Can weather generation capture precipitation patterns across different climates, spatial scales and under data scarcity?

Abstract

Stochastic weather generators can generate very long time series of weather patterns, which are indispensable in earth sciences, ecology and climate research. Yet, both their potential and limitations remain largely unclear because past research has typically focused on eclectic case studies at small spatial scales in temperate climates. In addition, stochastic multi-site algorithms are usually not publicly available, making the reproducibility of results difficult. To overcome these limitations, we investigated the performance of the reduced-complexity multi-site precipitation generator TripleM across three different climatic regions in the United States. By resampling observations, we investigated for the first time the performance of a multi-site precipitation generator as a function of the extent of the gauge network and the network density. The definition of the role of the network density provides new insights into the applicability in data-poor contexts. The performance was assessed using nine different statistical metrics with main focus on the inter-annual variability of precipitation and the lengths of dry and wet spells. Among our study regions, our results indicate a more accurate performance in wet temperate climates compared to drier climates. Performance deficits are more marked at larger spatial scales due to the increasing heterogeneity of climatic conditions.

Full text

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Scientific RepoRts | 7: 5449 | DOI:10.1038/s41598-017-05822-y
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Can weather generation capture
precipitation patterns across
dierent climates, spatial scales
and under data scarcity?
Korbinian Breinl1, Giuliano Di Baldassarre1, Marc Girons Lopez
2, Michael Hagenlocher
3,
Giulia Vico
4 & Anna Rutgersson1
Stochastic weather generators can generate very long time series of weather patterns, which are
indispensable in earth sciences, ecology and climate research. Yet, both their potential and limitations
remain largely unclear because past research has typically focused on eclectic case studies at small
spatial scales in temperate climates. In addition, stochastic multi-site algorithms are usually not publicly
available, making the reproducibility of results dicult. To overcome these limitations, we investigated
the performance of the reduced-complexity multi-site precipitation generator TripleM across three
dierent climatic regions in the United States. By resampling observations, we investigated for the
rst time the performance of a multi-site precipitation generator as a function of the extent of the
gauge network and the network density. The denition of the role of the network density provides new
insights into the applicability in data-poor contexts. The performance was assessed using nine dierent
statistical metrics with main focus on the inter-annual variability of precipitation and the lengths of
dry and wet spells. Among our study regions, our results indicate a more accurate performance in wet
temperate climates compared to drier climates. Performance decits are more marked at larger spatial
scales due to the increasing heterogeneity of climatic conditions.
Precipitation is a key component of the water cycle, which in turn aects terrestrial ecosystems, agricultural pro-
duction and human well-being. Access to long precipitation time series is crucial for many ecological, agricultural
or hydrological studies, as well as for public health and climate research1, 2. Many regions lack such wealth of data,
so that realistic simulations of precipitation patterns are needed. Simulations have to preserve the spatial and
temporal dynamics as well as the correlation structures of precipitation patterns and their variability as they are
fundamental for impact analyses3.
Precipitations patterns can be simulated either with numerical weather prediction models or stochastic algo-
rithms. ese methods are complementary and have specic advantages and drawbacks. Numerical weather pre-
diction models include a physical description of the entire atmosphere and its interaction with the land surface,
oen also including oceans and vegetation, making the simulated elds physically consistent. is however leads
to high computational costs and potential limitations in both the number of simulations that can be generated
and their spatial resolution. Typically, the feasible spatial resolution is coarser than required for most impact
assessments. Moreover, the accuracy of precipitation elds produced by such models can suer from spatiotem-
poral and amplitude errors depending on the model physics, dynamics and model conguration4, 5.
