Space–Time Characteristics of Areal Reduction Factors and Rainfall Processes

Abstract

We estimate areal reduction factors (ARFs, the ratio of catchment rainfall and point rainfall) varying in space and time using a fixed-area method for Austria and link them to the dominating rainfall processes in the region. We particularly focus on two sub-regions in the West and East of the country, where stratiform and convective rainfall processes dominate, respectively. ARFs are estimated using a rainfall dataset of 306 rain gauges with hourly resolution for five durations between 1 hour and 1 day. Results indicate that the ARFs decay faster with area in regions of increased convective activity than in regions dominated by stratiform processes. Low ARF values occur where and when lightening activity (as a proxy for convective activity) is high, but some areas with reduced lightning activity exhibit also rather low ARFs as, in summer, convective rainfall can occur in any part of the country. ARFs tend to decrease with increasing return period, possibly because the contribution of convective rainfall is higher. The results of this study are consistent with similar studies in humid climates, and provide new insights regarding the relationship of ARFs and dominating rainfall processes.

Full text

Space–Time Characteristics of Areal Reduction Factors and Rainfall Processes
KORBINIAN BREINL
a
Institute of Hydraulic Engineering and Water Resources Management, Vienna University of Technology, Vienna, Austria,
and Centre of Natural Hazards and Disaster Science, Uppsala, Sweden
HANNES MÜLLER-THOMY
b
AND GÜNTER BLÖSCHL
c
Institute of Hydraulic Engineering and Water Resources Management, Vienna University of Technology, Vienna, Austria
(Manuscript received 24 September 2019, in final form 12 February 2020)
ABSTRACT
We estimate areal reduction factors (ARFs; the ratio of catchment rainfall and point rainfall) varying in
space and time using a fixed-area method for Austria and link them to the dominating rainfall processes in the
region. We particularly focus on two subregions in the west and east of the country, where stratiform and
convective rainfall processes dominate, respectively. ARFs are estimated using a rainfall dataset of 306 rain
gauges with hourly resolution for five durations between 1 h and 1 day. Results indicate that the ARFs decay
faster with area in regions of increased convective activity than in regions dominated by stratiform processes.
Low ARF values occur where and when lightning activity (as a proxy for convective activity) is high, but some
areas with reduced lightning activity exhibit also rather low ARFs as, in summer, convective rainfall can occur
in any part of the country. ARFs tend to decrease with increasing return period, possibly because the con-
tribution of convective rainfall is higher. The results of this study are consistent with similar studies in humid
climates and provide new insights regarding the relationship of ARFs and dominating rainfall processes.
1. Introduction
Various applications in hydrology require an under-
standing of the spatial and temporal behavior of extreme
rainfall over a catchment as it impacts the runoff behav-
ior and its scaling characteristics (Allen and DeGaetano
2005a). Research on this topic refers to problem number
6 ‘‘What are the hydrologic laws at the catchment scale
and how do they change with scale?’’ of the 23 unsolved
problems in hydrology (Blöschl et al. 2019). In engi-
neering practice, point rainfall intensity is only appli-
cable to very small catchments, as already pointed out
by Marston (1924) in the early twentieth century. For
this reason, areal reduction factors (ARFs) are applied
to transform point rainfall into average areal rainfall.
The ARF is defined as the ratio between the areal
rainfall and the point rainfall, usually using the annual
maximum rainfall depths over a given time interval of a
couple of hours. ARFs are typically used to generate
input for rainfall–runoff modeling with areal design
rainfall of a certain return period on an event basis
(Müller and Haberlandt 2018). ARFs are typically
presented as so-called ARF curves that represent the
relationship between ARFs and catchment area. The
estimates of ARFs are influenced by 1) the rainfall
processes, 2) the magnitude of the events as charac-
terized by the return period, 3) any biases in the rainfall
data used, and 4) the estimation method.
1) Various authors describe a relationship between
the ARF and different rainfall processes. According to
Skaugen (1997), ARFs of spatially small-scale rain-
fall events in southern Norway recorded at daily
resolution decay more rapidly with increasing area
compared to large-scale rainfall events. By analyzing
rain gauge data in Illinois (United States) at high
temporal resolution, Huff and Shipp (1969) revealed
Denotes content that is immediately available upon publica-
tion as open access.
a
ORCID: 0000-0003-0489-4526.
b
ORCID: 0000-0001-5214-8945.
c
ORCID: 0000-0003-2227-8225.
Corresponding author: Korbinian Breinl, breinl@hydro.tuwien.
ac.at
APRIL 2020 B R E I N L E T A L . 671
DOI: 10.1175/JHM-D-19-0228.1
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that correlation with distance decayed quickly for
thunderstorms and rain showers, whereas the decay
was lower for steady rain and passing low pressure
centers. Similar results were obtained by Wright
et al. (2014).Allen and DeGaetano (2005a) found
smallerARFsinsummerthaninwinter,whichthey
attributed to a higher frequency of convective
events. Similar findings were reported by Huff and
Shipp (1969). Various authors reported a depen-
dency between the ARFs and the geographical lo-
cation, which is similarly related to the climate and
the dominating local rainfall processes (Asquith
and Famiglietti 2000;Omolayo 1993;Skaugen 1997;
Zehr and Myers 1984). For short durations, ARFs
tend to decay faster with area than for long dura-
tions (Mineo et al. 2018;NERC 1975;Ramos et al.
2005), usually due to the predominantly convective
nature and small spatial extent of short duration
events (Sivapalan and Blöschl 1998). Research also
suggests a potential difference in ARFs within ur-
banized areas and in the countryside as convective
processes may be amplified in major metropolitan
regions (Huff 1995).
2) The findings on the relationship between ARFs
and the rainfall return period are mixed: Skaugen
(1997),Sivapalan and Blöschl (1998),Asquith and
Famiglietti (2000),Allen and DeGaetano (2005a),
Mailhot et al. (2012),andLe et al. (2018) reported
that the ARF decreases with increasing return
periods, usually due to a higher contribution of
convective activity. These results are in contrast
with studies in Switzerland by Grebner and Roesch
(1997), who did not find a relationship between
ARFs and the return period for areas greater than
500 km
2
. There were variations for smaller areas,
though, which the authors explained with the low
density of the rain gauge network being unable to
capture convective events, and the relatively short
length of the observation records. Wright et al.
(2014) did not find a significant relationship either.
3) In terms of the rainfall data, the period of the
rainfall time series used can influence estimated
ARFs due to the temporal variability of rainfall
(Asquith and Famiglietti 2000;Svensson and Jones
2010). Also, the combination of different rain gauge
networks to reach an acceptable spatial coverage
canleadtobiasoftheARFvalues(Asquith and
Famiglietti 2000). The station density and the in-
terpolation techniques have however little influ-
ence on ARFs according to Allen and DeGaetano
(2005a).Allen and DeGaetano (2005a) state that effects
of mountains on rainfall can theoretically affect ARFs.
Interpolation methods such as Thiessen polygons
do not usually account for the fact that rain gauge
networks are sparser at higher altitudes. Considering
this effect in spatial interpolation techniques did
however not significantly impact the ARFs in the
study of Allen and DeGaetano (2005a).
4) Various methods have been proposed to estimate
ARFs. Svensson and Jones (2010) classify the dif-
ferent methods into (i) general empirical methods,
(ii) specific empirical methods, (iii) spatial correla-
tion structure, (iv) crossing properties, (v) scaling
relationships, (vi) storm movement, and (vii) radar
data. General empirical methods include fixed-area
and storm-centered approaches. As for the first,
the areal rainfall from a fixed area and for a spe-
cific return period is divided by the point rainfall of
the same return period. Storm-centered approaches
are similar but with the differences that the area
changes with each storm, that the point rainfall is
estimated from the highest value of the storm, and
that point and areal rainfall are estimated from the
same storm. The method of the U.S. Weather Bureau
(1957,1958) is similar to the fixed-area method with
the difference that the ratio between areal and point
rainfall is not based on the same return period but the
mean of areal and point annual maximum time series.
NERC (1975) suggest a simplification of the U.S.
Weather Bureau (1957,1958) method, which like-
wise ignores return periods. Bell (1976) proposes
ARFs based on the ratio between annual maximum
areal rainfall from Thiessen polygon interpolation
and annual maximum point rainfall, thereby con-
sidering return periods. Methods based on correla-
tions include the one by Rodriguez-Iturbe and Mejía
(1974) who related the ARF to a ‘‘characteristic
correlation distance’’ between station pairs, thereby
assuming Gaussian point rainfall and a specific cor-
relation structure. Sivapalan and Blöschl (1998) built
on the method of Rodriguez-Iturbe and Mejía (1974)
but additionally considered the transition from the
population of events to extreme values, and thus the
return period. Bacchi and Ranzi (1996) proposed a
stochastic derivation of ARFs based on crossing
properties of the rainfall process aggregated in space
in time. The method is suitable for small areas
and short durations (Svensson and Jones 2010). De
Michele et al. (2001) and Veneziano and Langousis
(2005) estimated ARFs based on the scale-invariant
behavior of rainfall with a possibility to take return
periods into account. Bengtsson and Niemczynowicz
(1986) proposed a method using the movement of con-
vective storms. Various authors applied different types
of methods using radar data (Allen and DeGaetano
2005b;Durrans et al. 2002;Lombardo et al. 2006;
672 JOURNAL OF HYDROMETEOROLOGY VOLUME 21

