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RESEARCH ARTICLE
10.1002/2014WR015918
Key Points:
Statistics for downscaling monthly
precipitation over Africa
Spatiotemporal variability of rainfall
(amount, frequency, duration, rate)
Statistics for the assessment of
rainfall from climate models
Supporting Information:
Supporting Information S1
Correspondence to:
A. Kaptue,
[email protected]
Citation:
Kaptu
e, A. T., N. P. Hanan, L. Prihodko,
and J. A. Ramirez (2015), Spatial and
temporal characteristics of rainfall in
Africa: Summary statistics for temporal
downscaling, Water Resour. Res., 51,
2668–2679, doi:10.1002/
2014WR015918.
Received 29 MAY 2014
Accepted 16 MAR 2015
Accepted article online 23 MAR 2015
Published online 20 APR 2015
C 2015. The Authors.
V
This is an open access article under the
terms of the Creative Commons Attribution-NonCommercial-NoDerivs
License, which permits use and distribution in any medium, provided the
original work is properly cited, the use
is non-commercial and no modifications or adaptations are made.
KAPTUE ET AL.
Spatial and temporal characteristics of rainfall in Africa:
Summary statistics for temporal downscaling
1, Niall P. Hanan1, Lara Prihodko1, and Jorge A. Ramirez2
Armel T. Kaptue
1
Geospatial Science Center of Excellence, South Dakota State University, Brookings, South Dakota, USA, 2Department of
Civil and Environmental Engineering, Colorado State University, Fort Collins, Colorado, USA
Abstract An understanding of rainfall characteristics at multiple spatiotemporal scales is of great importance for hydrological, biogeochemical, and land surface modeling studies. In the present study, patterns of
rainfall are analyzed over the African continent based on 3 hourly 0.25 Tropical Rainfall Measuring Mission
(TRMM) estimates between 1998 and 2012 to produce monthly statistical summaries. The selected rain
event properties are multiyear means of precipitation total amount (mm), event frequency (number), rate
(mm/h), and duration (h) calculated independently for each calendar month. Analysis of 3 hourly and daily
events in the 1998–2012 period suggests that rainfall amount can be summarized using gamma probability
density functions. Assuming stationarity, gamma probability density functions of the total depth of 3 hourly
and daily events are estimated and then used for temporal downscaling of monthly rainfall estimates (past
or future). As a result, we generate 3 hourly and daily rainfall estimates that are pixelwise statistically indistinguishable from the observations while preserving monthly totals. Example scripts are provided that can
be used to access monthly statistics and implement downscaling using archival (or projected) monthly rainfall estimates. These statistics could also be utilized for the assessment of rainfall from atmospheric models.
1. Introduction
In Africa, particularly in the seasonal deserts, grasslands, and savannas that constitute most of the continental area, rainfall is the key environmental constraint for hydrological, biogeochemical, agronomical, and ecological processes [Dunkerley, 2008]. While rainfall amount, expressed as an annual, monthly, or (less
frequently) daily precipitation depth is a key variable, other characteristics of rainfall, including rainfall intensity and event duration can significantly influence the partitioning of water among canopy interception, surface ponding, evapotranspiration, shallow, and deep soil infiltration [Wang et al., 2005; de Wit and
Stankiewiciz, 2006]. Moreover, properties of single rain events such as event intensity and duration directly
affect runoff mechanisms and related processes [Haile et al., 2011]. For instance, intraevent variability (i.e.,
variability of intensity within a rain event) can significantly impact runoff processes and flood generation
[e.g., Kusumastuti et al., 2007], while runoff strongly controls water available for plant growth and soil biogeochemical processes in water-limited environments [Sun et al., 2006]. A thorough understanding of rain
event properties and their changes in space and time is essential to correctly characterize the earth system
and is of particular importance for models of land surface hydrology, biogeochemistry, and ecology that are
used to simulate human systems in a context of global and environmental changes. This is especially true
for communities in arid and semiarid Africa (Figure 1) who depend on rain-fed agriculture and rangelands
for their livelihoods [Cooper et al., 2008].
