International Journal of Basic & Applied Sciences IJBAS-IJENS Vol: 11 No: 06
153
Critical Evaluation of Dry Spell Research
SC Mathugama and TSG Peiris
Abstract— Prior knowledge on the starting dates and lengths
of dry spells has a significant importance in rain-fed
agriculture, irrigation planning, and various decision making
processes related to climate. When reviewing statistical
analyses already in existence to study the above aspects on
dry spell events, it was evident that some analyses worth
further review. The statistical and mathematical algorithms
used to analyze dry spells varied from empirical frequency
distributions to more complex statistical distributions.
Empirical frequency distributions involve comparison of
frequencies of dry spells at different lengths and probabilities
of maximum and conditional dry spells exceeding a user
specified threshold. Markov models of order 1-10, Negative
Binomial, Truncated Negative Binomial, Eggemberg-Polya
and Weibull distributions have been used to find the
probabilities of a dry spell event greater than or equal to a
certain value. Some of the fitted models were not compared
with the results obtained from empirical statistical models.
Also, such models have not attempted to forecast length and
starting dates of dry spells and thus the information derived
from past studies cannot be used efficiently for planning
purposes. Further, as forecasting of all the dry spells is
impossible, it is also recommended to concentrate only on the
most critical dry spells.
I. INTRODUCTION
D
roughts and dry spells
Plants, animals and human beings need water to survive. If
there is not enough water they will eventually die from
dehydration. Drought usually occur due to no rainfall or
minimal rainfall. Drought is a common phenomenon that
takes place nearly every year in many areas of the world,
which influences the whole society. Dry spell is a period
where the weather has been dry, for an abnormally long time,
shorter and not as severe as a drought1.
Manuscript received November 9, 2011. This project was funded by the
National Research Council (NRC Grant No. 2009-16) of Sri Lanka, 380/72,
Baudhaloka Mawatha, Colombo 07, Sri Lanka. The authors wish to thank
NRC for funding the project.
S.C. Mathugama is with the Institute of Technology University of
Moratuwa, Katubedda, Moratuwa, Sri Lanka (e-mail: mathugama@
yahoo.com).
T.S.G. Peiris is with the Department of Mathematics, Faculty of
Engineering, University of Moratuwa , Katubedda, Moratuwa, Sri Lanka (email: [email protected]).
Benefits of the dry spell analysis
Studies on earth’s global climate show an increasing trend on
average air temperature. Consequently, the vegetation period
is expected to become shorter and even more irregular
distribution of rainfall is expected2. It has been noted that the
long dry spells incur heavy costs to the affected communities.
In humid countries the success or failure of the crops,
particularly under rainy conditions is highly related with the
distribution of dry spells. For achieving maximum benefits
from dry land agriculture the knowledge of distribution of dry
spells within a year is useful.
Dry spells affect not only in agriculture but also other sectors
such as fisheries, health, electricity etc. Long dry spells may
physically weaken the people which could cause mental
degradation due to the lowering of their status. The fish
productivity from fresh water is likely to be stricken by longer
dry spells3. Longer dry spells also interrupt generating
electricity using hydroelectric power4. Therefore, the effects
of dry spells in various sectors as described above ultimately
has a direct impact on the economy of a country.
The information on the length of dry spells could be used for
deciding a particular crop or variety in a given location, and
for breeding varieties of various maturity durations5.
Information on dry-spell lengths could be used in decision
making with respect to supplementary irrigation and field
operations in agriculture6. Prior knowledge of dry spell
studies can be applied to generate synthetic sequences of
rainfall and to the estimation of the irrigation water demand7.
Crops are more likely to do well with uniformly spread ‘light’
rains than with a few ‘heavy’ rains interrupted by dry periods.
The timing of breaks in rainfall (dry spells) relative to the
cropping calendar rather than total seasonal rainfall is
fundamental to crop viability8. The longest period of several
long spells is of crucial importance in planning agricultural
activities and managing the associated water supply systems9.
As drying (the dry period) in one year is not necessarily the
same as drying in another year the knowledge of behavior of
these patterns has become increasingly important to
understand. A major challenge of drought research is to
develop suitable methods and techniques for forecasting the
onset and termination points of droughts10. Past studies done
on dry spells in Sri Lanka highlighted the heavy losses in
paddy production caused by prolonged dry spells and the
importance of studying the temporal and spatial variability of
dry spells11,12,13,14.
114806-7575 IJBAS-IJENS © December 2011 IJENS
IJENS
International Journal of Basic & Applied Sciences IJBAS-IJENS Vol: 11 No: 06
Objective of the study
In view of the above, prior knowledge of the occurrence of
dry spell analysis would be beneficial to minimize unexpected
damage due to long dry spells and to have effective and
efficient planning for various stake holders. With this view in
mind, past work on dry spell analyses were reviewed in order
to identify methodologies used in analysing dry spell
properties to predict the commencement and the length of dry
spells.