Olivera et al. 2008). More details on various methods
can be found in the comprehensive review by Svensson
and Jones (2010). The large number of approaches for
deriving ARFs and the large number of eclectic case
studies make it difficult to critically examine each
method and come up with general recommenda-
tions about their applicability. It is therefore of
interest to connect ARFs with the predominant
hydrometeorology of the region of interest. Not
only will an understanding of the hydrometeorology
help assess the plausibility of the ARFs estimated,
but also the ARFs will contribute to a better un-
derstanding of the hydrometeorology as they are a
fingerprint of the spatial statistical behavior of ex-
treme precipitation. Additionally, they can help in
the testing of spatial statistical models of rainfall
(e.g., Müller and Haberlandt 2018).
The aim of this paper is to link dominating rainfall
processes to ARFs over all of Austria, by analyzing their
space–time distribution for different rainfall durations.
Our study includes, but is not limited to, the mapping of
ARFs in space. To support this goal, we use countrywide
lightning data as a proxy for convective activity, and
particularly focus on two regions of Austria dominated
by stratiform and convective rainfall processes. To the
best of our knowledge, (i) countrywide analyses of
ARFs have not yet involved a mapping of the ARFs for
improved understanding of the link between ARFs and
rainfall processes and (ii) not yet examined the potential
of using regional lightning data in such space–time in-
vestigations. Our main hypothesis is that the different
rainfall processes should be reflected in both the dif-
ferences in the intensity–duration–frequency (IDF)
curves and the ARFs in space and time. In other words,
we expected the spatial distribution of ARFs to follow
similar spatial patterns as the distribution of lightning
activity, that is, a fast decay of the ARFs with area in the
predominantly convective regions compared to regions
dominated by stratiform rainfall. We use an hourly rain
gauge dataset to estimate ARFs across the country, us-
ing an empirical fixed-area method.
2. Study area and data
Austria is a predominately mountainous country in
central Europe with an area of about 84 000 km
2
. There
are three major ranges of the Alps running from west
to east, including the Northern Calcareous Alps, the
Central Alps, and Southern Calcareous Alps. The an-
nual mean temperature ranges from above 118Cinthecity
of Vienna to 298C at the highest Alpine summits, which
exceed 3500 m MSL (Fig. 1a). The complex mountainous
environment comprises temperate oceanic climates,
humid continental climates, and subarctic and tundra
climates (Peel et al. 2007). The total annual precipi-
tation reaches up to 3000 mm in the High Tauern
mountain range in the Central Eastern Alps and is
below 500 mm in the north of the province of Lower
Austria (see Fig. A1 in appendix A).
To estimate the ARFs, we used hourly rainfall time
series from 306 rain gauges across Austria covering a
simultaneous recording period of 20 years (1995–2014),
hereinafterreferredtoasraingaugedata.Gapsinthe
recordings were interpolated using kriging with exter-
nal drift with elevation as an external drift variable
(e.g., Haberlandt 2007). The rain gauge data went
through comprehensive quality checks before inter-
polation. The spatial density of the rain gauge dataset
turned out to be too low to support more detailed an-
alyses to examine the relationship between rainfall
extremes and lighting information (section 4c). We
thus used an additional gridded rainfall dataset from
the Integrated Nowcasting through Comprehensive
Analysis (INCA) system (3354 grid points). INCA was
provided for the years 2003–18 by the Central Institute
for Meteorology and Geodynamics (ZAMG) with a 1-h
temporal resolution and 5-km spatial resolution. As the
INCA algorithms have been changed over time causing
inhomogeneities in the time series, ZAMG provided a
consolidated dataset for this research, where the most
recent INCA algorithm was applied to all available
years. The original (i.e., unconsolidated) data are avail-
able at 1-km spatial resolution; each grid cell in the con-
solidated dataset represents the mean rainfall of the grid
cell area. INCA is a composite product consisting of nu-
merical weather prediction (NWP) output, surface sta-
tion observations, and radar rainfall and satellite data. It
has been specifically developed for the mountainous do-
main of Austria by considering topographic effects in the
analysis methods (Haiden et al. 2011). While the INCA
analyses of some parameters such as temperature or hu-
midity do include numerical weather prediction data,
INCA analyses of precipitation are solely based on rain
gauge and radar data. Furthermore, the averaged 5-km
grid cells and the short time series of only 16 years from
INCA do not necessarily allow for reliable analyses of
ARFs, but they are considered appropriate to better
understand the rainfall–lightning relationship.
We analyzed the rain gauge data for Austria with
an additional emphasis on two regions with contrast-
ing dominating rainfall processes: one area domi-
nated by stratiform orographic rainfall in the west of
Austria (province of Vorarlberg, about 1600 km
2
—blue
rain gauges and INCA grid in Fig. 1), and one area
in the central parts of the province of Styria (about
1800 km
2
—magenta rain gauges and INCA grid in Fig. 1),
APRIL 2020 B R E I N L E T A L . 673