Stochastic algorithms, in contrast, require considerably less computational eort and can therefore easily pro-
vide long time series. Multi-site stochastic precipitation generators are mathematical algorithms for producing
synthetic precipitation based on multiple ground observation sites (i.e. precipitation gauges). ey can simu-
late precipitation patterns in space and time similar to the actual observations. Several algorithms exist, oen
1Department of Earth Sciences, Uppsala University, Villavägen 16, 75236, Uppsala, Sweden. 2Department of
Geography, University of Zurich, Winterthurerstrasse 190, 8057, Zurich, Switzerland. 3Institute for Environment and
Human Security, United Nations University (UNU-EHS), UN Campus, Platz der Vereinten Nationen 1, 53113, Bonn,
Germany. 4Department of Crop Production Ecology, Swedish University of Agricultural Sciences, Ulls väg 16, 75007,
Uppsala, Sweden. Correspondence and requests for materials should be addressed to K.B. (email: korbinian.breinl@
geo.uu.se)
Received: 15 March 2017
Accepted: 20 June 2017
Published: xx xx xxxx
OPEN

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embedded in weather generators for various climate variables. Stochastic precipitation generators can be used for
downscaling of numerical weather models and for climate projections6–15, ood and drought assessments16–21,
agricultural studies22–24, food security25, 26, as well as public27–29 and veterinary health30. e main drawback of
statistical methods is that, while spatial and temporal correlation structures are kept, unlike numerical weather
prediction, they cannot simulate the associated large-scale dynamics leading to temperature and precipitation
variabilities.
Despite this limitation, there is undoubtedly potential for more extensive application of multi-site precipi-
tation generators. Yet surprisingly little knowledge is available regarding their application across spatial scales,
in dierent climates and under conditions of data scarcity. So far, stochastic multi-site precipitation generators
have been primarily applied at small spatial scales (not exceeding some tens of kilometers), with only a handful
of sites10, 13, 31–37. Only very few authors have focused on larger spatial scales38–41. e majority of these studies has
been carried out in temperate and precipitation-rich climates in developed countries, where dense observation
networks and long time series of reliable climate data are the norm8, 9, 19, 20, 22, 24, 39, 42, 43. Furthermore, the wide-
spread application of precipitation generators has been limited by the lack of publicly available transparent source
codes and the mathematical complexity of many models, so that setting up a model still requires major eorts.
e complexity of algorithms has been recently identied as an issue by Apel et al.44 and the fragmented body of
knowledge has been critically reviewed by Ailliot et al.45.
Towards an easier and more widespread use of stochastic multi-site precipitation generation, here we rst
assess the performance of multi-site precipitation generation across three dierent climatic zones and across
spatial scales in the United States, from about thirty kilometers to over one thousand kilometers of maximum
extent. Second, we link the density of the observation network to the performance of the precipitation generation,
to provide new insights into model performance under conditions of data scarcity and thus into the applicability
in data-poor regions, such as in emerging economies and developing countries.
Addressing these multiple aspects requires the generation of very large data amounts that go far beyond what
has yet been presented: for our study we generated almost 1.5 million years of synthetic precipitation. For this
reason, we use the latest version of the very fast reduced-complexity stochastic multi-site precipitation generator
TripleM (Multisite Markov Model), which requires only two key parameters for simulating any gauge network
in its simplest setup. Other algorithms require a very large number of parameters that grow exponentially with
the number of gauges42, making comprehensive studies not feasible. To our knowledge, TripleM is the most
straightforward multi-site precipitation generator currently available and thus probably one of very few models
that allows for comprehensive studies.
Data and Experiments
Station-based climate observations. In order to fulll the objectives of the study, a homogeneous data-
set covering dierent climatic zones and providing a suciently dense observation network is needed. For these
reasons, we use the dataset of daily precipitation observations available for the United States from the Global
Historical Climatology Network - Daily (GHCN-Daily)46 for the 30-year period 1986–2015, which has been
compiled by the National Climatic Data Center (NCDC) (https://www.ncdc.noaa.gov/oa/climate/ghcn-daily/).
