Atmospheric Research 132-133 (2013) 12–21
Contents lists available at SciVerse ScienceDirect
Atmospheric Research
journal homepage: www.elsevier.com/locate/atmos
Estimation of dry spells in three Brazilian regions — Analysis
of extremes
José Ruy Porto de Carvalho a,⁎, Eduardo Delgado Assad a,
Silvio Roberto Medeiros Evangelista a, Hilton da Silveira Pinto b
a
b
Embrapa Informática Agropecuária, CP: 6041, CEP: 13083-886 Campinas, SP — Brazil
CEPAGRI-Universidade Estadual de Campinas, CEP: 13083-970 Campinas, SP — Brazil
a r t i c l e
i n f o
Article history:
Received 22 October 2012
Received in revised form 6 February 2013
Accepted 16 April 2013
Available online 1 May 2013
Keywords:
Dry spells
Generalized extreme value distribution
Number of rainy days
Return period
a b s t r a c t
The aim of this study was to model the occurrence of extreme dry spells in the Midwest,
Southeast and Southern regions of Brazil and estimate the return period of the phenomenon
indicating the time when the occurrence is more severe. The generalized extreme value
distribution was the best fit for a series of maximum dry spell number and the parameters
estimated by the maximum likelihood method. The data series adherence to the probability
distribution was verified by the Kolmogorov–Smirnov test and the percentile–percentile charts.
The positive trend of dry spells was verified by the Mann–Kendall test and non-stationarity
rejected by Dickey–Fuller and augmented Dickey–Fuller tests. The irregular distribution of
rainfall in the growing season for the Midwest region has increased the number of dry spells.
The increase of rainy days in the Southeast and the South resulted in a decrease of dry spells in
these regions. Regarding the return period of one year, dry spells occurred from 5 to 25 days in
the Midwest region meaning a loss of productivity for Brazilian agriculture if it happened
between the flowering and grain filling phases, making it, therefore the region with the largest
agricultural risk. When the intensity of the dry spells was analyzed for different return periods,
the Southern region was the most vulnerable.
© 2013 Elsevier B.V. All rights reserved.
1. Introduction
The rainfall in most of the Brazilian states is not uniformly
distributed in all seasons. The rainy and dry periods alternate.
The major crops are usually growing under rain-fed conditions which make them rely solely on natural rainfall. The
irrigated crops are still in the minority (4% of total agricultural
production), especially with regard to large planted areas.
Thus, farming becomes exclusively seasonal, being practiced
mainly in the rainy season known as “rain-fed crops” (Sousa
and Frizzone, 1997). In the Midwest, Southeast and Southern
regions, the summer period is characterized by the occurrence of rainfall which normally meets the hydration needs of
developing major crops. However, the rainfall data does not
⁎ Corresponding author.
E-mail address: [email protected] (J.R. Porto de Carvalho).
0169-8095/$ – see front matter © 2013 Elsevier B.V. All rights reserved.
http://dx.doi.org/10.1016/j.atmosres.2013.04.003
provide precise information on the climate or on the existence
of adequate conditions for the cultivation of certain plant
under rain-fed conditions. The parameters of water storage
in the soil and the gains should be analyzed along with
the losses of humidity from the soil–plant–atmosphere
system (Mota et al., 1992). A phenomenon known as dry
spell, a sequence of dry days during the rainy season, is still a
common occurrence. The occurrence of dry spells can be
extremely harmful for agriculture, especially in periods in
which the plants have need of water, that is, at critical periods
such as flowering and fruiting which will have a serious effect
on the final yield (Assad et al., 1993).
To reduce the economic losses in agriculture, the knowledge of the temporal and spatial distribution of precipitation
and consequently periods of dry spell is important. A dry spell
affects not only agriculture, but also other sectors such as
fisheries, health, electricity, and so on. Electricity generation
J.R. Porto de Carvalho et al. / Atmospheric Research 132-133 (2013) 12–21
in hydroelectric facilities may suffer interruptions in dry
periods (Marengo, 2009). Information about the intensity of a
drought could be used to determine which crop or variety
should be used in a particular location. Therefore, the effects
of a dry spell ultimately have a direct impact on the economy
of a nation.