hereinafter referred to as the ‘‘orographic rainfall
region’’ and ‘‘convective rainfall region.’’ The number
of rain gauges is 16 in the orographic rainfall region (105
INCA grid points), and 19 in the convective rainfall
region (115 INCA grid points). The orographic rainfall
region is characterized by three dominant weather pat-
terns with northwesterly flow causing heavy rainfall
(Seibert et al. 2007). These weather patterns [called
northwesterly flow, westerly ‘‘Stau’’ (the German syn-
onym for orographic lift), and north-northwesterly flow;
Seibertetal.2007] cause high orographic rainfall
amounts north of the Alpine divide, which can be de-
picted from the annual rainfall patterns (Fig. A1). The
convective rainfall region is dominated by heavy rainfall
from summer thunderstorms. The central-eastern part
of the province of Styria as well as the eastern parts of
Carinthia are the regions of Austria with the highest
frequency of thunderstorms (Fig. 1b). The selection of
the rainfall data in the convective rainfall region was
conducted using spatial lightning information from the
Austrian Lightning Detection and Information System
(ALDIS; Schulz et al. 2005). ALDIS includes intracloud
lightning as well as cloud-to-ground lightning, which we
used as a proxy for convective activity (Fig. 1b). Hence,
the two different regions described above with their
different dominating rainfall processes were ideal for
our analyses.
3. Methodology
In the analysis, we estimated IDF statistics and ARFs
in space and time for five rainfall durations (d51, 3,
6, 12, 24 h).
a. Estimation of IDF statistics
For constructing IDF curves at each location we fitted
the generalized extreme value (GEV) distribution to the
annual maximum (AMAX) rainfall of duration dusing
the method of maximum likelihood. The areal IDF curves
were estimated similarly by fitting the GEV distribution
FIG. 1. (a) Distribution of the rain gauges and INCA grid across Austria including the two
areas in focus (blue and magenta colors) and the province borders. Numbers refer to the
provinces (1 5Vorarlberg, 2 5Tyrol, 3 5Salzburg, 4 5Carinthia, 5 5Styria, 6 5upper
Austria, 7 5lower Austria, 8 5Vienna, and 9 5Burgenland. (b) Average annual number of
flashes of lightning per square kilometer according to the Austrian Lightning Detection and
Information System (ALDIS) for the period 1992–2018 (www.aldis.at) including information
on rain gauges and the INCA grid. Rain gauges were used for the ARF analyses, while the
INCA grid was used to support additional analyses on the rainfall–lightning relationship.
674 JOURNAL OF HYDROMETEOROLOGY VOLUME 21