From this dataset we selected three study areas representing dierent climatic conditions, located in the North-
East (NE), South-East (SE) and West (W) of the United States (Fig.1).
e NE is dominated by a relatively cold climate without any dry season, with evenly distributed monthly
precipitation and warmer summers. e SE is dominated by a temperate and tropical monsoon climate without
any pronounced dry season and moist, hot summers. e W is dominated by an arid and semi-arid climate with
frequent droughts47. It has a marked seasonality in precipitation and includes a temperature gradient with warmer
(South) and colder regions (North). While the inter-annual variability of precipitation is more evenly distributed
over the year in the NE and SE, it has a pronounced annual cycle in the W, reaching comparatively high values in
the winter months. e lengths of dry and wet spells show similar annual cycles in the NE and SE with dry spells
peaking in winter/spring and in the fall. Dry spells are longest in the summer in the W. For the period 1986–2015,
72 precipitation gauges with complete time series are available for the NE, 111 for the SE and 98 for the W.
Design of the experiments. To evaluate model performance under diering conditions of data availability/
scarcity, we investigated four dierent levels of gauge network densities (Table1).
e density scenarios are based on actual precipitation gauge network densities of the GHCN-Daily dataset
(1986–2015) in two of the three study areas in the United States (“very high”), the average density over Europe
(“high”) as well as China (“medium”) as an orientation for emerging economies, and the average density on the
African continent (“low”) as an orientation for developing countries (see FigureS1 in the Supplementary mate-
rial). e distribution of precipitation gauges in each scenario was conducted subjectively, aiming for equally
spatially distributed networks. As each density scenario required a comparable network density for each study
area, a high-density scenario could not be examined for the W.
For each density scenario, we conducted four separate experiments, each starting at one of the four so-called
‘starting sites’ (located in four dierent regions of each study area; see red gauges in Fig.1). e four starting sites
(i.e. four experiments starting in dierent regions of the study area) were introduced to capture the obviously not
fully homogenous climate of each study area. Each experiment began by considering a minimum precipitation
gauge network of three sites (i.e. the starting site and its two closest sites), continuously widening up the pre-
cipitation gauge network by adding the next closest precipitation gauge up to the maximum number of gauges
available for each density scenario. For each precipitation gauge network, we simulated 30 dierent ensembles to
obtain stable results, each time over the 30-year period (i.e. 900 years). In other words, for each density scenario
and starting site in each study area, we performed 30 runs for a network of three gauges, 30 runs for four gauges
and so on, up to 30 runs for all available gauges. For example, the total number of simulated years in the NE for

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the density scenario “very high” is 4 (experiments using the four starting sites) × 70 (dierent precipitation gauge
networks between three and 72 sites) × 30 (dierent ensembles) × 30 (observation years) = 252,000 years. e
combination of all three study areas (i.e. climates), precipitation gauge network sizes (i.e. spatial scale) and num-
ber of sites (i.e. network density) led to a total of 1,461,600 generated precipitation years.
We used the semi-parametric multi-site precipitation generator TripleM21, 48, which we optimized for large
gauge networks to perform the experiments (see Methods). We simulated daily precipitation amounts by a pure
resampling of the observations (bootstrap) to eliminate uncertainties arising from parametric precipitation sam-
pling. A detailed description of the TripleM algorithm is available in the Methods.
Results and Discussion
We focus on four key metrics relevant for climate change and climate change impacts studies, namely: (i) the
inter-annual standard deviation of precipitation, (ii) the average maximum length of dry spells (dry periods), (iii)
the mean length of dry spells, and (iv) the average maximum length of wet spells (wet periods). e intra-annual
distribution of precipitation and inter-annual variability in precipitation amounts are key drivers of the function-
ing of terrestrial ecosystems, and hence local carbon balance, agricultural production, natural hazards such as
oods and droughts, and have both direct and indirect impacts on human health and well-being49–51. Inter-annual
variations in precipitation and temperature explain on average a third of the global crop yield variability50. Mean
dry spells represent continuing water stress of plants52, while maximum dry spells (ii) are of relevancy for drought
Figure 1. e three study areas in the North-East (NE), South-East (SE) and West (W) of the United States,
including the location of all precipitation gauges available for the period 1986–2015 (grey dots), the starting
sites of the experiments (see section ‘Design of the experiments’ below), plots of the mean precipitation, annual
standard deviation of precipitation, mean length of dry and wet spells, averaged over all gauges for each month.