The occurrence of prolonged periods of drought is particularly common in the Midwest, but also occurs in the
Southeast and Southern regions. According to Barbosa (1986)
and Affolder et al. (1997), the occurrence of irregular dry
spells from year to year makes the corn crop vulnerable to
water stress at any stage of development with visible damage
on its production. For maize, Espinoza et al. (1980) found
reductions of up to 60% in crop yield when water deficits
occurred between the stage of flowering and grain filling, and
40% when it occurred during floral initiation.
Sousa and Frizzone (1997) simulated production declines
in corn growth of up to 65% for dry spells occurring at
flowering stage in Piracicaba, Brazil. A major limitation to
soybean production in the cerrado (Brazilian Savanna) is
due to the dry spell, which significantly affects production.
Espinoza (1982) found yields higher than 24–55% with
irrigated soybean crops in relation to which water was the
limiting factor. Magalhães and Millar (1978) found reductions of 20%, 38% and 52% in the production of beans
undergoing dry spell of 14, 17 and 20 days occurring from
the start of flowering, respectively. Guimarães et al. (1982)
observed reductions of 49% in the cultivation of beans
subjected to water stress. Stone et al. (1986), in studying
the effects of dry spells in rice, concluded that the production
of grain and dry matter yields were negatively affected by
increases in periods of dry spells.
Numerous studies have been conducted to model the
distribution of rainfall in several regions of the world. The
result of such studies is useful in decisions regarding planning
and management of water resources of the involved countries. Statistical distributions are used to model the long-term
characteristics of precipitation parameters and knowledge
of how the dry spell distribution plays an important role in
this respect.
Several authors have used different statistical distributions to model the behavior of a dry spell. The Extreme
Value distribution was used in the work of Assad et al.
(1993), Henriques and Santos (1999), Lana et al. (2008),
Vicente-Serrano and Beguería-Portugués (2003) and Lana
et al. (2006); the Gamma distribution in Ribeiro et al. (2007)
and Dan'azumi and Shamsudin (2011); the Geometric
distribution in Mahamud et al. (2011) and Deni and Jemain
(2009), Log-Normal distribution in Dan'azumi and Shamsudin
(2011), the Pareto distribution in Lana et al. (2006) and
Dan'azumi and Shamsudin (2011) and the Truncated Negative Binomial distribution in Deni et al. (2008) and Giuseppe
et al. (2005). Arruda and Pinto (1980a, 1980b) developed
simplified models for the distribution of rainfall and distribution of drought periods which were applied to agricultural
production in Brazil.
The objective of this study was to model the occurrence of
extreme dry spells in the Midwest, Southeast and Southern
regions of Brazil and estimate the return period of the
phenomenon, indicating the region and frequency where the
occurrence is more severe.
13
2. Material and methods
The daily rainfall dataset was obtained from rainfall records
of weather stations belonging to the National Water Agency
ANA (2011) covering the three Brazilian macro regions for the
period between 1940 and 2011. The data refer to the Midwest
region with 1,228,562 observations from 559 rainfall stations,
the Southeast region with 6,474,928 observations from 2941
rainfall stations and the Southern region with 2,917,075
observations from 1325 rainfall stations. Several studies
published in Brazil, such as Farias et al. (2001), Pinto et al.
(2001), among others, worked together to establish the best
planting dates to coincide with the non-occurrence of dry
spells during grain filling for soybean, maize and rice,
respectively. The correct identification of these periods in
terms of space-time, that is location and duration of periods of
sporadic droughts defining the policy of climate risks agricultural zoning (Rossetti, 2001), has been widely adopted by the
Ministry of Agriculture, Livestock and Supply of Brazil. The data
refer to the maximum number of dry spell length per year for
the months of October, November, December, January and
February summer season for the three regions.
A computer program was developed, written in SAS (SAS,
2010) for calculating the number of dry spells, with at least
10 consecutive days without rainfall for each rainfall station.
Every time a dry spell is recorded with rainfall greater than
zero, verification is made to be sure the number of days
without rain has exceeded ten. If this occurs, a dry spell is
registered with the number of days without precipitation.
The counter will then be cleared and will continue to account
for the other records. When a new year begins, the method
also resets the counter and the dry spell number. For the set
of dry spells calculated for each rainfall station, the maximum
number of dry spell length per year was used, totaling 71
observations (one for each year) for Midwest, Southeast and
Southern regions. The graphs were constructed using the
SGPLOT procedure (SAS, 2008).