to areal AMAX rainfall. The cumulative distribution
function (CDF) of the GEV distribution is defined as
F(x;m,s,z)5expn2h11zx2m
si21/zo,
where the parameters m,s, and zrepresent the location,
the scale, and the shape of the distribution, respectively.
Koutsoyiannis (2004a,b) analyzed global rainfall ex-
tremes and demonstrated that they are more adequately
described by a GEV rather than a Gumbel distribution.
Notwithstanding the difficulties with estimating the
shape parameter zfor records smaller than 100 years
related to estimation bias and sampling variability,
Koutsoyiannis (2004a,b) therefore recommend the use
of the GEV distribution over alternative distributions
such as the Gumbel distribution. In that context, typical
annual maximum rainfall time series with a length be-
tween 20 and 50 years hide the GEV distribution and
often display Gumbel behavior, although the real be-
havior of rainfall maxima can be better described by a
GEV distribution (Koutsoyiannis 2004b). This is not a
peculiarity of the examined records but a generalized
statistical effect (Koutsoyiannis and Baloutsos 2000).
We also applied model selection using the Akaike in-
formation criterion (AIC
c
) for short sample sizes (e.g.,
Burnham and Anderson 2004;Okoli et al. 2018) for the
Gumbel and GEV distributions. The AIC
c
analysis
can be found in appendix B. Based on the analysis of
AIC
c
and the studies by Koutsoyiannis (2004a,b) and
Koutsoyiannis and Baloutsos (2000), we used the GEV
distribution for all stations (periods, durations, area
sizes). Given the uncertainty of the shape parameter z,
we did not examine return periods beyond 30 years due
to the relatively short length of the time series available
(20 years of rain gauge data).
b. Estimation of ARFs
Figure 2 provides an overview of the three steps
conducted to estimate the ARFs.
1) Variogram modeling (Fig. 2, right): We fitted vario-
gram models for all of Austria, that is, considering
all 306 gauge locations g. The procedure was con-
ducted for the five different durations das well as five
periods s, annual (January–December), spring (March–
May), summer (June–August), autumn (September–
November), and winter (December–February). As the
ARFs refer to annual maxima and to ensure that the
variogram models better represent extreme rainfall
events, we fitted variogram models only taking into
account time steps/durations with high areal rainfall
amounts. The latter were estimated by computing
the arithmetic mean over the entire country for each
duration time step from which we only took the
upper 10% of for estimating empirical variograms.
These empirical variogram models were then aver-
aged over all locations g, and a theoretical variogram
model was fitted. We used the exponential model as
the theoretical variogram model, which has been
proven to be robust across rainfall of different dura-
tions in Austria (Skøien and Blöschl 2006), and visual
inspection of the resulting variograms confirmed its
suitability. The models were fitted without a nugget
to avoid steps in the ARF curves for small areas and
thus allow for smooth ARF curves across all area
sizes. That is, as a result, we obtained 25 variogram
models (from five durations for five periods). In a
sensitivity study, we conducted the whole study using
the rain gauge data fitting variograms to the upper
1% (very extreme events but small sample ratio) and
the upper 90% (most types of rainfall events, very
high sample ratio) of rainy durations. The final
results turned out to be very similar.
2) Block kriging (Fig. 2, top-left area): The estimated
variogram models for the rain gauge data served as
input for the block kriging methodology (Fig. 2,left
area ‘‘block kriging’’). To the best of our knowledge,
block kriging has not yet been applied in the context of
ARF research but is an efficient way of achieving this
task. To do so we used the statistics package ‘‘gstat’’
version 2.0.2 in the statistical computing software R
version 3.6.0 (Pebesma and Wesseling 1998). Block
kriging is similar to more commonly applied ordinary
kriging (OK) but allows for the estimation of average
values over a surface, segment, or volume of any shape
and size (e.g., Goovaerts 1997) without interpolating
point values over a grid. Gstat assumes the block to
have a square shape of a given area, which we assume
to approximately represent the shape of catchments.
We likewise tested block kriging with external drift
(with elevation as drift variable), but differences in the
results were negligible. We limited the number of
(spatially) nearest observations used for the kriging
predictions to 30 for numerical efficiency. Test sim-
ulations showed that the results are almost identical
with those when using a larger number of observa-
tions (see appendix C). Annual maximum point
rainfall was estimated at each rain gauge location g,
in each period s, for each duration d, and for each
year m. Areal annual maximum rainfall for each rain
gauge location g, each period s, each duration d, and
for each year mwas then estimated by block kriging
for nine different square block sizes b(1, 3, 5, 10, 30,
50, 100, 300, 500 km
2
), using the related variogram
models estimated in step 1. The annualmaxima for the
point and areal rainfall were estimated independently,
APRIL 2020 B R E I N L E T A L . 675

FIG. 2. Schematics of the framework for deriving the areal reduction factors (ARFs), split into (i) the block kriging
methodology, (ii) variogram modeling, and (iii) the estimation of the final ARFs.
676 JOURNAL OF HYDROMETEOROLOGY VOLUME 21

that is, the spatial annual maxima do not necessarily
coincide with a point annual maximum. As result, we
obtained 225 vectors of length n520 (from 5 periods,
5 durations, and 9 block sizes) for each rain gauge of
the rain gauge data.
3) Deriving ARFs (Fig. 2, bottom-left area): To both
resulting vectors of the point and areal rainfall max-
ima we fitted a GEV distribution using the method of
maximum likelihood (see section 3a for details on the
GEV). Based on the GEV parameters for g,s,d,andb
we computed point and areal rainfall for five different
return periods (RP; 2, 5, 10, 20, and 30years). The final
ARFs for each return period RP, rain gauge location
g, each season s, each duration d, and each area (i.e.,
block) size bwere then computed by the ratio of the
areal rainfall P
areal
and the point rainfall value P
point
,
that is, P
areal
/P
point
.
A limitation behind fitting countrywide variograms to
the upper 10% of rainy duration time steps is that strong
localized storms may not be represented with this ap-
proach as they occur locally, when the rest of the country
is relatively dry. By this, the spatial extent of small-scale
rainfall events of small durations may be overestimated,
which may also overestimate ARFs. We investigated the
possibility of fitting variograms separately centered on
each single rain gauge to address this issue, varying the
number of nearest observations from 10 to 50 gauges. In
the majority of cases these local empirical variograms
had a very high scatter (especially when using a smaller
number of nearest neighbors) and did not give robust fits
of the theoretical variogram models. A sensitivity study
comparing the local and countrywide variograms at se-
lected rain gauges demonstrated that the global vario-
grams produce lower interpolation biases across all
periods and durations and are thus recommended (see
appendix C).
As for the block kriging methodology, generally
speaking, some kind of interpolation is always needed to
estimate the ARFs for different area sizes. As an alter-
native to our proposed approach, one could interpolate
the station values for each time step and each duration
over a very fine grid (to be able to estimate small areas),
and then average over the areas to estimate the areal
rainfall. However, the computational costs become very
large. Block kriging does not require the interpolation
over a grid but gives identical results. The so-called
kriging weights for the rain gauges and each (block-)
area size under consideration can be estimated from the
variogram models in a much more efficient way.
FIG. 3. IDF estimates for different durations (1 and 24 h) and frequencies (2- and 30-yr return periods) across Austria, estimated from the
entire time series (i.e., entire year) of the rain gauge data. Maps are based on nearest neighbor interpolation with five nearest neighbors.
APRIL 2020 B R E I N L E T A L . 677