Rainfall characterization has been the focus of many studies performed at various scales using direct or indirect rainfall measurements [e.g., Kruger, 2006; New et al., 2006; Lebel and Ali, 2009; Haile et al., 2011; Frappart
et al., 2009; Ngongondo et al., 2011; Pierre et al., 2011]. However, these studies tend to focus on areas where
high quality climatic data are available at the desired temporal scales. One of the main challenges in the
analysis of risk and vulnerability in hydrologic and agricultural studies in Africa lies in the scarcity of accurate
long-term rainfall information. Historical rainfall estimates for Africa are available via spatial interpolation of
rain gauge measurements (‘‘gauge interpolations’’) or satellite-based estimates (‘‘satellite retrievals’’) [Trenberth et al., 2007]. Leading gauge interpolation data sets include the monthly 1 3 1 Global Precipitation
Climatology Project (GPCP) from 1979 to present [Adler et al., 2003], the monthly 0.25 3 0.25 Global Precipitation Climatology Centre (GPCC) data set from 1901 to present [Meyer-Christoffer et al., 2011] and the
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monthly Climatic Research Unit
(CRU) product at 0.5 3 0.5 from
1901 to 2012 [Harris et al., 2013].
However, for many applications, the
coarse temporal and spatial resolution of existing gauge interpolations
reduces their utility. In recent years,
this has stimulated the use of high
spatial (0.25 ) and temporal (1
day) resolution hybrid products
which combine satellite infrared and
microwave retrievals with groundbased gauge measurements, such as
the daily African Rainfall Climatology
data set from 1983 to present (ARC,
0.1 3 0.1 ) [Novella and Thiaw,
2013] and 3 hourly Tropical Rainfall
Measuring Mission data from 1998
to present (TRMM, 0.25 3 0.25 )
[Huffman et al., 2007].
Rainfall data covering longer time
periods are generally only available
at a low temporal resolution, while
data sets with a high temporal resolution cover a short time period.
Rainfall data sets generated by genFigure 1. Bioclimatic regions of Africa derived from SPOT/VEGETATION [Kaptue et al.,
eral circulation models (GCM)—
2011].
important tools to simulate time
series of global climate variables accounting for the effects of greenhouse gases in the atmosphere—can
be used as scenarios in climate variability assessment studies. Bridging the gap between the temporal resolution (monthly or annual) of observed/predicted long-term rainfall data and the relevant time scales (daily
or subdaily) at which hydrological processes are modeled is known as temporal downscaling of rainfall
[Prudhomme et al., 2002; Fowler et al., 2007].
€schl, 2005]. The first
There are two generic approaches to temporal downscaling: dynamical and statistical [Blo
type involves physically based atmospheric models that simulate rainfall events based on atmospheric physics
that can be scaled to match observations. The second downscaling approach consists of stochastic models of
rainfall that can be used to generate high-resolution rainfall time series at the desired scale that are statistically
indistinguishable from the observations at both the original scale and the desired scale. In general, statistical
temporal downscaling is achieved by partitioning the longer-time scale rainfall amounts through a recursive
rule as in the case of multiplicative cascades where rainfall depth over a period of time is split into two subperiods and these subperiods are split again and so forth [e.g., Kang and Ramirez, 2010], or by a two-step process of repeated adjustments of stochastic model runs where the occurrence of a rainfall event is first
determined and then the amount of rainfall in each event is generated [e.g., Wilks and Wilby, 1999]. The most
popular approach used in the literature for statistical temporal downscaling of rainfall is a two-step process
where (i) the occurrence of a rain event follows a first-order Markov chain, where the occurrence of rainfall in
any period depends on rainfall in the previous period and transition probabilities that define the likelihood
that wet/dry conditions will persist or transition [e.g., Gabriel and Neumann, 1962] and (ii) the total event
depth follows a gamma distribution [Thom, 1958; Katz, 1977; Coe and Stern, 1984; Wilks, 2011].