e bar/line plots also contain information on the mean annual precipitation (MAP). e map was generated in
ArcGIS 10.2 (http://www.esri.com/), related bar/line plots in MATLAB 2016a (http://www.mathworks.com/).
Study area/number
of gauges very high
(5,200 km²/gauge) high (11,400 km²/
gauge) medium
(48,000 km²/gauge) low (94,400 km²/
gauge) Maximum gauge
network extent (km)
NE 72 32 8 4 1,173
SE 111 52 12 6 1,161
Wnot available 98 23 12 1,167
Table 1. Number of gauges for the three study areas, the four simulated precipitation gauge density scenarios
referred to as ‘very high’, ‘high’, ‘medium’ and ‘low’, and the maximum extent of the networks in each scenario.

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studies. Maximum wet spells (iv) inuence oods and, depending on climatic conditions, have a direct impact
on the prevalence of water-related vector-borne diseases, such as Chikungunya53 or ri valley fever54, 55, but also
on agriculture56. We assessed the performance of the precipitation generator with focus on the average annual
performance and on the summer (Jun, Jul, Aug) and winter (Dec, Jan, Feb) seasons separately. e precipitation
generator performance was characterized as the relative error between the mean of the 30 simulations for all sites
of each precipitation gauge network and the observations. Since we conducted four experiments with four start-
ing sites in each study area, in the gures below we show the mean of these four simulations.
For a more in-depth assessment, we examined ve additional standard hydrological metrics: (i) the simulated
mean precipitation, (ii) the daily standard deviation of precipitation, (iii) the mean length of wet spells, (iv) the
lag1 autocorrelation of precipitation occurrence as well as (v) the cross-correlation of precipitation occurrence
lagged by one day as a proxy for the persistence of weather situations. e results for these metrics are reported
in the Supplementary material.
Climate and spatial scale. We present the impact of the spatial scale (Figs2 and 3) for the high density
scenario (see Table1) for the three climates. e performance generally decreases with increasing gauge network
size. is is expected as in TripleM daily snapshots of precipitation occurrences are rst clustered according
to their similarity and then simulated based on a univariate Markov process. A larger extent means larger, less
homogenous precipitation snapshots and lower performance. For the four metrics, increasing the network size
Figure 2. Relative error for all sites in the North-East (blue), the South-East (green) and the West (magenta) for
all months. e error is plotted for four metrics against the maximum extent of each simulated gauge network.
For each study area, the lines show the mean of the four simulations (using four dierent starting sites; see
Fig.1).

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increases the mean error on average by 3.7% in the NE (from 0.0% with three sites), 6.2% in the SE (from −1.0%
with three sites) and 4.3% in the W (from −5.7% with three sites).