To provide further validation of our methodology, we
compared interpolation results at six exemplary sites
using kriging with local and countrywide variograms as
well as (alternative) inverse distance weighting (IDW)
interpolation. The results from this sensitivity study
justify the block kriging approach with (i) countrywide
variograms and (ii) 30 nearest observations (corre-
sponds to a mean maximum distance of 59.5 km over all
sites) for the kriging predictions (see appendix C).
4. Results and discussion
a. IDF statistics
IDF rainfall for different durations and frequencies
from the rain gauge data are presented in space and time
for the entire year (Fig. 3). For 1-h duration and a return
period of 2 years (Fig. 3a), the highest rainfall occurs in
eastern Styria (see Fig. 1 for the Austrian provinces). The
pattern is similar with a higher return period of 30 years
(Fig. 3b), differences can be seen for example along the
northern border with relatively higher rainfall amounts.
For a rainfallduration of 24 h the pattern across Austria is
again very similar for low and high return periods (2 and
10 years, Figs. 3c and 3d, respectively), but it differs sig-
nificantly from the pattern identified for rainfall with 1-h
duration (Figs. 3a,b). Regions of high rainfall include the
province of Vorarlberg in the west (orographic rainfall
region), in the south of Carinthia along the southern
Austrian border, and along the north of the Alpine divide
in the central parts of Austria.
The high rainfall intensities in eastern Austria (Figs. 3a,b)
are in line with high lightning activity (Fig. 1b), which
FIG. 4. IDF curves for the (left) orographic and (right) convective rainfall region (right) (see Fig. 1 for regions) in
(a),(b) summer and (c),(d) winter. IDF curves are the averages of all rain gauges of the rain gauge data.
678 JOURNAL OF HYDROMETEOROLOGY VOLUME 21

suggests convective rainfall as their likely cause. Flash
floods are frequent in eastern Austria, especially in
southeastern Austria and in northeastern Austria
(Merz and Blöschl 2003). The hilly terrain enhances
vertical motion in the boundary layer and increases
the likelihood of convective storms (Merz and Blöschl
2003). Additionally, the southerly location and thus
closeness to the Adriatic Sea, that is, very warm sum-
mer temperatures and high atmospheric humidity, may
contribute to the high intensities. The spatial distribu-
tion of the 24-h rainfall can be related to the dominant
circulation patterns, that is, mainly synoptic systems
and stratiform rainfall. The regions in Vorarlberg and
in central Austria are characterized by heavy rainfall
from three different dominant synoptic patterns called
northwesterly flow, westerly ‘‘Stau’’ (the German syn-
onym for orographic lift), and north-northwesterly flow
(Hofstätter et al. 2018;Seibert et al. 2007), that is,
stratiform orographic rainfall from air masses from
predominantly northwest directions. The most fre-
quent pattern is the northwesterly flow, where low level
trajectories come from the Atlantic Ocean, thus trans-
porting humid air. The westerly Stau and the north-
northwesterly flow are characterized by higher wind
speeds compared to the northwesterly flow. The high
rainfall across the southern border in Carinthia is to a
large degree related to the southerly Stau pattern, that
is, southerly flow at higher and lower levels (Seibert
et al. 2007). Airflow at low levels supports advection of
humidity from the Mediterranean Sea, which is precip-
itated over the Alps (Seibert et al. 2007). As the four
synoptic patterns mentioned above are the most fre-
quent ones across Austria causing most of the rainfall,
the pattern of the 24-h IDF estimates (Figs. 3c,d) show
clear similarities with the pattern of annual rainfall in
Austria (Fig. A1).
Figure 4 presents the IDF curves in the two regions
with dominant convective and orographic rainfall,
stratified by season and averaged over all rain gauges
of the related region. In summer, rainfall intensities
are lower in the orographic rainfall region across all
return periods (Fig. 4a) for short durations (1, 3 h)
compared to the convective rainfall region (Fig. 4b).
While intensities are similar for a duration of 6 h, in-
tensities become higher in the orographic rainfall re-
gion with long durations (12, 24 h) compared to the
convective rainfall region. The IDF curves for sum-
mer thus show the dominant convective activity in
the convective rainfall region in summer, while oro-
graphic processes and long-duration storms are less
relevant than in the orographic rainfall region. The
precipitation is generally lower in winter, and in par-
ticular in the convective rainfall region compared
to the orographic rainfall region (Figs. 4c,d). In win-
ter, there is almost no lightning activity. According to
the monthly ALDIS statistics, only 0.14% of all
flashes recorded in the period 1992–2018 were recor-
ded in winter, while 81.4% were recorded in summer
(www.aldis.at).
We also examined how the IDF statistics relate to the
characteristics of wet spell intensities in the different
regions. Figure 5 summarizes the results. We present
results for the intensities of wet spell lengths up to 24h
on an annual basis (Fig. 5a), for the summer period
(Fig. 5b) and for winter (Fig. 5c). In general, intensities
decrease with longer durations, a phenomenon that has
been observed in other studies (Haddad and Rahman
2014;Poduje and Haberlandt 2018). On an annual basis
(Fig. 5a), the intensity of wet spells is on average higher
in the convective rainfall region compared to the oro-
graphic rainfall region, for short durations. This is re-
latedtomoreintensedownpours from convective activity.
FIG. 5. Intensity of all wet spells recorded in the time series, for the (a) entire year, (b) summer, and (c) winter. Solid lines represent the
mean of all wet spells of all gauges in the region. Shaded areas denote the 10th and 90th percentiles of temporal and spatial variability.
APRIL 2020 B R E I N L E T A L . 679

On average, intensities are 25.2% higher in the con-
vective rainfall region for durations up to 5 h, and
12.1% between 6 and 10 h. Beyond 10 h duration, in-
tensities become very similar. The effect is even more
pronounced in the summer period for shorter spell
lengths (Fig. 5b), where intensities are generally higher
in both study areas. On average, in the convective
rainfall region, intensities are 33.3% higher for lengths
up to 5 h and 16.5% for length between 6 and 10 h.
Intensities are similar for the winter period (Fig. 5c),
but on average, intensities are 10.7% lower in the
convective rainfall region compared to the orographic
rainfall region for wet spell lengths up to 24 h. This is
most likely related to the lack of convective storms
in this season. In summary, the characteristics of wet
spells to a large degree confirm the rainfall processes in
the two regions in focus as discussed above.
b. Areal reduction factors in space and time
Figure 6 shows some of the ARF results. It is clear that
the ARFs change with the return period of the rainfall.
For example, for a duration of 1 h, the ARF for a 2-yr
rainfall and 100 km
2
is 0.84 while the corresponding es-
timate for a 30-yr return period is 0.78. Several authors
have detected decreasing ARFs with increasing return
periods (e.g., Allen and DeGaetano 2005a;Asquith and
Famiglietti 2000;Le et al. 2018;Mailhot et al. 2012),
although they do not provide precise numbers and focus
on considerably larger areas. The differences are as-
sumed to be related to the areal rainfall becoming
FIG. 6. Areal reduction factors (ARFs) for different return periods, seasons, for all of Austria and the two study
regions based on the rain gauge data. Comparisons are shown for (a),(b) two return periods, (c),(d) two regions, and
(e),(f) summer and winter.
680 JOURNAL OF HYDROMETEOROLOGY VOLUME 21