Here we present monthly and annual summary statistics for African rainfall using the Tropical Rainfall and
Measuring Mission (TRMM) data [Huffman et al., 2007]. We discuss the spatial and temporal patterns and
trends in rainfall during the 1998–2012 period and provide transition probabilities (or first-order Markov
parameters) and gamma-distribution parameters (shape and scale) of daily and 3 hourly events. Processing
scripts that could be modified by users for their own purposes are made available to assess summary
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rainfall statistics and implement downscaling of monthly rainfall data (e.g., the CRU, GPCC, and GPCP rainfall
data sets) to daily and 3 hourly timescales.
2. Materials and Methods
Tropical Rainfall Measuring Mission (TRMM 3B42v.7) data at 0.25 3 0.25 grid size and 3 h sampling frequency for the 1998–2012 period were used in this study. These data are widely used to understand the
impact of rainfall for water resources management, land surface, and hydrological modeling [e.g., Huffman
€ro
€smarty, 2004; Boone et al., 2009; Kaptue et al., 2013]. Over Africa TRMM retrievals
et al., 2007; Fekete and Vo
have been validated with rain gauge data sets [Nicholson et al., 2003] and they have also been intercompared with other continental rainfall products [Novella and Thiaw, 2013; Sylla et al., 2013]. The TRMM3B42
algorithm, also called ‘‘TRMM and other satellites,’’ combines data streams from multiple satellites, including
data from geostationary infrared (IR) satellite sensors (from the Geostationary Meteorological Satellite, GMS;
the Geostationary Operational Environmental Satellite, GOES; and the National Oceanic and Atmospheric
Administration; NOAA-12) and passive microwave (PM) data (from the TRMM microwave imager, TMI; the
Special Microwave Imager, SSM/I; the Advanced Microwave Scanning Radiometer, AMSR; and the Advanced
Microwave Sounding Unit, AMSU). Monthly rain gauge observations from the Climate Assessment and Monitoring System (CAMS) produced by the NOAA Climate Prediction Center (CPC) and the Global Precipitation
Climatology Center (GPCC) are used to rescale the combined IR and PM estimates [Huffman et al., 2007].
TRMM 3B42 only provides mean rain rates over each nonoverlapping 3 h period. Though a fixed rainless
period (e.g., the minimum interevent time) is required to be equaled or exceeded before and after each
event to identify a single event [Eagleson, 1978; Dunkerley, 2008], in the following, a daily event or storm
corresponds to a wet day and a 3 hourly event to a wet 3 h period of a daily event. After defining a rainy
day as a day with a minimum precipitation depth of 1.0 mm and a rainy 3 h period of a rainy day as a 3 h
period with a precipitation depth >0 mm, we estimate for each of the 12 calendar months multiyear averages of: (i) the monthly rainfall amount (mm) defined as the total amount of rainfall accumulated within a
month, (ii) the rainfall frequency (number) defined as the number of rainy days within a month, (iii) the daily
rainfall duration (h) defined as the number of 3 hourly events in a rainy day multiplied by three (i.e., the frequency of 3 hourly events within a rainy day multiplied by three), and (iv) the daily rainfall rate (‘‘intensity’’;
mm/h) defined as the ratio of the total rain depth in a daily event to the duration of the event. Perpixel correlations between monthly time series (number of rainy months 3 15 years) of these rain event properties
were then estimated with the alternative hypothesis that the correlation values were not equal to zero.
Temporal correlations of the occurrence of daily and 3 hourly events were modeled using a first-order Markov
process. A first-order Markov process can be fully defined by the transition probability of a wet period to a
wet period (w) and the transition probability of a dry period to a wet period (d). We estimated transition probabilities by computing the conditional relative frequencies and validated the assumption that the rainfall
occurrence follows a first-order Markov process by checking whether the unconditional transition probabilities
are sufficiently close to those conditioned on past states using a paired sample t test [Wilks, 2011]. The statistical distribution of the total depth of daily/3 hourly events was also investigated using the gamma probability
density function (PDF) (see supporting information Figure S1). The shape (a) and scale (b) parameters of the
gamma distribution were obtained as maximum likelihood estimates. Once the shape and scale parameters
were estimated, the confidence in accepting or rejecting the theoretical gamma distribution was measured
by the Kolmogorov-Smirnov test [Wilks, 2011]. The relationship between the shape and scale parameters of
the daily and 3 hourly events was then investigated. The parameters (a, b, w, and d) of daily and 3 hourly
events were computed separately for each of the 12 calendar months. Rainfall statistics were calculated for
each month of the year. For brevity and illustration purposes in this paper, we mostly show annual and 3
monthly (‘‘seasonal’’) summaries. Full monthly statistics are available as part of Supporting Information S1. All
the analyses were performed within the R environment [R Development Core Team, 2011].