For the annual performance (Fig.2), the inter-annual standard deviation tends to be underestimated. is is a
typical phenomenon of daily weather generators, referred to as overdispersion. e underestimation is generally
low for the NE and the SE and higher for the W. On average, the underestimation reaches a maximum of −6.4% in
the NE (starting at −3.0% for the smallest network size) and −6.2% (starting at −4.3%) in the SE, with a slightly
decreasing performance towards larger gauge networks. e underestimation in the W increases from −16.7%
for three gauges to −19.6% for all sites. Daily weather generators rely on daily weather scenarios and have thus a
limited capability for reproducing the inter-annual variability. e underestimation is predominately caused by
the resampling approach. e bootstrap only takes into account observations and cannot generate very extreme
events, which underestimates the sampling distribution, especially for small sample sizes. e latter explains
the higher overdispersion in the W where precipitation events are rare. Attempts have been made to overcome
this shortcoming57–59. For example, overdispersion could be further reduced also in TripleM-type models by
introducing parametric precipitation sampling with heavy tailed distributions as suggested by Wilks39. However,
tting of parametric precipitation curves in dry areas may be infeasible due to the limited number of precipitation
observations. Seasonal dierences are shown in Fig.3. In the NE and SE, the variability is more underestimated
in the summer. e observed annual standard deviation averaged over the entire precipitation gauge network is
43.8% higher in summer than in winter in the NE and 30.1% higher in the SE. In the W, it is 8.7 times higher in
Figure 3. Relative error for all sites in the North-East (blue), the South-East (green) and the West (magenta)
for the summer (solid lines) and winter season (dashed lines). e error is plotted for four metrics against the
maximum extent of each simulated gauge network. For each study area, the lines show the mean of the four
simulations (using four dierent starting sites; see Fig.1).

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winter than in summer due to the predominantly arid summer. It is an inherent property of the bootstrap that
the underestimation decreases exponentially with an increasing variability of the observations. is explains
why seasons with a relatively high inter-annual variability are more strongly underestimated than seasons with a
relatively low variability.
e length of maximum dry spells is slightly overestimated in the NE with 2.1% for three gauges and under-
estimated for larger extents, reaching −1.6% at full extent (Fig.2). e trend is similar for the SE and W, starting
with an overestimation of 1.3% and underestimation of −3.5% respectively and reaching −4.3% and −5.2% at
full extent. Mean dry spells are likewise least underestimated in the NE (−0.3% to −4.3%). e underestimation
in the SE and W starts with −1.4% and −3.2%, reaching −9.8% and −7.9% at full extent. e bias for simulating
maximum wet spells is smallest in the NE (1.2% to −2.5%). Maximum wet spells are less well reproduced in the
SE and W and follow similar trends (0.5% and 0.6% to −8.3% and −7.4%).
In the NE and the SE maximum dry spells are better reproduced in winter compared to summer (Fig.3).
is is related to the persistence of weather events, expressed by the lagged cross-correlation of the precipitation
occurrences, which is 6.4% higher in winter in the NE and 13.8% higher in the SE compared to summer. e clus-
tering approach performs better when precipitation events are predominantly of frontal nature. e convective
systems that are common in summer are more variable, with smaller scales in time and space, thus leading to
more distinctive precipitation patterns and reducing the clustering performance. e performance for mean dry
spells is similar with almost equal performance in summer and winter in the NE.
Performance dierences between the NE and the SE are related to the strong impact of convective systems in
the SE, particularly in summer: Florida is the state in the United States with the highest thunderstorm activity60, 61.