In this work, four models, generalized extreme value
distributions — GEV, Lognormal, Pareto and Gamma distributions (Johnson and Kotz, 1970) were fitted to the observed
values x corresponding to the period of maximum annual dry
spell length, with the following probability density functions
and cumulative distribution functions below:
1 — Generalized extreme value
• Domain
1þk
ðx−μ Þ
> 0 para k ≠ 0
σ
−∞ b x b þ ∞ para k ¼ 0
• Probability density function
8
!
1
>
>
> 1 exp −ð1 þ kzÞk ð1 þ kzÞ−1−1=k
<
f ðxÞ ¼ σ
>
>1
>
: expð−z− expð−zÞÞ
σ
k≠0
k¼0
ð1Þ
14
J.R. Porto de Carvalho et al. / Atmospheric Research 132-133 (2013) 12–21
• Cumulative distribution function
F ðxÞ ¼
8
< exp −ð1 þ kzÞ−1=k
k≠0
: expð− expð−zÞÞ
k¼0
ð2Þ
where k = shape parameter, σ = scale parameter μ = location parameter and z≡ x−μ
σ . The shape parameter k can be used
to model a large number of distribution tails. The case where
k b 0 corresponds to the Weibull distribution; for k > 0
corresponds to the Fréchet distribution; when k = 0 on the
Gumbel distribution.
2 — Lognormal
• Domain
−y b x b þ ∞
• Probability density function
lnðx−y−μ Þ z
exp −1=2
σ
pffiffiffiffiffiffi
f ðxÞ ¼
ðx−yÞσ 2π
• Cumulative distribution function
lnðx−yÞ−μ
F ðxÞ ¼ Φ
σ
ð3Þ
ð4Þ
where σ and μ are continuous parameters (σ > 0) and
γ = location parameter and Φ is the Laplace integral.
3 — Pareto distribution
• Domain
β≤xb þ∞
• Probability density function
f ðxÞ ¼
α
αβ
xαþ1
• Cumulative distribution function
α
β
F ðxÞ ¼ 1−
x
ð5Þ
ð6Þ
where α = shape parameter (α > 0) and β = scale parameter (β > 0).
4 — Gamma distribution
• Domain
γ≤xb þ∞
• Probability density function
f ðxÞ ¼
ðx−γ Þα−1
−ðx−γ Þ
exp
α
β Γðα Þ
β
ð7Þ
• Cumulative distribution function
F ðxÞ ¼
Γðx−γ Þ=β ðα Þ
Γðα Þ
ð8Þ
where α = shape parameter (α > 0), β scale parameter
(β > 0) e and γ = location parameter.
The Kolmogorov–Smirnov adherence test (Wilks, 2006)
was used to verify the degree of adjustment of the x series
to the probability density function. The statistic D of
Kolmogorov–Smirnov test is based on the largest vertical
difference between the functions of cumulative theoretical
and empirical distributions.
D¼
max
i−1 i
F ðxi Þ−
; −F ðxi Þ
1≤i≤n
n n
ð9Þ
where F′(x) is the cumulative empirical frequency of the
maximum annual dry spell length values and F(x) is the
cumulative frequency given by Eq. (9).
The null and alternative hypotheses are set out below:
• H0: the observed data follow a particular distribution;
• HA: the observed data do not follow a particular distribution.
The hypothesis is rejected on the fit of a particular
distribution at a level of statistical significance α if D is greater
than the critical value obtained from theoretical tables.
To check the fit of the distribution with respect to
the observed data, percentile–percentile (PP) graphs were
presented factoring in the abscissas the empirical cumulative
probability on the ordinate, the cumulative theoretical probability estimated through the density function. The graph is
approximately linear if the theoretical distribution is the
correct model.
According to Sansigolo (2008) and Blain and Moraes
(2011), an important feature of the extreme value distribution is that it assumes that there are no systematic variations
in the observed period. The nonparametric Run or Wald–
Wolfowitz test (Wald and Wolfowitz, 1940; Thom, 1966), the
Mann–Kendall test proposed by Mann (1945) and studied by
Kendall (1975) and improved by Hirsch and Slack (1984) and
the Pettitt test (Pettitt, 1979), are the most widely used to
study the characteristics of the series of daily maximum
rainfall.