3. Results
3.1. Annual and Seasonal Average Rainfall Amount, Frequency, Duration, and Rate
Figure 2 illustrates TRMM mean annual precipitation (MAP) and annual frequency of rainy days over Africa
from 1998 to 2012, showing the familiar continental-scale distribution of African climates with the annual
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Figure 2. Annual average total rainfall (a) amount (mm) and (b) frequency (number) of rainy days from 1998 to 2012 (for the seasonal averages, see supporting information Figures S2
and S3). A rainy day is a day with a total precipitation depth 1 mm/d.
precipitation being the total accumulated rainfall during the year. MAP values greater than 2500 mm are
observed in the Eastern part of Democratic Republic of Congo, in Sierra Leone, around Lake Victoria, and
also along the coastal Gulf of Guinea region. MAP values less than 100 mm are observed in the Sahara
Desert and the Namib Desert along the coastal sector of Namibia. Timing of rainfall (see supporting information Figures S2 and S3) highlights the seasonal variations between West, East, and Southern Africa. Frequency of rainy days (Figure 2b) shows broad patterns that are similar to patterns of rain amount (Figure
2a). The smallest average number of rainy days per year (<30) is found in the hyperarid areas (MAP
<100 mm) but the Sahara desert shows regional differences, with the eastern Sahara having the lowest
number of rainy days (<5), while the Northwestern and Southwestern regions have a higher frequency of
rainy days (6–10 and 11–30, respectively).
Figures 3 and 4 show mean rain event duration and rate for the 3 monthly seasons defined by the months
DJF (December to February), MAM (March to May), JJA (June to August), and SON (September to November). Most of the daily rain events lasted 6 h (68% in DJF, 81% in MAM, 60% in JJA, and 74% in SON). During each season, very light storms (rate 1.5 mm/h) occur over 37–54% of the rainfed lands while light
storms (1.5 < rate 2.5 mm/h) occur over 35–50% of the rainfed lands. Heavy (5 < rate 10 mm/h), very
heavy(10 < rate 20 mm/h), and extreme (rate > 20 mm/h) storms contribute 20%, 11%, and 5%, respectively, to the 3 monthly seasonal mean rainfall (supporting information Figure S4). Moreover, heavy to
extreme storms mainly occur over lands with infrequent rainfall (Figure 4 and supporting information Figure
S3). This is in agreement with previous studies that found that the most intense storms occur over lands
with infrequent rainfall [e.g., Zipser et al., 2006; Yang and Nesbitt, 2014].
Relationships between rain event properties were evaluated by correlating monthly rain amount with event
frequency, duration, and rate (Figure 5). Over most of the pixels, there is a positive and significant linear
relationship (i.e., the coefficeint correlation r > 0 and the p-value p < 0.05) between the total rainfall volume
and the other investigated rain event properties (96% for the frequency, 60% for the duration, and 96% for
the rate). In general, the linear relationship between frequency and amount is stronger (i.e., higher r-values)
than the rate-amount relationship which is also stronger than the duration-amount. Subhumid areas (900–
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Figure 3. Monthly average rainfall duration (h) from 1998 to 2012 for the seasons (a) DJF, (b) MAM, (c) JJA, and (d) SON). Areas with no rain events during the corresponding 3 month
period were masked.
1500 mm of MAP; 24% of the continent) and semiarid areas (600–900 mm of MAP; 12%) have the largest frequency-amount r-values (average of 0.85) while the hyperarid areas (MAP < 100 mm; one fourth of
the continent) have the smallest r-values with an average of 0.58 (Figure 5a). For the rate-amount (durationamount) relationship, hyperarid areas exhibit large r-values with an average of 0.67 (0.38) while an average
r-value of 0.49 (0.37) is observed over subhumid areas. Low r-values indicate areas where the monthly precipitation does not change much with rainfall duration or rainfall rate (see, for example, an almost continuous belt roughly from the Congo basin to Ethiopian Highlands in Figure 5b or a latitudinal belt from the
coastline of the Gulf of Guinea to Lake Victoria in Figure 5c). These results reinforce the findings that frequency of events is the primary driver of variations in monthly precipitation in these regions.