relatively smaller due to increasing convective activity.
ARFs differ between the orographic and convective
rainfall regions (Figs. 6c and 6d, curves from different
study areas averaged). For example, for a duration of
1 h, the ARF in the orographic rainfall region for a 2-yr
rainfall and 100 km
2
is 0.78 while the corresponding es-
timate in the convective rainfall region for a 30-yr return
period is 0.75. The smaller ARFs in the convective study
area (Fig. 6d) would be expected due to the dominance
of strong convective events. As convective events tend
to be smaller than stratiform rainfall events, stronger
decays of the ARFs with increasing catchment area will
result. The results from other return periods (e.g., 30
years, not shown here) are very similar in respect due to
the relative differences between the orographic rainfall
region and the convective rainfall region. ARFs are
smaller in summer than in the winter (Figs. 6e,f). This
would be expected due to the dominance of convective
rainfall processes in summer and the dominance of
synoptic precipitation processes in winter in Austria.
Overall, the ARF estimates are similar to fixed-area
related results from other humid climates across the
globe (see Table 1).
Figure 7 showsmapsoftheARFsfortworainfall
durations (1 and 24 h) and two area (block) sizes (50
and 500 km
2
), for a return period of 10 years. The
maps were generated by nearest neighbor interpola-
tion with five nearest neighbors for visualization
purposes. As can be seen, the ARFs show little spatial
variability for 1-h duration and an area of 50 km
2
(Fig. 7a). For example, for the duration of 1 h, which is
relevant for convective events, there is no noticeable
difference between the orographic rainfall region and
convective rainfall region for 50 km
2
(Fig. 7a). The
pattern becomes patchier for a catchment of 500 km
2
(Fig. 7b). The general pattern shows similarities with
the distribution of the lightning frequency as an in-
dicator of convective activity with smaller ARFs in
regions of higher lightning frequency, also see
Fig. 1b), such as Carinthia and Styria. However, there
are also low values in the western parts of Austria with
less lightning activity, which we discuss in more detail
in section 4c.
The spatial distribution of the ARFs is similar for 24 h
(Figs. 7c,d) and 50 km
2
with little spatial variability
(Fig. 7c). For a catchment area of 500 km
2
the region of
Styria gives particularly low ARFs, which is likely re-
lated to the dominance of convective rainfall (Fig. 7d).
However, the relative spatial differences in ARFs are
lower for 24 h than for 1 h. For example, the ARFs de-
crease on average by 21.1% when increasing the area
from 50 to 500 km
2
for a 1-h duration (Figs. 7a,b) (av-
erage computed over the entire interpolated grid), while
TABLE 1. ARFs from fixed-area methods for different area sizes and durations from the present study in comparison with results from other regions. Numbers refer to different
(or no) return periods (RPs) in years (yr).
Area (km
2
)/
duration (h)
Austria (present
study)—RP 52yr
South Korea
(Kang et al.
2019)—RP 520 yr
Germany
(Verworn
2008)—RP 510 yr
Australia without
dry inland area
(Myers and Zehr
1980)—RP 52yr
United Kingdom
(NERC 1975)—no RP
United States (Ohio Valley, Southeastern
U.S., Middle Atlantic region,
Northeastern U.S., Great Lakes region)
(U.S. Weather Bureau 1957)—no RP
100/1 0.84 0.85 0.74 0.80 0.85 0.85
500/1 0.67 0.71 0.61 0.72 0.72 0.72
100/6 0.92 0.96 0.95 0.92 0.94 0.94
500/6 0.83 0.90 0.88 0.88 0.87 0.87
APRIL 2020 B R E I N L E T A L . 681

they only decrease by 6.0% for a 24-h duration (Figs. 7c,d).
The differences suggest that, at 1-h duration, con-
vective events dominate, while at 24-h duration syn-
optic weather systems and stratiform rainfall are more
important.
c. ARFs in context of lightning data
To better understand the situation in the west of
Austria with its smaller-than-expected ARFs for 1 h in
both analyses, the lightning data were analyzed in more
detail. While we received the aggregated lightning data
with the average number of flashes of lightning for the
period 1992–2018 at a 5-km grid from ALDIS (Fig. 1b)
to support the identification of the main rainfall pro-
cesses across the country, we also received a detailed
dataset for the year 2012 (5-km grid, lightning infor-
mation for every ALDIS grid cell and day). We linked
the (spatially more dense) annual maxima of INCA
rainfall to lightning information, to examine their rela-
tionship. To do so, we assigned the maximum number
of flashes from ALDIS on the date of the maximum
rainfall to each INCA grid cell (INCA to have a very
high spatial coverage), using a 10-km radius. Lightning
can strike at some distance from the core of a convective
cell and 10 km is a typical rule of thumb used by weather
forecasters (Walsh et al. 2013). That is, for the year 2012
we got one data point for each grid cell.
Figure 8 provides an overview of the association
of INCA annual maximum rainfall with lighting for
Austria (Fig. 8a), the orographic rainfall region (Fig. 8b)
and the convective rainfall region (Fig. 8c). Specifically,
the figure shows the percentage of annual rainfall max-
ima associated with lightning (i.e., at least one flash
within 10 km from the rain gauge). As can be seen, the
lightning activity decreases with increasing duration,
indicating a change in rainfall processes. Overall, the
lightning activity is higher in the convective rainfall re-
gion compared to the orographic rainfall region. The
slight increase in winter (Figs. 8a,b) for long durations
and the absence of lightning in the convective rainfall
region may be an artifact of the small sample size, as
only one year of daily lightning data could be obtained.
The detailed lightning data provide an explanation of
the relatively small ARFs in the west of Austria despite
the general dominance of stratiform orographic rainfall
in the region. One explanation is that strong Stau events
may lead to sharp small-scale contrasts in rainfall totals,
such as is typically the case in the orographic rainfall
region. However, convective activity provides another
explanation: in the orographic rainfall region, 84.8% of
FIG. 7. ARFs for a return period of 10 years, two durations (1 and 24 h) and two catchment sizes (50 and 500 km
2
), estimated from the rain
gauge dataset.
682 JOURNAL OF HYDROMETEOROLOGY VOLUME 21