Let's examine the series x = {x1,x2,…,xn}. The nonparametric Mann–Kendall (MK) test is defined as: T =
∑j bisignal(xi − xj), with i,j ∈ n where signal(xi − xj) = {1 for
(xi − xj) > 0; 0 for (xi − xj) = 0 and −1 for (xi − xj) b 0}.
Considering temporal independence between observations
under the null hypothesis, there is no presence of trends, T is
normally distributed with mean E(T) = 0 and variance Var(T)
T−1
T þ1
MK ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffi for T > 0; 0 for T ¼ 0; pffiffiffiffiffiffiffiffiffiffiffiffiffiffi for T < 0 ð10Þ
Var ðT Þ
Var ðT Þ
Þð2nþ5Þ
:
where Var ðT Þ ¼ nðn−118
Coles (2004), Fawcett and Walshaw (2008), Gilleland and
Katz (2005) and Marty and Blanchet (2011) state that for the
types of data to which the extreme value theory is applied,
the temporal independence is an unrealistic assumption.
Extreme conditions persist over many consecutive observations. A detailed investigation requires a mathematical treatment with a high level of sophistication that can be found
in Leadbetter et al. (1983). The generalization of a sequence of
independent random variables is the stationary series. Stationarity is a more realistic assumption for most of the physical
process and corresponds to a series where the variables may
be mutually independent, but the stochastic properties are
J.R. Porto de Carvalho et al. / Atmospheric Research 132-133 (2013) 12–21
Fig. 1. Maximum yearly dry spell in days for the Midwest, Southeast and Southern Brazilian regions.
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J.R. Porto de Carvalho et al. / Atmospheric Research 132-133 (2013) 12–21
homogeneous over time. That is, if a series x1,x2,…,xn is a
stationary series then x1 has the same distribution of x100 and
the joint distribution of (x1, x2) has the same distribution
of (x100, x101) although x1 need not be independent of x2 or x100.
Wilks (2006) states that the data comprising a series of
extreme values do not come from the same distribution.
However empirically, this theoretical distribution is often
appropriate even when not all assumptions are met.
To test the stationarity of the observed series, the unit
root test was applied (Dickey and Fuller, 1979, 1981). We
considered the stochastic model based on the difference of
a first order autoregressive process, the difference of which
generates the model described below:
∇xt ¼ φxt−1 þ εt
ð11Þ
where:x0 = fixed initial value; ∇xt = (xt − xt − 1) → is the
difference operator; ϕ = ρ − 1 → time series autoregressive
coefficient; εt = sequence of random variables independently
and identically distributed.
The null hypothesis is that xt is non-stationary, that is,
there is an autoregressive unit root and φ = 0, against the
alternative hypothesis that xt is an AR (1) process, in this case
there is no unit root and consequently φ b 0. To conduct this
hypothesis test, an estimation process is used, the Ordinary
Least Square model. In the presence of trend and intercept,
the equation to be used is as follows:
∇xt ¼ α þ βt þ φxt−1 þ εt
ð12Þ
where α is the intercept and the linear trend t.
With the Dickey–Fuller test, part of the assumption is
that error terms in the above equations are independently
and identically distributed, that is, they do not exhibit
autocorrelation. The Dickey–Fuller augmented test (Dickey
and Fuller, 1979, 1981) incorporates lags in relation to the
variable which is being analyzed so that the errors do not
exhibit autocorrelation. The equation is:
p−1
∇xt ¼ α þ βt þ φxt−1 þ ∑j¼1 ρjþ1 ∇xt−j þ εt :
ð13Þ
The parameters of Eqs. (11), (12) or (13) and its significances can be estimated by the autoregressive procedure
(SAS, 2010).
After calculating the distribution function associated with
the maximum annual dry spell length for the three regions
studied, their return period was estimated by:
RðF ðxÞÞ ¼
1
1
ð1−F ðxÞÞ
year
ð14Þ
where F(x) is the cumulative probability of occurrence of a
given value of the maximum dry spell period for each region
and 1/year denominator is the average sample frequency of
the annual dry spell maximum period.