3.2. Rainfall Modeling
3.2.1. Transition Probabilities of a Two-State First-Order Markov Chain: w and d
Figure 6 shows wet-after-wet (w) and wet-after-dry (d) probabilities that define the extent to which wet/dry
conditions will persist or transition at a daily and 3 hourly time scale in January and July; w is the probability
of having a wet day/3 h after a wet day/3 h and d is the probability of having a wet day/3 h after a dry day/
3 h. As expected, daily precipitation is more persistent than 3 hourly precipitation (i.e., daily w-values >3
hourly w-values). Similarily, wet-after-wet probabilities are higher than wet-after-dry probabilities at daily
and 3 hourly time scales except over some areas with a monthly precipitation amount less than 50 mm;
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Figure 4. Monthly average rainfall rate (mm/h) from 1998 to 2012 for the seasons (a) DJF, (b) MAM, (c) JJA, and (d) SON). Areas with no rain events during the corresponding 3 month
period were masked. See supporting information Figure S4 for details about the contribution of heavy to extreme storms to the monthly precipitation.
these areas mainly consist of the hyperarid areas and/or the transition areas between rainfed and nonrainfed lands during the selected month. Though the mean monthly rainfall volume of all of Africa is
approximately the same in January and July, daily precipitation is more persistent in January with large daily
w-values (>0.6) over approximately 29% of the rainfed land compared to only 9% in July. Subdaily precipitation is also shown to be more temporally organized in wetter areas than it is in drier areas. For example,
higher 3 hourly w values in the Miombo region of south Central Africa indicate more consecutive rainy
hours whereas higher 3 hourly d values in the Sahel indicate nonconsecutive rainy hours.
Results of the paired-sampled t test establish the Markov chain as an adequate model for the process of
daily precipitation occurrences (supporting information Figure S5). For instance, the wet-after-wet and wetafter-dry probabilities are suitable to fit the occurrence over at least 50% of the daily events with a confidence level of 0.05. For those instances where the null hypothesis is rejected, the use of a Poisson distribution or higher-order Markov chains might be preferable [e.g., Ramirez-Cobo et al., 2014].
3.2.2. Shape (a) and Scale (b) Parameters of the Gamma Distribution
Figure 7 shows the spatial distribution of the shape (a) and scale (b) parameters of the 3 hourly and daily
precipitation depths in January and July. a determines the form of the distribution from a highly skewed
exponential distribution for low values (1) to a more Gaussian-like distribution for high values (>4)
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Figure 5. Correlation coefficient (r) between the 1998 and 2012 monthly time series of rainfall amount (mm) and (a) frequency (number), (b) duration (h), and (c) rate (mm/h) of the rainy
days with the alternative hypothesis that the correlation values were not equal to zero. Nonsignificant (p 0.05) and negative (r < 0) values were masked.
Figure 6. Wet-after-wet (w) and wet-after-dry (d) probabilities of daily (top) and 3 hourly (bottom) rain events in (left) January and (right) July across the African continent. Gray colors
show areas where the transition probabilities were not computed because their multiyear monthly average rain event frequency is less than 1 (see supporting information Figure S3).
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Figure 7. Shape (a) and scale (b) parameters for the gamma distribution of the total daily (top) and 3 hourly (bottom) precipitation depths in (left) January and (right) July across the African continent. Gray colors show areas where the gamma parameters were not computed because their multiyear monthly average rain event frequency is less than 1 (see supporting
information Figure S3).