the hourly annual maxima were associated with light-
ning activity in 2012 (Fig. 8b),intheconvectiverainfall
region these were 98.3% (Fig. 8c). The corresponding
average number of flashes per annual maximum was 31.7
and 5.1. That is, it is valid to assume that convective ac-
tivity is associated with summer extremes in both areas.
To gain further insights into the role of convective
activity, we investigated the synchronicity of the dates of
the annual rainfall maxima in both regions across all grid
points. A large number of annual maxima occurring si-
multaneously would point toward stratiform events, as
events covered a larger area. It turned out that annual
maxima in the orographic rainfall region can be related
to eleven different dates (and thus most likely different
events), while annual maxima in the convective rainfall
region can be related to seven different dates. The areas
covered by the events on each date were also similar. On
average, the annual maxima in the orographic rainfall
region were related to a maximum distance between
grid points of 26.4 km, while in the convective rainfall
region the average maximum distance was 27.1 km. The
small sample size from only one year of detailed light-
ning data does not allow us to draw final conclusions but
does provide a plausible indication of convective activity
in both regions on the dates of annual maxima. This
would explain the similarity of the ARFs in the two re-
gions despite different (generally) dominating rainfall
processes.
d. Limitations
One limitation in this study is the variograms used.
As described above (section 3b), to reach stable fits
of the theoretical variogram models, we estimated
the empirical variograms for the upper 10% of rainy
duration time steps based on the countrywide (and thus
based on a large sample size) mean rainfall. Using the
same variogram throughout the country may lead to
underestimating the spatial variability of the ARFs, but
fitting local variograms to address this limitation ten-
ded to result in higher interpolation biases, very likely
resulting from less robust fits of the theoretical vario-
gram models. In general, longer rainfall time series
would probably allow more robust fits of the extreme
value distributions (section 3a). Finally, additional
detailed lightning data would help better understand
the detailed rainfall processes behind annual rainfall
extremes (section 4c).
5. Conclusions
The findings of the paper allow us to draw the fol-
lowing conclusions:
dWe proposed a new method of estimating ARFs based
on block kriging, which is computationally more effi-
cient than interpolating each duration time step and
each area size of the entire time series across the
domain at high resolution to estimate the ARFs.
dARFs tend to decay faster in areas with dominant
convective activity than in areas with dominating
stratiform rainfall, visible in both classic (regional)
IDF curves and in space (maps). This finding is con-
sistent with the original hypothesis of the paper as well
as with findings from numerous authors (e.g., Allen
and DeGaetano 2005a;Huff and Shipp 1969;Skaugen
1997;Wright et al. 2014).
dLightning information can be a useful proxy for
convective activity and thus the magnitude of areal
reduction factors in space and time, which was likewise
related to our main hypothesis. However, the usefulness
FIG. 8. Percentage of AMAX with different durations associated with lightning for (a) Austria, (b) the orographic rainfall region, and
(c) the convective rainfall region, for the entire year, summer, and winter. The lightning statistics are estimated from the ALDIS dataset
for the year 2012.
APRIL 2020 B R E I N L E T A L . 683

of lightning data in ARF analyses is also limited, at least
in the case of Austria, as relatively low ARFs can also
occur in areas with relatively low lightning activity, for
example in the orographic rainfall region in the west. As
the detailed analysis of lightning data for one year
revealed, there is a general tendency across Austria
that annual maxima are associated with convective
activity, leading to reduced ARF values.
dThe (countrywide) magnitudes of the ARFs estimated
in Austria are similar to those from other studies con-
ducted in humid climates using fixed area methods (e.g.,
Kang et al. 2019;Myers and Zehr 1980;NERC 1975;
U.S. Weather Bureau 1957,1958;Verworn 2008). For
example, for 1-h duration and an area of 100 km
2
(RP 52 years), we estimated an ARF of 0.84 while the
mean of five other studies was 0.82. For 6-h duration
and 500 km
2
(RP 52 years), we estimated an ARF of
0.83 (mean of other studies 0.88).
dThe areal reduction factors decrease with the return
period, which matches findings of other authors (e.g.,
Allen and DeGaetano 2005a;Asquith and Famiglietti
2000;Le et al. 2018;Mailhot et al. 2012;Sivapalan and
Blöschl 1998). This decrease is most pronounced for
durations shorter than 24 h. This decrease may possi-
bly be observed because the contribution of convec-
tive rainfall is higher.
dFor future research, it would be interesting to inves-
tigate how the process links of the ARFs analyzed
here relate to the space–time scaling of floods, which is
the main natural hazard in terms of monetary losses in
Austria.
Acknowledgments. This research has received fund-
ing from the European Union’s Horizon 2020 re-
search and innovation programme under the Marie
Sklodowska-Curie Grant Agreement STARFLOOD
793558 (www.starflood.at). Hannes Müller-Thomy ac-
knowledges the funding from the Research Fellowship
(MU 4257/1-1) by DFG e.V., Bonn, Germany. This
research has received funding from the Austrian
Federal Ministry for Sustainability and Tourism and
the Bavarian Environment Agency in the framework
of the project WETRAX1. We thank the Central
Institution for Meteorology and Geodynamics for
providing the rain gauge and INCA data. We thank
Wolfgang Schulz and ALDIS for providing the lightning
data. Helpful discussions with Thea Turkington, David
Lun, and Jürgen Komma on our work as well as detailed
comments from two anonymous reviewers and the edi-
tor are gratefully acknowledged.
Data availability statement: The rain gauge data
used in this study can be obtained from the Central
Institution for Meteorology and Geodynamics (ZAMG)
FIG. C1. Six rain gauges (three in the west, three in the east)
selected for additional tests to validate the kriging interpolation
method.
FIG. A1. Annual average rainfall in Austria derived from the
rain gauge data. The map is based on nearest neighbor interpola-
tion with five nearest neighbors.
FIG. B1. Analysis of AIC
c
.PlotshowsD
i
5AIC
GEV
2AIC
Gumbel
of all distribution fits (all periods, all durations, all area sizes,
all years) sorted by value. Values below 0 indicate selection of the
GEV, and values above 0 indicate selection of the Gumbel.
684 JOURNAL OF HYDROMETEOROLOGY VOLUME 21