February are representative of the rainy season (growing
season) as shown in Fig. 1. Visually, some important features
may be observed. There is a clear, increased trend in the
annual maximum dry spell period of the Midwest region but
a decreased dry spell period in the Southeast and Southern
regions. The increasing trend associated with the migration
for medium and early crop cycles increases the chances for
productivity losses, particularly for soybean and corn.
The statistical confirmation of the trend obtained in Fig. 1
can be seen in Table 1 which shows the results of applying
the nonparametric Mann–Kendall according to Eq. (10).
As the calculated p-value for the three regions is less than
the significance level (α = 0.05), the null hypothesis (H0:
there is no trend in the series) should be rejected in favor of
the alternative hypothesis (HA: There is a positive trend in
the series). The risk of rejecting the null hypothesis when it is
true for the Midwest, Southeast and South is, respectively,
0.21%, 4.97% and 1.51%.
The number of non-consecutive rainy days for each
precipitation station was calculated and the maximum yearly
rainy days for the growing season can be seen visually in
Fig. 2. There is an increasing trend in the number of rainy
days for these three Brazilian regions.
Table 2 shows the descriptive statistics for the maximum
yearly rainy days for the three regions. The data series seen
here presents similar mean values for the Midwest and
Southern regions despite the dispersion around the mean
being greater for the South. The average for the maximum
yearly rainy days increases around 19% for the Southeast in
relation to the other two regions. The dispersion around the
mean can be considered low for all three regions.
The statistical confirmation of the trend obtained visually
in Fig. 2 can be seen in Table 3 where the results of applying
the nonparametric Mann–Kendall test according to Eq. (10)
are presented.
As the p-value calculated for the Southeast and Southern
regions is less than the significance level (α = 0.05), the null
hypothesis is rejected (H0: there is no trend in the series) in
favor of the alternative hypothesis (HA: There is a positive
trend in the series). The risk of rejecting the null hypothesis
when it is true for the Southeast and Southern regions is
respectively 4.69% and 2.13%. For the Midwest region, the
p-value is greater than calculated, so the null hypothesis
(H0: there is no trend in the series) should not be rejected
with a risk of 8.20%. Several authors have used nonparametric Mann–Kendall test to study trends in climatic variables,
such as Subash et al. (2011a, 2011b), Casa and Nasello
(2012), and Shahid et al. (2012).
As the amount of maximum yearly rainy days for the
growing period is increasing significantly in the Southeast
and Southern regions, the maximum number of dry spell
length is decreasing. As there is no significant trend in the
Table 1
Mann–Kendall test for the maximum number of days of dry spell for the
three regions (Midwest, Southeast and Southern).
3. Results and discussion
Periods of maximum annual dry spells observed for the
three Brazilian regions (Midwest, Southeast and Southern)
for the months of October, November, December, January and
Region
Mann–Kendall
p-value
Midwest
Southeast
Southern
0.252
−0.161
−0.118
0.002
0.049
0.015
J.R. Porto de Carvalho et al. / Atmospheric Research 132-133 (2013) 12–21
Fig. 2. Maximum yearly rainy days for the Midwest, Southeast and Southern Brazilian regions.
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J.R. Porto de Carvalho et al. / Atmospheric Research 132-133 (2013) 12–21
Table 2
Descriptive statistics for the maximum yearly rainy days for all three regions
(Midwest, Southeast and Southern) at growing time.
Region
Mean
Std deviation
Std error mean
Midwest
Southeast
Southern
104.98
122.62
102.59
10.26
13.94
16.27
1.22
1.68
1.94
Table 4
Maximum likelihood estimation of the parameters of GEV, Lognormal,
Pareto and Gamma distributions and Kolmogorov–Smirnov adherence test D
for the three regions.
Region
Midwest
series of maximum yearly rainy days for the Midwest region,
the increasing maximum number of dry spell length is
associated with the uneven distribution of rainfall in the
growing period for this region. Similar result was found in
Assad et al. (1993), Barbosa (1986), Espinoza (1982), and
Farias et al. (2001). These studies indicate the high coefficient
of variation for the growing season in the Midwest region.