(supporting information Figure S1a) and b stretches the distribution out while maintaining the skew given
by a. Since large b-values correspond to areas having large variance in comparison to the mean [Groisman
et al., 1999; Husak et al., 2007; Becker et al., 2009], large b-values can indicate areas prone to heavy and
extreme precipitation. While there are very few areas of large daily b-values (>30), large 3 hourly b-values
(>30) are predominant over the wet areas (56% in January and 42% in July) indicating that heavy 3
hourly events tend to be relatively short-lived. As expected, low daily (3 hourly) precipitation depths dominate in Africa given that a 1 over at least 63 (76) % of the rainfed lands in January and July. Consequently,
3 hourly precipitation depth distributions are more strongly skewed (i.e., have a smaller shape parameter)
than daily precipitation depth distributions. Values of a > 1, both for daily and 3 hourly precipitation depth
distributions, occur mainly over areas with a monthly precipitation amount less than 50 mm (Figure 4 and
supporting information Figure S2). By excluding such areas, there is little spatial variability in a for both the
daily and 3 hourly distributions. This has already been shown for monthly precipitation amounts over the
entire continent [Husak et al., 2007; Naumann et al., 2012].
Results of the Kolmogorov-Smirnov test reveal that the gamma distribution is adequate for modeling the
total daily precipitation depths (supporting information Figure S6). In this statistical test, the null hypothesis
is that the observed data are drawn from the fitted gamma distribution and therefore, a small p-value is
cause for rejection of the null hypothesis, while a p-value larger than the selected significance level means
that the null hypothesis cannot be rejected. In our analysis, less than 3% of the points with daily events had
a calculated p-value smaller than 0.05 indicating a rejection of the gamma distribution for those sites.
Nevertheless, the use of other distributions where the gamma distribution is not suitable (i.e., where the
null hypothesis that the rainfall depths follow a gamma distribution is rejected) is out of the scope of this
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Figure 8. Count frequency distribution of an example of downscaling of monthly GPCP data to daily data for two locations along a latitudinal rainfall gradient in August 1997 (i.e., prior
to the TRMM era). Daily GPCP data were downloaded from ftp://rsd.gsfc.nasa.gov/pub/1dd-v1.2, summed to monthly totals, and then downscaled. Downscaled (blue) and observed
(red) daily data are shown along with their total monthly amount (amnt, mm), the number of events (freq), their mean event size (mesz, mm/d), their wet-after-wet probability (w), and
their wet-after-dry (d) probabilities. Yellow colors indicate overlapping of downscaled (blue) and observed (daily) data. The multiyear monthly or climatology (1998–2012) of these statistics (gray) is also shown for comparison purposes. Note that by construction (i) the frequency of simulated rainy days is equal to the multiyear monthly frequency multiplied by the ratio
of the multiyear monthly amount by the monthly amount to be downscaled, (ii) the simulated mean event size is equal to the multiyear monthly mean event size, (iii) the absolute difference between the simulated and climatological transition probabilities is less than 0.05, and (iv) the simualted daily amounts are extracted from a gamma distribution defined by the climatological shape and scale parameters (p < 0.01, Kolmogorov-Smirnov test).
paper. It is worth noting that when the parameters of the distribution have been estimated from all available data, it is recommended to use the Lilliefors test which is a modification of the Kolmogorov-Smirnov
test because the probability of accepting a false null hypothesis becomes high with the KolmogorovSmirnov test [Wilks, 2011]. Nevertheless, the Kolmogorov-Smirnov is usually applied in practical meteorological or climatological application where the gamma distribution is the one to be tested because of the nonexistence of the counterpart of the Lilliefors test for the gamma distribution and building this counterpart is
out of the scope of this paper.
To complete the validation process, we test wether (ad, bd) and (an, bn/n) are statistically the same with ad,
bd (an, bn) being the scale and shape of daily (hourly) events and n the frequency of 3 hourly events (i.e.,
one third of the duration of daily events displayed in Figure 3). The patterns of rejection of the null hypothesis (supporting information Figure S7) strengthen the conclusion of supporting information Figure S6 that
daily events are gamma-distributed for any month in regions like the Sahel-Sudan, Horn of Africa, and North
Africa. In such regions, the gamma distribution parameters of 3 hourly events can be scaled to describe rainfall for events of longer duration. That is, the gamma distribution is suitable for representing rainfall at a
variety of timescales from subdaily accumulation to daily and hence monthly and seasonal accumulations.
