend
of 1%–3% per century
Moderately heavy pr. became less frequent
compared to days with pr. above 25.4 mm.
Significant increases occurred in the frequency of
very heavy and extreme pr. events with up to
40% increase between the two time periods.
Notes:
1. Conterminous US
2. National Climatic Data Center
3. US Historical Climatology Network
4. Global Historical Climatology Network
References:
Groisman PY, Karl TR, Easterling DR, Knight RW, Jamason PF, Hennessy KJ, Suppiah R, Page CM, Wibig J, Fortuniak K (1999) Changes in the probability of
heavy precipitation: important indicators of climatic change. Climatic Change 42:243-283.
Groisman PY, Knight RW, Karl TR. (2001) Heavy precipitation and high streamflow in the Contiguous United States: trends in the twentieth century. Bulletin of
the American Meteorological Society. 82(2):219-246.
Groisman PY, Knight RW, Easterling DR, Karl TR, Hegerl GC, Razuvaev VN (2005) Trends in intense precipitation in the climate record. Journal of climate 18.
Groisman PY, Knight RW, Karl TR (2012) Changes in Intense Precipitation over the Central United States. Journal of Hydrometeorology 13.
Karl TR, Knight RW, Easterling DR, Quayle RG (1995) Trends in U.S. Climate during the Twentieth Century. Consequences 1 (1).
Karl TR, Knight RW (1998) Secular trends of precipitation amount, frequency, and intensity in the United States. Bulletin of the American Meteorological society
79:231-241.
Kunkel KE, Andsager K, Easterling DR (1999) Long-term trends in extreme precipitation events over the Conterminous United Sates and Cananda. Journal of
Cliamte. 12: 2515-2527.
McRoberts DB and JW Nielsen-Gammon (2011) A New Homogenized Climate 1 Division Precipitation Dataset for Analysis of Climate Variability and Climate
Change. Journal of Applied Meteorology and Climatology 50: 1187-1199.
S2: Non-parametric trend estimate
A nonparametric approach to estimate the slope of a line that best “fits” all data points is to calculate the slopes of all pairs of data points and then calculate
some kind of average or median of these slopes (Birkes and Dodge 1993). The slope of the line joining data points (𝑥𝑖 , 𝑦𝑖 ) and (𝑥𝑗 , 𝑦𝑗 ) is
𝑦𝑖 − 𝑦𝑗
𝑏𝑖𝑗 =
𝑥𝑖 − 𝑥𝑗
In the case of a time series, y indicates values of the time series variable (e.g. precipitation variables in our study), and x indicates values of time (years in
our case of annual series). Slopes (𝑏𝑖𝑗 ) are calculated for all pairs of n points (1 ≤ 𝑖 < 𝑗 ≤ 𝑛). The commonly used Thiel-Sen slope estimator is defined as
the median of all 𝑏𝑖𝑗 . However, there is one problem with the Thiel-Sen slope estimate when the data has many ties, i.e. 𝑦𝑖 = 𝑦𝑗 , yielding many slopes of zero.
This could happen with count-based data (e.g. precipitation frequency). It could also happen with all precipitation variables in an extremely dry area where
many observations were zero. In these cases, the Theil-Sen slope could equal to zero even when the Mann-Kendall test detects a statistically significant nonzero trend. In order to counter this problem, we used a weighted average as an alternative to median of all slopes as the estimate for the trend. It is defined as
follows.
𝛽 = ∑ 𝑤𝑖𝑗 𝑏𝑖𝑗
Where
𝑤𝑖𝑗 =
(𝑥𝑖 − 𝑥𝑗 )2
∑(𝑥𝑖 − 𝑥𝑗 )2
Weights are introduced based on the assumption that pairs of samples whose x-coordinates (years) differ more, are more likely to have an accurate slope and
therefore should receive a greater weight (Scholz 1978; Sievers 1978). This method generated slope values comparable to those derived from linear regression
for most stations, but more muted values for stations with precipitation outliers. Slopes produced by this method were also close to Theil-Sen slopes when they
were not zero, but gave higher regional mean values, because they were more likely to be positive when Theil Sen results were zero.
References:
Birkes D, Dodge Y (1993) Nonparametric Regression. Alternative Methods of Regression:111-141.
Scholz F-W (1978) Weighted median regression estimates. The Annals of Statistics:603-609.
Sievers GL (1978) Weighted rank statistics for simple linear regression. Journal of the American Statistical Association 73:628-631.
S3: Bootstrap method to assess the statistical significance of changes in extreme precipitation events.
For each region, the regionalized annual maximum daily precipitation data was divided into two groups (samples) based on two time periods: 1951-1980 and
1981-2013. The bootstrap method involves taking the original data set of N observations, and sampling from it to form a new sample (called a bootstrap sample)
that is also of size N. The bootstrap sample is taken from the original using sampling with replacement so, assuming N is sufficiently large, there is virtually zero
probability that it will be identical to the original “real” sample. This process is repeated 1000 times, and for each of these bootstrap samples we fit the Extreme
Value Distribution, based on which we calculate the precipitation magnitudes of various return intervals. Distribution parameters (location, shape and scale) are
also generated. We now have a 1000 bootstrap estimates for each of the variables (i.e. precipitation of various return intervals) and the distribution parameters for
each of the two time periods. We then use Student’s t-test to assess the statistical significance of their means. More details about the bootstrap method is provided
in Efron and Tibshirani (1993).
Reference:
Efron, B. and Tibshirani, R. (1993) An Introduction to the Bootstrap. Chapman and Hall, New York, London