Supplementary Information (SI) for:
Increased record-breaking precipitation events
under global warming
Jascha Lehmann*, Dim Coumou, Katja Frieler
*Corresponding author. Email: jlehmann@pik-potsdam.de
S1: ‘Testing the iid assumption’
The iid assumption for the stationary model is justified because the detrended observational
time series are close to iid. To show this, we first detrend the original Rx1day time series by
subtracting the smoothed mean value calculated using singular spectrum analysis with
window length of 15 years. The residuals contain the year-to-year variability for a specific
month. We then test whether the residuals are temporally independent by calculating the serial
correlation in the detrended Rx1day time series for each calendar month. We find that the
correlation values are randomly distributed over all land areas and are generally small and
within -0.2 and 0.2 for all months (Fig. S13). Some outliers reach values of -0.6 and +0.5, but
for such relatively small values of serial correlation the 1/n solution holds1.
1
Coumou, D., A. Robinson, and S. Rahmstorf (2013), Global increase in record-breaking monthly-mean
temperatures, Clim. Change, 118(3-4), 771–782, doi:10.1007/s10584-012-0668-1.
S2: Additional figures
Fig. S1 Location of the 11 391 observing weather stations used to create the HadEX2 data set which is
given on a 3.75° x 2.5° grid.
Fig. S2 Number of grid points with monthly maximum 1-day precipitation data for each point in time
given in (a) absolute numbers and (b) relative to the total number of grid points with data.
Fig. S3 Same as Fig. 1, but for the GHCNDEX data set.
Fig. S4 Same as Fig. 2, but for the GHCNDEX data set.
Fig. S5 Time series of the annual record-breaking anomaly calculated from HadEX2 (grey bars) and
GHCNDEX (pink bars) shown for (a) Global, (b) northern extratropics, (c) northern subtropics, (d)
tropics, and (e) southern subtropics. The long-term non-linear trend of the record-breaking anomaly
(solid line) is calculated using singular spectrum analysis with window length of 15 years. (f)-(j) and
(k)-(o) are the same as (a)-(e), respectively, but for seasonal record-breaking anomalies representing
NDJFM (middle panel) and MJJAS (right panel).
To ensure comparability between both data sets record-breaking anomalies were only calculated for
the period 1951-2010, where both data sets provide data. For each region and season, we computed the
Pearson correlation coefficient (  XY ) between the record-breaking anomaly time series of both data
sets, which is shown in the corresponding panels in Fig. S5. In general, results are in good agreement
between the two data sets indicated by high positive Pearson correlation coefficients implying that the
variables are positively linearly related. In the tropics, correlation coefficients are in a range of 0.070.20 indicating a positive but weaker linear relationship. This could be due to larger uncertainties due
to sparse data coverage in this region. A large and consistent increase in record-breaking anomaly can
be found in both data sets over the northern extratropics, northern subtropics, and on the global scale.
However, the increase is slightly stronger in GHCNDEX compared to HadEX2. Over the southern
subtropics both data sets show no trend in record-breaking anomaly.
Fig. S6 Schematic illustration of (Step 1) finding region specific time boundaries for the shuffling
process, (Step 2) computing the observed regional record-breaking anomaly, and (Step 3) computing a
set of modeled record-breaking anomalies based on the iid-model.
In the first step, for each month, the Rx1day data is organized in a p x n matrix where the number of
rows p equals the number of grid cells and thus denotes the location and the number of columns n
refers to the number of years. For each region a time period is defined for which this regions provides
data (see colored rectangles). The following steps are applied to each individual region in the limits of
these time boundaries.
To compute the observed record-breaking anomaly (“Step 2”), record-breaking events are counted in
each row, i.e. for each grid cell in the given region with a value of 1 denoting that this value has set a
new record and a value of 0 that this particular value was not a record-breaking event (see upper
matrix in middle panel of Step 2). We subsequently sum up all values of this matrix along the p grid
cells which leaves a vector of length n giving the total number of record-breaking events per year in
the given region. This vector is normalized with the number of expected record-breaking events (lower
matrix in middle panel) using eq. [1] to come up with a time series of the regional record-breaking
anomaly (right panel). The black dashed vertical lines in the middle panel of “Step 2” denote the time
period which fulfills the applied data requirements.
“Step 3” explains how the iid-model is computed. First, the n columns are randomly shuffled in which
process the order in time is lost, but the spatial correlation within the given region is kept. From the
shuffled matrix a time series of simulated regional record-breaking anomaly is computed in the same
way as described for the observational data. The resulting record-breaking anomaly refers to one
realization of the iid-model. The full procedure described in “Step 3” is repeated 10.000 times to
create a set of possible record-breaking anomalies under the Null hypothesis of the iid assumption.