3.2.3. DownScaling Application
The downscaling process of monthly precipitation to daily is performed in two-steps. First, a sequence of
dry and wet days according to the two-state first-order Markov process characteristic of that month is generated (see supporting information Text S1 for theoretical details and synthetic generation procedure).
Then, for each wet day, the precipitation amount is obtained by sampling from the gamma probability distribution function characteristic of that month. Figure 8 shows the count frequency distribution of an example of downscaling monthly total GPCP rainfall for August 1997 to daily events for two locations along a
rainfall latitudinal gradient. The daily GPCP were summed to monthly totals and the monthly totals were
then downscaled where (i) the rainfall occurrence modeling step was repeated until the simulated transition
probabilities resemble the climatological one (i.e., the absolute value of their difference is lower than a
specified limit) and the frequency of simulated rainy days resembles the multiyear monthly frequency multiplied by the ratio of the multiyear monthly amount by the monthly amount to be downscaled, and (ii) the
rainfall amount modeling step—that consists in selecting daily events that follow the gamma distribution
with the climatological shape and scale parameters—was repeated until the simulated mean daily precipitation depth resembles the multiyear monthly mean daily precipitation depth. Hence, by construction, the
simulated data preserve the statistical characteristics (event size, transition probabilities, gamma
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parameters) of the TRMM rainfall data acquired during the period 1998–2012. However, instead of adjusting
only the frequency of rainfall and keeping the mean daily precipitation depth constant as in this example,
the user could also adjust both the frequency and the mean daily precipitation depth, or adjust only the
mean daily precipitation depth. Such choices could be based either on the correlation maps (Figure 5) or on
the user knowledge of the region of interest.
4. Discussion
4.1. Limitations of the Definition of a Rain Event
Conventionally, a given day (or 3 hourly period) is declared rainy if the observed rainfall depth equals or exceeds
an arbitrary threshold. However, if the threshold is set too low, there will be many hydrologically meaningless
data retained for the analysis while if the threshold is too high, there will be only few events left for analysis and
the results may be highly sensitive to the depths recorded in such few events [Reiser and Kutiel, 2009]. The fixed
thresholds for daily (1 mm/d) and 3 hourly (>0 mm/h) precipitation events used in this study are not necessarily meaningful for all modeling purposes of African climates. Over hyperarid areas (MAP < 100 mm), for instance,
total daily precipitation depths of <l mm/d represent 7.7% of total of daily precipitation depths 0 mm/d (supporting information Figure S8). This is in agreement with the findings of Dai [2006] who showed that days with
rainfall less than 1 mm contribute 8% of the average of the total rainfall in the tropics (50 S–50 N). To account
for the spatial heterogeneity of rainfall, some researchers have proposed the use of spatially varying thresholds
as a percentage of the annual or seasonal mean daily rainfall [e.g., Ratan and Venugopal, 2013]. However, the
challenge remains in defining the appropriate percentage. Consider, for instance, the month of January and two
locations: one in Tunisia (36.5N, 9.5W) where it rained 90 mm in 3 days and the other one in Zambia (13.25N,
26.5E) where it rained 300 mm in 20 days. Five percent of the multiyear monthly amount will yield two threshold values of 1.5 mm in Zambia and 0.15 mm in Tunisia which are not meaningful given that the mean event
size (calculated on rainy days) is 15 mm in Zambia and 30 mm in Tunisia.
Although a sequence of consecutive rainy days (or rainy 3 hourly periods) could constitute the same meteorological event, this work assumes that each rainy period is a different event. This may not be necessarily
adequate in each climate region. The more humid a region and/or season, the more likely it is that consecutive wet periods might correspond to the same meteorological event.