Precipitation contribution of tropical cyclones to the seasonal precipitation totals can reach up to 20% in the
coastal regions, with comparatively high inter-annual variabilities depending on whether a year has hurricane
observations or not62. According to the International Best Track Archive for Climate Stewardship IBTrACS63
(release version v03r09), the South-East study area as presented in this research has been hit by 23 named and
three unnamed tropical cyclones between 1986 and 2015 in the summer season. Conversely, precipitation in the
NE is predominately of frontal nature. According to a study by Hawcro et al.64 using two dierent reanalysis
datasets the contribution of extratropical cyclones to the total precipitation in the NE study area reaches over
80% in the winter season and over 65% in the summer season with uncertainties of up to about 20% depending
on the reanalysis dataset under investigation. In the W, the precipitation climatology is much more complex,
with a pronounced spatial heterogeneity of precipitation with a large impact of smaller-scale climatic controls in
the mountainous areas65. e region is also strongly inuenced by the El Niño–Southern Oscillation (ENSO). In
the Great Basin, which covers most of the Western study area except for California, above normal precipitation
between October and March is predominately associated to ENSO years66. e inter-annual variability is also
linked to ENSO67. e mountain ranges of the Sierra Nevada in California receive high precipitation amounts due
to orographic eects, which also explain the dry conditions in the Great Basin because of a rain shadow eect. e
lagged cross-correlation of observed precipitation occurrences (i.e. weather persistence) is three times higher in
winter than summer, due to the dominant inuence of midlatitudinal synoptic-scale storms68, 69. e still better
performance for dry spells in the W in summer is related to the arid summer (recorded precipitation on only
4.3% of all days), making a pronounced underestimation of dry spells unlikely. e performance for maximum
wet spells in the NE and in the SE is similar to the performance in regard to dry spells. e performance is better
during winter with higher persistence of weather events. Maximum wet spells are equally reproduced in both
seasons in the W. e 90% condence intervals (see Supplementary material) show similar spreads across seasons
and study areas. e most signicant dierences are related to the W: For summer, condence intervals are signif-
icantly wider for the majority of metrics, which is related to the low number of precipitation days. e results for
the medium and low gauge density scenarios (Table1, not shown here) showed comparable results.
Network density and spatial scale. e gauge network density impacts the performance. Here, for all
available density scenarios (Table1), we focus on the annual performance only (Fig.4), but seasonal perfor-
mances are comparable.
Deviations between network densities can be encountered. For the majority of the metrics, the model bias
decreases with a reduced network density, with dierences between about one to ve percent, depending on
the study area and maximum extent. However, low density does not always mean better performance, primar-
ily for the inter-annual standard deviation of precipitation in the SE and W. To reach the same xed duplication
rate of observations, fewer clusters of daily precipitation snapshots are required for small networks. us, the
clusters represent weather situations less well, which eectively reduces the model performance. e opposite
applies to dry and wet spells where the bias decreases with a reduced network density. e most pronounced
dierences can be recognized for maximum and mean dry spells in the SE, and maximum wet spells in the SE
and in the W. e phenomenon is likewise caused by the clustering algorithm. A smaller number of gauges
leads to a better distinction between the clustered daily precipitation snapshots and therefore higher similarity
within these clusters, which improves the performance. Deviations are higher in the less homogenous climates
of the SE and W. e slightly better performance may give the impression that a lower dense gauge network
may likewise be preferable, but (i) dierences in the performance do not exceed dierences of one to ve per-
cent and (ii) most applications require the interpolation of the simulated precipitation patterns, where a high
number of stations is desirable.
Conclusion
is study is a rst step towards overcoming the fragmented, eclectic knowledge in stochastic generation of pre-
cipitation patterns and is thereby a call for testing multiple, and possibly publicly available, model codes across
dierent climate types, spatial scales and network densities. e comparison of 30-year long observed daily

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precipitation patterns with generated precipitation across three dierent climates shows a general adequate agree-
ment when considering relatively small regions, although key metrics such as dry or wet spells are oen underes-
timated. Larger spatial scales lead to reduced performance in reproducing the observations. e simulations are
Figure 4. Relative error for all sites in the North-East (NE), the South-East (SE) and the West (W) for all
months. e error is plotted for four metrics against the maximum extent of each simulated gauge network and
density scenario. For each study area and scenario, the lines show the mean of the four simulations (using four
dierent starting sites; see Fig.1). e solid line represents the results for the very-high gauge network density
(not available for the West), the dashed line for the high-density, the dash-dot line for the medium density and
the dotted line for the low gauge density scenario.

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less biased in wet temperate climates than in dry climates. Seasons and locations that are dominated by frontal
precipitation are better reproduced than seasons with a more pronounced impact of convective systems. is
explains the dierent performance obtained in the temperate North-East and subtropical South-East. Seasons
with a higher inter-annual variability of precipitation are less well reproduced, as demonstrated with the Western
study area, which is inuenced by ENSO.