From this set of time series the 90th and 95th confidence intervals are determined.
Fig. S7 Same as Fig. 2 in main manuscript but, here, confidence ranges are estimated using a shuffling
process which does not account for spatial correlation but therefore keeps missing values fixed in
space and time and thus conserves trends in the number of data points per year. This leads to generally
smaller confidence ranges compared to Fig. 2, where spatial correlation within each region is taken
into account at the expense of neglecting changes in the number of data points per year.
Fig. S8 Temporal heterogeneity in monthly data coverage, exemplarily shown for January. In the left
panel (a), for each grid point, years with data are colored corresponding to the region the grid point
belongs to. Only those values are colored for which the data requirements for the full year are fulfilled,
i.e. minimum time series length of 30 years and minimum 100 time series per year. For each region,
grid points are sorted by the start year of the given time series to illustrate temporal heterogeneity
within individual regions. For each year we sum up all grid points with data which results in a time
series with the total (global) number of available data as depicted by black circles in panel (b).
Randomly shuffling years of each time series in each region within its individual time boundaries
leads – on average – to a nearly equal distribution of data coverage in the given region. This is shown
by red circles in panel (b). This time series is characterized by steplike increases in years where new
regions start to supply data as indicated by the vertical dashed lines. The shuffling method is thus able
to reproduce a similar curve of changes in the amount of data over time.
Fig. S9 same as Fig. S8 but with data requirements applied to winter season.
Fig. S10 Same as Fig. S8 but shown for June and with data requirements applied to summer season.
Fig. S11 Same as Fig. 2 in main manuscript but, here, confidence ranges are estimated using a blockshuffling method with a fixed block size of 2 years.
Fig. S12 Same as Fig. S11 but with a fixed block size of 3 years.
Fig. S13 Serial correlation in the non-linear detrended HadEX2 Rx1day time series for each calendar
month.
Fig. S14 Time series of annual record-breaking anomaly shown for the global mean (black line).
Colored bars represent the ENSO time series (nino3.4 index) with positive values indicating El Niño
years (blue bars) and negative values corresponding to La Niña years (red bars).
S3: Additional tables
#
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
Region Label
Alaska
Australia
Central Africa
Central
America
CGI
Central North
America
Central West
Asia
Eeastern Asia
Eastern North
America
Europe
Mediterranean
Northern Asia
Northern
South America
Southern
Africa
Sahara
Southern Asia
Southern
South America
South East
Asia
Tibetan
Tableau
Western South
America
Western North
America
Northern
Extratropics
Northern
Subtropics
Tropics
Southern
Subtropics
lat1 lon1 lat2 lon2 lat3 lon3 lat4 lon4 lat5 lon5 lat6 lon6 lat7 lon7
60 105 60 168 73 169 73 105
-------50 110 -10 110 -10 155 -30 155 -30 180 -50 180 ---11 -20 15 -20 15 52 -11 52
------11
50
-68
-10
-1
50
-80
105
29
85
118
105
29
85
-90
-10
---
---
---
---
---
---
50
-85
29
-85
29
105
50
105
--
--
--
--
--
--
15
20
40
100
50
50
40
100
50
50
75
145
30
20
75
145
30
--
60
--
15
--
60
--
---
---
25
45
30
50
-60
-10
-10
40
25
75
45
70
-85
-10
-10
40
50
75
45
70
-85
40
40
180
50
45
30
50
-60
40
40
180
-----
-----
-----
-----
-----
-----
-20
-66
-1
-80
11
-69
11
-50
0
-50
0
-34
-20
-34
-35
15
5
-10
-20
60
-11
30
30
-10
-20
60
-11
30
30
52
40
100
-35
15
20
52
40
100
--20
--95
--5
--95
----
----
-20
-39
-57
-39
-57
-67
-50
-72
-20
-66
--
--
--
--
-10
95
20
95
20
155
-10
155
--
--
--
--
--
--
30
75
50
75
50
100
30
100
--
--
--
--
--
--
-1
-80
-20
-66
-50
-72
-57
-67
-57
-82
1
-82
--
--
29
105
29
130
60
130
60
105
--
--
--
--
--
--
40
180
90
180
90
180
40
180
--
--
--
--
--
--
20
-20
180
180
40
20
180
180
40
20
180
180
20
-20
180
180
---
---
---
---
---
---
-40
180
-20
180
-20
180
-40
180
--
--
--
--
--
--
Table S1 Coordinates of corners of regions displayed in Fig. 4 and Fig. 5. Values are given in degrees
North (for latitudes) and in degrees East (for longitudes). Regions 1-21 are used to compute the global
aggregate.