II. DRY SPELLS INDICATORS
A. Dry day
In general, the definition of a ‘dry day’ is zero rainfall per day
taking threshold value of zero rainfall. However, different
authors have used different threshold values to define a dry
day. Rainfall amount 0.1 mm per day was used as the
threshold because it is often used with respect to the usual
precision of rain gauges15,16. Some studies employed a
threshold of 1.0 mm, on the assumption that rainfall less than
this amount is evaporated off directly17,18,19,20,21. Few authors
employed 1.5 mm22 and 2 mm23 as threshold values
respectively in order to remove any events featuring less
rainfall. A range of threshold values between 1-25 mm was
also used6 while 10 mm24,25,26 was used by some studies. The
use of longer threshold eliminates the excessive weight that
some isolated rainy days with small amounts have in breaking
the long dry spell. However, the threshold value should not be
selected in a subjective manner, but it should be related to the
type of the application.
B. Dry spell
Although the definition of a dry spell may vary depending on
the aims and methodology used in each study, the definition of
a dry spell is based on the length of the consecutive dry days.
A “dry spell” was first defined and used in British Rainfall in
1919 as a dry spell being a period of at least 15 consecutive
days to none of which was credited ≥ 1.0 mm18. Thereafter,
various versions of definitions of a dry spell were used by the
different authors along with different threshold values8,25. It
should be pointed out that, unlike a dry day, the minimum
number of consecutive dry days required to define a dry spell
has to be identified in a meaningful manner depending on the
practical problem.
For example, in Sri Lanka in paddy cultivation, it is
reasonable to consider dry spell of 7 or more days while in
coconut cultivation the corresponding value would be of 30
days. In studying drought effect, longer dry spells (> 40 days)
would be more effective.
C. Dry spell indicators
In the analyses of dry spells, various statistical indicators have
been used by different authors6,9,12,20,21,26. The two common
indicators used are the length of the dry spell (LDS) and the
frequency of a dry spell (FDS). Few studies used the length of
critical dry spells (CDS) 6,12. Among these, one study
considered CDS as length of dry spells greater than of a
154
specified length6. Another study defined critical dry spells as
the lengths of the three longest dry spells in a year among the
dry spells greater than seven days12. One drawback of this
selection is that it allows the analysis of the three longest dry
spells separately without considering the time of occurrence of
the dry spells. Further, some authors6,15,26,27 have studied the
maximum dry spells (MDS) in a year irrespective of the time
of the occurrence. Each indicator of the length of dry spell has
advantages as well disadvantages from a practical point of
view. Nevertheless, the analysis of the first few CDS in a
year, according to the time of occurrence would be more
useful.
Unlike the length of dry spells, less attention has been given to
study the time of the start of dry spells (SDS). For any
applications, both the length and time of start of critical dry
spells would be equally important from both the practical and
statistical point of views.
III. ANALYSES ON DRY SPELLS
A. Use of frequency distributions
A study done in India investigated the mean starting date of
critical dry spells (SDCDS) and the mean duration of the
critical dry spells (LCDS) using empirical frequency
distribution6. The study used daily rainfall data of 22 years
(1965-1986) for nine stations in Vidarbha region in India.
They related the duration of dry spells for different crops and
thus a critical dry spell was defined based on the water
requirement of the crop. The study found that when the dry
day was taken as 0.1 mm (irrespective of the crop), the first
critical dry spell commenced on the first or second week of
July with the length varying from 12 to 25 days. The second
critical dry spell started during the second week of August
with length varying from 18 to 40 days. The corresponding
value for the third critical dry spell was the first week of
September with the length ranging between 15 to 50 days.
The results obtained through this study provided a good
indication about the level of dry spells in the Vidarbha region.
Although the study provided predictions on SDCDS and
LCDS, these were not based on a specific statistical
distributions or statistical methodology. Further obtained
predictions were not validated for an independent set of data.
A Sri Lankan study analysed the properties of the three most
critical dry spells (CDS1-CDS3) in the Hambantota district of
Sri Lanka based on daily rainfall data from 1951 to 200112.
The three critical dry spells were taken as the three longest dry
spells in a year irrespective of the starting time. Since
Hambantota is a major coconut growing district, a threshold of
5 mm was taken for a dry day with reference to coconut
cultivation. The mean lengths of the CDS1-CDS3 during 1951
to 2001 were 55, 35 and 25 days respectively. The variability
was the highest in CDS1 (CV=35%). The probabilities of the
length of a dry spell in a year being more than 60, 45 and 30
days were found as 31%, 25% and 45% using normality
assumption. The normality was confirmed by AndersonDarling statistic. The computed probabilities were not
compared with the corresponding values using empirical
114806-7575 IJBAS-IJENS © December 2011 IJENS
IJENS
International Journal of Basic & Applied Sciences IJBAS-IJENS Vol: 11 No: 06
distributions. Though the study provided some useful
information on dry spells, no attempt was made to forecast the
dry spell length or any other properties of the dry spells.