Table 4 presents the parameter estimates of the generalized extreme value distributions (Eq. (1)), Lognormal
(Eq. (3)), Pareto (Eq. (5)) and Gamma (Eq. (7)) obtained by
the maximum likelihood method, and the result of the
Kolmogorov–Smirnov adherence test (Eq. (9)) to check the
degree of adjustment of the x series to the probability density
function.
Table 4 shows that the generalized extreme value distribution is the analytic function that best describes the frequency of
occurrence of the annual values of maximum periods of dry
spell length for the three regions with the lowest values of the
statistic D for α = 5%. In the Midwest, as k > 0, the Fréchet
distribution was the best fit. In the Southeast and Southern
regions, as k b 0, the distribution that best represents the
observed data was the Weibull.
Fig. 3 shows the percentile–percentile (PP) graphs used
to check the fit of theoretical distributions in relation to
the observed data. You can visually check the good fit of the
annual maximum dry spell length data series in relation to
the accumulated probability (theoretical and empirical)
estimated by the density function. Even the extreme upper
and lower distributions of the graph show approximately
linear, indicating that the adjusted distribution is correct.
As Coles (2004) states, the stationarity assumption is more
realistic for most physical processes and sufficient for the
application of the extreme value theory. The non-stationarity
was rejected by the Dickey–Fuller test and Dickey–Fuller
augmented test (Dickey and Fuller, 1979, 1981) according
to Eqs. (12) and (13) respectively, through the use of the
autoregressive procedure of SAS software (SAS, 2010) using
stationarity option = (ADF). Table 5 shows the results obtained
for the Dickey–Fuller test.
For the three regions, setting a 5% level of significance, the
probabilities obtained are lower indicating therefore that the
series with the maximum dry spell length values are apparently stationary, that is, the classical statistical assumptions
Table 3
Mann–Kendall test for the maximum number of rainy days for all three
regions (Midwest, Southeast and Southern) at growing time.
Region
Mann–Kendall
p-value
Midwest
Southeast
Southern
0.321
0.324
0.505
0.082
0.047
0.021
Southeast
Southern
Distributions parameters/D test
GEV
Lognormal
Pareto
Gamma
k = 0.2221
σ = 12.539
μ = 28.344
D = 0.0962
k = −0.0435
σ = 15.443
μ = 47.42
D = 0.0759
k = −0.0224
σ = 10.899
μ = 39.467
D = 0.0569
σ = 0.63401
μ = 3.2678
γ = 7.0316
D = 0.1173
σ = 0.3069
μ = 4.0387
γ = −3.7681
D = 0.0778
σ = 0.2257
μ = 4.0305
γ = −12.227
D = 0.0765
α = 0.9492
β = 12
D = 0.2994
α = 1.8269
β = 15.273
γ = 11.166
D = 0.1504
α = 4.2761
β = 9.098
γ = 16.791
D = 0.0817
α = 11.258
β = 3.9129
γ = 1.4683
D = 0.0818
α = 1.4916
β = 27
D = 0.316
α = 161.21
β = 5175.7
D = 0.5515
remain valid. As the unit root tests have low power, the
inclusion or removal of parameters, such as the constant and
the trend may change the results and thus a conclusion about
stationarity of this variable cannot yet be considered definitive.
Table 6 shows the results for the three regions of the
Dickey–Fuller augmented test with the average and trend
included (ADF1) and (ADF2), respectively.
From Table 6, the inclusion of parameters for the three
regions does not change at the 5% significance level. The null
hypothesis rejects that the series is non-stationary, that is,
there is an autoregressive unit root in favor of the alternative
hypothesis that the series studied presents stationarity and in
this case there is no unit root.
The previous analysis provides technical support for using
the extreme value theory, employing the density distribution
function defined in Eq. (1) to describe the frequencies of
the occurrence of dry spell maximum annual values for the
three regions studied. Several studies have shown good
results by adjusting the generalized extreme value distribution to dry spell data, such as Assad et al. (1993), Henriques
and Santos (1999), Lana et al. (2008), Vicente-Serrano and
Beguería-Portugués (2003) and Lana et al. (2006).
The return period expressed in years (Eq. (14)) for the
three Brazilian regions is presented in Table 7 based on the
maximum estimated annual period.