4.2. Gamma Distribution in Rainfall Modeling
Using the gamma distribution, it was possible to fit the distribution of rainfall depths at a monthly time
scale except over a very small number of pixels for 3 hourly precipitation (8 and 56 pixels in January and
July, respectively). However, given the versatility of the gamma distribution (e.g., supporting information
Figure S1), our results indicate that it is well-suited to represent the distribution of rainfall depths in Africa,
as shown for other areas in previous observations and simulations [see, for example, Groisman et al., 1999;
Becker et al., 2009]. Nonetheless, gamma distributions underestimate the probabilities of extreme storms
(rate > 20 mm/h) for a > 1 (i.e., for distributions that exibits a peak; see supporting information Figure S1b)
making such gamma-PDFs not suitable for the representation of extreme events. These findings are in
agreement with various observation-based and simulation-based studies [see, for example, Osborn and
Hulme, 2002; Panorska et al., 2007].
4.3. Application
There are various possible applications of these rainfall statistics (transition probabilities and gammadistribution parameters). For example, they could be used in ecological modeling where total rainfall
amount exerts a strong control on the vegetation structure [Sankaran et al., 2005; Good and Caylor, 2011],
ecosystem carbon fluxes respond to changes in rain pulses [Williams et al., 2009] and perturbations of the
interannual rainfall variability can induce shifts of the vegetation into alternative states [Higgins and Scheiter,
2012]. They could also be used by soil scientists to model soil erosion [Wei et al., 2009; Symeonakis and
Drake, 2010] which is one of the most serious environmental problems in Africa with respect to its negative
impacts on agricultural production, infrastructure, and water quality [Vrieling, 2006]. It is worth recalling that
water erosion accounts for 55% of total continental soil erosion [Lal, 1995]. It is also possible to use these
rainfall statistics in epidemiological research to simulate potential areas for malaria transmission (i.e., wet
areas where anopheles mosquito may breed) by analyzing surface runoff [Yamana and Eltahir, 2011]. This is
very important because malaria is one of the most deadly diseases in Africa [Rao et al., 2006]. Further, the
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dependency between the scale and shape parameters (their product is equal to the mean) could be used to
identify areas where occasional drought (large scale, small shape) or heavy rainfall (large shape, small scale)
may have a significant impact on agriculture [Husak et al., 2007]. Because it is best practice to use statistical
properties for comparison of meteorological variables obtained from two different sources [Liebmann et al.,
2012; Yang and Nesbitt, 2014], our approach would also allow for an evaluation of satellite products available at various temporal resolutions over Africa [e.g., Jobard et al., 2011].
5. Conclusions
This study presents an analysis of rainfall characteristics (amount, frequency, duration, and intensity) over
Africa from 1998 to 2012. The estimated properties of observed precipitation reveal many interesting features such as the spatial and temporal variability of light and heavy rainfall. High values of correlations
between the monthly time series of the rainfall amount and frequency, rainfall amount and duration, rainfall
amount, and rate were found and mapped. These correlation maps could be used to indicate areas where
the same amount of rainfall with different frequency and rate could lead to different surface runoff, evaporation, and soil condition.
The Kolmogorov-Smirnov test showed that the gamma distribution is adequate for fitting the observed distribution of rain amounts at daily and 3 hourly scales. In addition to the parameters of the gamma distribution of storm depth, other statistical rainfall descriptors such as the wet/dry transition probabilities of a firstorder Markov chain were also estimated. Such information on the disaggregation of rainfall could be used
in a variety of tools to estimate the impacts of historic rainfall events or future rainfall scenarios in order to
help decision makers respond to extreme climatic events (droughts, floods). However disaggregation of
rainfall at the local scale would not be possible because of the spatial resolution of the TRMM data sets
(0.25 ). In this regard, future work by the authors will focus on merging these summary statistics with high
spatial resolution rainfall estimates (5 km) like TRMM 2B31 to account for the subgrid spatial distribution
of rainfall within the coarse-scale 0.25 grid cells [e.g., Tarnavsky et al., 2012].
Acknowledgments
The authors would like to
acknowledge the staff of the Goddard
Earth Sciences Data and Information
Services Center (GES DISC) responsible
for customizing and sharing the TRMM
data sets. The rainfall statistics can be
downloaded from Distributed Active
Archive Centers (DAAC) at ftp://daac.
ornl.gov/data/global_climate/African_
Rainfall_Patterns. This research was
supported by the NSF Coupled Natural
and Human systems grant DEB
1010465.
KAPTUE ET AL.
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