In this research, we focused on the current climate. ere are dierent approaches to parameterize precipita-
tion generators to simulate climate change, for example by altering the precipitation values using output from cli-
mate models as for example suggested by Turkington et al.13. However, as the pure alteration of the precipitation
amounts ignores potential future changes in dry and wet spells, another promising avenue could be to condition
the clustering of the daily precipitation snapshots in the TripleM model to the distribution of current and future
circulation patterns to incorporate changes of dry spells, wet spells and also in the autocorrelation of precipita-
tion. Simulating climate change with weather generators however has inherent limitations in regard to decadal
variabilities and long-term trends. e consideration of other climate types beyond the three of this study would
be another interesting topic for investigation.
e development of common evaluation standards as for instance information on relative errors for better
comparability is highly desirable. Additional comparative studies particularly in countries with lower network
densities (FigureS1, Supplementary material) would be useful to validate the ndings of this study. Further, the
proposed methodology should be complemented to enable simulating projected precipitation patterns that can
be used for climate change impact studies, ideally in the developing world, where impacts of climate change are
oen most signicant. At this point in time, facing numerous published types of algorithms, eclectic case studies,
a very limited number of transparent publicly available source codes and a lack of common evaluation standards,
the full potential of stochastic multi-site weather generation remains unclear. e issue magnies when dierent
model types are parameterized for simulating future climate scenarios. We made a rst step towards closing this
gap by demonstrating that – if there is awareness and knowledge of stochastic approaches and model type specic
opportunities and shortcomings – stochastic multi-site precipitation generation has the potential to support a
variety of societally and ecologically relevant issues in dierent climates, at dierent spatial scales and under
diering conditions of data availability.
Methods
e reduced complexity multi-site precipitation generator TripleM (Multisite Markov Model) applied here works
as follows: First, daily snapshots of the precipitation occurrences (i.e. catchment-wide precipitation patterns) are
clustered according to their similarity. e model uses the non-hierarchical k-means clustering method70, 71 and
the hamming distance (equation (1)).
∑
=≠
=
distance xy pIx y(, )
1
{},
(1)
j
p
jj
1
where I is the indicator function.
In the original version of TripleM48, the k-means clustering was applied to daily snapshots of precipitation
amounts that were rst standardized using the z-score transformation in order to take into account the het-
eroscedastic nature of the precipitation. is led to a satisfying performance in a comparatively small Alpine
precipitation gauge network not exceeding a maximum distance between sites of about 150 km. For this research,
we ran multiple experiments with dierent clustering methods and it turned out that the performance increases
signicantly for large gauge networks when applying the hamming distance to binary precipitation occurrences.
Second, the clustered occurrence vectors are simulated with a Markov process (equation (2)), where the tran-
sition probabilities depend on m previous days, i.e.,
…= …<−
+−−+−−
XXXX XXXX XmtPR{,,,,} PR{,,, }with1 (2)
tttt ttttm1121 11
Once the synthetic time series of clusters are simulated, each cluster is replaced by a random amount vector
(i.e. daily snapshot of precipitation amounts) belonging to the same cluster. In a last step, which is optional, the
model introduces sampling of parametric precipitation amounts in combination with an adapted version of a
resampling approach by Clark et al.41, to account for unobserved precipitation extremes. e method is shown
in Fig.5, using a hypothetical example of three sites and a ten states Markov chain: Aer generating synthetic
time series of clusters using the Markov process (a), amount vectors are randomly drawn from all observations
that t the corresponding cluster (b). Following this, synthetic precipitation amounts are sampled independently
for each site from parametric curves (c) optionally using correlated uniform random numbers from a Cholesky
decomposition72. e use of correlated random numbers avoids the generation of signicantly dierent precipi-
tation amounts across sites, which becomes increasingly important when generating short synthetic time series.