(www.zamg.ac.at). The lightning data used can be ob-
tained from the Austrian Lightning Detection and
Information System (ALDIS) (www.aldis.at).
APPENDIX A
Additional Figure
Figure A1 shows the annual average rainfall in Austria
derived from the rain gauge data.
APPENDIX B
AIC
c
Analysis
The Gumbel distribution produced the lowest AIC
c
in the majority of the rain gauges (77%). However, ac-
cording to Burnham and Anderson (2004), one must also
consider the AIC differences, that is, D
i
5AIC
i
2AIC
min
over all candidate models examined. Models with D
i
#2
have substantial support, models with 4 #D
i
#7 have
considerably less support, models with D
i
.10 have es-
sentially no support (Burnham and Anderson 2004). For
our time series, we plotted D
i
5AIC
GEV
2AIC
Gumbel
,
that is, a positive value means selection of Gumbel and a
negative value means selection ofGEV. As can be seen in
Fig. B1,whenAIC
c
suggests Gumbel, both the Gumbel
and GEV are essentially valid according to Burnham
and Anderson (2004) with D
i
not exceeding a value of
2.9. The opposite does not apply, that is, the GEV is
considerably more supported when suggested by AIC
c
as D
i
can get negative values of a much larger magnitude.
Based on the analysis of AIC
c
and the studies by
Koutsoyiannis (2004a,b);Koutsoyiannis and Baloutsos
(2000), we used the GEV distribution for all stations
(periods, durations, area sizes) (i) for the sake of con-
sistency and (ii) to address the issue of the Gumbel not
being supported for a larger number of fits.
FIG. C2. RMSE from three different interpolation methods averaged over the three gauges and all years in the west of Austria, namely,
IDW and OK with local and global (i.e., countrywide) variograms. Results are presented across the different periods (a) spring, (b) summer,
(c) fall, (d) winter, and (e) annual maxima from the entire year (AMAX). The number of neighbor sites is varied from 10 to 50 neighbors.
APRIL 2020 B R E I N L E T A L . 685

APPENDIX C
Validation of the Kriging Interpolation Method
We selected six rain gauges (three in the west and
three in the east of Austria, Fig. C1), to further validate
the kriging interpolation method.
We conducted interpolations with the point rainfall
across the different five periods (annual maxima and
four seasons), thereby using inverse distance weighting
(IDW), ordinary kriging (OK) with local variograms
fitted to the 10, 20, 30, 40, and 50 nearest neighbors
(corresponds to a mean maximum site distance over all
sites of 30.6, 46.3, 59.5, 72.0, and 84.1 km), and OK with
FIG. C3. RMSE from three different interpolation methods averaged over the three gauges and all years in the east of Austria, namely,
IDW and OK with local and global (i.e., countrywide) variograms. Results are presented across the different periods (a) spring, (b) summer,
(c) fall, (d) winter, and (e) annual maxima from the entire year (AMAX). The number of neighbor sites is varied from 10 to 50 neighbors.
FIG. C4. Change in the RMSE averaged over the three gauges and all years in the west, when varying the number of neighbors with OK
and global variograms from 10 to 50 neighbor sites. Results are presented across the different periods (a) spring, (b) summer, (c) fall,
(d) winter, and (e) annual maxima from the entire year (AMAX) and across different durations. Results are presented as changes in the
RMSE when using 20 instead of 10 nearest neighbors (10 to 20), 30 instead of 10 sites (10 to 30), and so forth.
686 JOURNAL OF HYDROMETEOROLOGY VOLUME 21

global (i.e., countrywide) variograms. The actual rainfall
value at the location of the rain gauge was left out
(leave-one-out analysis). The rainfall value at the loca-
tion was estimated with one of the methods and setups.
In the case of OK with local variograms, the OK itself
was conducted using the same number of nearest
neighbors as used for estimating the variograms, that is,
when a local variogram fitted to 10 nearest neighbors
was used, the same number of 10 nearest neighbors was
used in the OK procedure itself and so on. In case of the
global variograms, the number of sites considered in
the OK procedure itself was likewise varied between
10 and 50.
Figure C2 summarizes the results for the three rain
gauges in the west of Austria. As can be seen, across all
periods and durations (Figs. C2a–e), OK with the global
variograms produces the lowest RMSE. While the
number of neighbors considered for OK with local
variograms has noticeable influence (decreasing bias
with increasing number of neighbors), the number of
neighbors considered when conducting OK with global
variograms hardly influences the results. In general, the
results are comparable in the east of Austria (Fig. C3)
with OK producing the lowest bias when using global
variograms.
Our study confirms other studies on rainfall inter-
polation, which state that kriging is preferred compared
to other more simplistic methods such as IDW or nearest
neighbor interpolation (e.g., Haberlandt 2007;Mair and
Fares 2011;Wagner et al. 2012).
While the tests revealed that OK with global vario-
grams is the interpolation method producing the lowest
bias, we take a closer look into the number of gauges
considered in the OK itself. This information is con-
tained in Figs. C2 and C3 but hardly visible. Figure C4
shows the bias for the three gauges in the west from OK
with global variograms with varying number of neigh-
bors, in relation to the minimum number of 10 nearest
neighbors across the five periods and across durations.
As can be seen, the reduction of the bias reaches a
minimum when considering 30 nearest neighbors but
does not noticeably further decrease with 40 or 50 sites.
The results are similar for the three gauges in the east
(Fig. C5). In all periods and with all durations, 30
neighbor sites appear to be a reasonable number.
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