It can be seen in Table 7 that the Midwest region has a
higher probability of dry spells than the other regions. For
a 90-day dry spell, the return periods for the Midwest,
Southeast and Southern regions at the agricultural growing
season are respectively 28, 19 and 134 years. However, dry
spells over as little as 14 days can cause reduced productivity
of more than 20%, and in the case of the three regions for a
return period of one year, it is possible to have an occurrence
of dry spells between 5 and 25 days in the Midwest. This
situation, happening during grain filling season, increases the
productivity losses for the three Brazilian regions.
The dry spell length for the return period of one year is
higher in the Southeast, Southern and Midwest, respectively.
That is, the region most at risk of agricultural loss due to
dry spells for a return period of one year is the Midwest
region. If the dry spell length is from 40 to 50 days, the most
vulnerable is the Southern region.
J.R. Porto de Carvalho et al. / Atmospheric Research 132-133 (2013) 12–21
19
Fig. 3. Percentile–percentile graphs for the adjustment of the dry spell length extreme values observed within each year for the three regions under study.
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Table 5
The Dickey–Fuller test (ADF) for the three regions (Midwest, Southeast and
Southern).
Region
ADF
p-value
Midwest
Southeast
Southern
−5.4594
−6.3089
−5.1951
0.0002
b.0001
0.0003
observed for return periods of 4 to 5 years. To the Southeast,
dry spells of 35–40 days were observed for a return periods
of one year. That is, using the series from 1940 through
1975 (Assad et al., 1993) and within the current series from
1940 through 2011, the intensification of the dry spell length
doubles in the Midwest and increases ten times for Southeast.
The same distributions were adjusted in both works.
4. Conclusions
Table 6
The Dickey–Fuller augmented test (ADF) for the three regions (Midwest,
Southeast and Southern) with the average (ADF1) and trend (ADF2)
included.
Region
ADF1
p-value
ADF2
p-value
Midwest
Southeast
Southern
−3.7450
−6.0693
−4.8393
0.0257
b.0001
0.0011
−3.964
−4.687
−4.589
0.0435
0.0017
0.0024
Comparing these results with those found by Assad et al.
(2003), that used a rainfall series of 104 rainfall stations with
30 years of daily data, the observed maximum dry spell
length for the return period of 10 years in the Midwest was
45–50 days while in the Southeast it was 35–40 days. In this
work, the dry spells of 45–50 days for the Midwest region were
Table 7
Return period (years) for the three Brazilian regions based on the maximum
annual period of dry spell (days) during the growing season.
Return period (years)
X (day)
Midwest
Southeast
Southern
5
10
15
20
25
30
35
40
45
50
55
60
65
70
75
80
85
90
95
100
110
120
130
140
150
160
170
180
190
200
210
220
230
240
1
1
1
1
1
2
2
3
4
5
6
8
10
13
16
19
23
28
34
41
57
77
104
137
177
226
285
356
440
538
653
785
938
>1000
1
1
1
1
1
1
1
1
1
2
2
3
4
5
7
10
14
19
28
40
87
193
440
>1000
1
1
1
1
1
1
1
2
2
3
5
7
12
19
30
49
81
134
224
377
>1000
The dry spell length increased in growing seasons for the
Midwest region and decreased in the Southeast and Southern
regions. Although the number of rainy days has increased in
the growing season for the three regions, the distribution
of rainfall in the Midwest is irregular at growing time. This
result is similar to the results found by other authors for time
series with fewer years, indicating that the irregularity in the
distribution of rainfall for the Midwest region has intensified
in recent years. The average number of rainy days in the
growing season for the Southeast region is 19% higher than
for the Midwest and South regions.
For all distributions studied, the generalized extreme
value distribution was the best fit to the data series of dry
spell length for the three regions.
It has been showed that 5–25 day dry spell lengths occur
in return period of one year for the series analyzed from 1940
to 2011. Previous works clearly show that 5–25 day dry spell
lengths were observed in return periods of 10 years. The dry
spell length that has become more frequent in recent years
may bring loss of productivity for the Brazilian agriculture,
especially if it happens at the time of grain filling. The
greatest risk of loss due to agricultural dry spells for the
return period of one year is the Midwest region, and for the
return period of two years, the Southern. The rapid change in
dry spell length increases the vulnerability of agricultural
production in all regions.
Acknowledgments
We would like to thank the anonymous reviewer for
many helpful comments that greatly improved the quality of
this work.
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