In the last step (d), the parametric precipitation amounts are reshued according to the original ranks aer the
resampling in step (b), to maintain the inter-site correlations.
e entire simulation process can be depicted from Fig.6.
In its most simplistic setup (resampling i.e. bootstrap without parametric sampling of precipitation amounts as
applied in this study), TripleM has two key parameters the user has to dene: the duplication rate and the order of
the Markov chain. As for the duplication rate, an inherent characteristic of TripleM is that the clustering approach
will duplicate parts of the time series: A higher number of clusters will generally improve the reproduction of
various metrics of the observations such as the precipitation autocorrelation, but result in duplicated observa-
tions in the simulations, especially in large station networks. Here and elsewhere48, a maximum duplication rate
of only 1% produced satisfying results. Higher duplications rates increase the computational costs. e second

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Scientific RepoRts | 7: 5449 | DOI:10.1038/s41598-017-05822-y
key parameter is the order of the Markov chain used. In this study, a one-order Markov chain was used. For larger
observation networks, it is recommended not to increase the order due to the exponentially growing state-space
related to higher orders.
Another model specic characteristic is the reproduction of inter-site correlations. If long synthetic time series
are generated in combination with parametric precipitation sampling, inter-site correlations are better repro-
duced. is is caused by the reshuing method. With long synthetic time series, the pool of parametric precip-
itation amounts becomes more similar to the resampled precipitation amounts. In TripleM, the reshuing is
conducted over all generated years separately for all months or seasons depending on the chosen model setup.
e choice of parametric models for the synthetic precipitation amounts is another inuencing factor in general,
Figure 5. Key steps of precipitation generation in TripleM aer clustering of the daily precipitation snapshots
and Markov simulation (a), including resampling of amount vectors (b), parametric sampling (c) and
reshuing (d).
Figure 6. Schematic ow diagram of the TripleM precipitation generator. TripleM can be used as a bootstrap
model (Output 1) and a parametric precipitation model (Output 2). Parallelograms represent time series or
variables, boxes represent methods. Blue parallelograms represent input and output data. Cholesky matrices and
transition matrices are either derived monthly (12) or seasonally (4). e parametric distribution parameters
are either derived monthly (12) or seasonally (4) for the number of gauges simulated (n).

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Scientific RepoRts | 7: 5449 | DOI:10.1038/s41598-017-05822-y
which has been discussed in the past36, 73, 74 and is not specic to TripleM. e MATLAB code oers the Gamma
distribution, the Weibull distribution or a compound distribution of the Weibull distribution for lower and a
Generalized Pareto distribution for higher and extreme precipitation amounts with a user-dened threshold
between both curves.
TripleM oers monthly and seasonal setups. All steps, including the clustering of amount vectors, the tting of
the Markov chains, the simulation of the Markov process and the reshuing of parametric precipitation amounts,
can either be run monthly or seasonally. In this study we used a monthly setup.
Code and data availability. e MATLAB source code of TripleM, a user manual and a training dataset
are available from the github page, https://github.com/KBreinl/TripleM. e data used in this paper are available
from the NOAA websites.
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Acknowledgements
This research has been funded by the project STEEP STREAMS funded by the Swedish Research Council
FORMAS within WaterJPI, ERA-Net Cofund WaterWorks 2014. e data used in this paper are available from
the NOAA websites.
Author Contributions
K.B. and G.D.B. conceived the research; K.B. prepared the data, improved the MATLAB code for modelling large
gauge networks and ran the experiments; K.B. and G.D.B. designed the experiments with major contributions
by A.R., G.V. and M.H.; M.G.L. revised the MATLAB code for high performance; all authors contributed to the
interpretation and writing of the manuscript with major contributions by M.H.
Additional Information
Supplementary information accompanies this paper at doi:10.1038/s41598-017-05822-y
Competing Interests: e authors declare that they have no competing interests.
Publisher's note: Springer Nature remains neutral with regard to jurisdictional claims in published maps and
institutional aliations.

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