Another study analysed the lengths of dry spells (LDS) and
frequencies (FDS) in a year using rainfall daily data from
1951-1990 in 35 stations in Spain under three threshold values
of 0.1, 1.0 and 10.0 mm24. The longest dry spells during the
period 1951 – 1990 had a length of 169, 166 and 156 days, in
the stations of Andalusian, Malaga and Almeria respectively
when the threshold value was 0.1 mm. When the threshold of
10.0 mm was taken the corresponding values were 363, 360
and 343 days. Though it was obvious that the length of the
critical dry spells increases with the increase of threshold
values, no such statistical relationship has been established.
As the study was confined to compare the dry spells among
different locations in Spain, no attempt was made to fit
statistical distributions or to predict the length of the dry spells
to justify the results.
A similar study was carried out using rainfall data from 1967
to 2000 in three ecological regions consisting of 15 stations in
the Duero Basin Spain25. They too compared the mean length
and the maximum length of dry spells under two different
threshold values. In the analysis, 15 stations have been
clustered into three units based on water availability in the
soil. The study also claimed that for the threshold of 0.1 mm,
both the mean length of dry spells and maximum length of dry
spells were not significantly different among three clusters.
However, both statistics are significantly different (p < 0.05)
among the stations within a cluster. The study claimed that
there was an inverse relationship between the number of dry
days per year and the total annual rainfall, but no statistical
relationship has been established. Rainfall data from 1961 to
1990 in Argentina were used to study the temporal and spatial
distribution of the LDS and the FDS28. The study found that in
the driest parts in Western Argentina, the mean length of dry
spells was 60 days. As atmospheric circulation showed a
significant change around 1970, a comparison of dry spells
were done between prior to 1970 and after 1970 and indicated
that both the longest dry spells and the mean lengths of the
dry spells were smaller in the latter period. The spatial and
temporal distribution of dry spells was investigated using daily
data (1960–2000) in Tanzania under threshold value of 1.0
mm21. Results indicated that the longest dry spells in some
parts of Tanzania were 249 days in 1999 due to a cold El
Nino-Southern Oscillation (ENSO).
As discussed above, similar studies were carried out by some
other authors to compare basic properties of the dry spell
lengths and frequencies at spatial and temporal
levels29,30,31,32,33.
Using daily rainfall data (1950-1980) in West Africa, the
probabilities of the maximum dry spell lengths exceeding 7,
10 and 15 days over the next 30 days starting from the first
day of each decade during the period from May to October
were computed5. In this study five threshold values (1mm,
5mm, 10mm, 20mm and 25mm) were taken and compared the
drought risk for certain crops. Though the study indicated that
155
there were relationships between the mean annual rainfall and
the average frequencies of dry spells for the selected locations,
statistical models were not shown. Further, the probabilities
that the dry spells for durations <5, 5-10, 15-20 and >20 days
were computed using empirical frequency distributions.
In addition to the comparison of descriptive statistics of dry
spells based on empirical frequency distributions, probabilities
of the length of a dry spell greater than a specified value have
also been estimated using Markov process of order one by
some of the authors mentioned above24,25,28. The results
obtained under Markov process will be discussed in a
separate section.
B. Use of linear models
Some authors have developed simple linear regression models
between frequencies of the dry spells and the amount of
rainfall. The frequency of dry pentads (DSF) and mean
frequency of dry spells for summer (December to February)
season in southern Africa were studied using daily rainfall
data from 1979 to 20028. The particular period has been
selected as it is the peak of the growing season of many crops
and as El Nino Southern Oscillation (ENSO) impacts reach
maximum during that period. The study indicated that the
areas of low consistent rainfall showed a high number of dry
spells while high consistency in rainfall showed less number
of dry spells irrespective of time (p< 0.05). A similar study
was carried out for summer (May to September) in China
using daily precipitation during 1956 to 2000 in 30 stations33.
The results indicated that the short dry spells (< 10 days)
decreased remarkably at a rate of 2.0% per 10 years and the
long dry spells ( ≥ 10 days) became more frequent at a rate of
7.3% per 10 years. However, it indicated that even though
very high dry spell frequencies were recorded in certain years,
the precipitation amount in these years were not so low.
A significant correlation (p < 0.05) between extended dry
spells and a positive sea surface temperature (SST) anomaly
gradient in the east-central North Pacific was found in another
study34. A set of 17 rain gauges in Isfahan province in the
centre of Iran with daily rainfall recording for 30 years were
used to analyse spatial patterns of trends of two dry spell
magnitudes; annual maximum dry spell length (AMDSL) and
annual number of dry spell period (ANDSP). Time series
plots were used to explore temporal changes of dry spells and
the trend line. The linear regression model was used to
estimate the time trend of data35. Precipitation data from 104
stations in 100 years from 1901 to 2000 in Switzerland were
analysed to identify trends of wet and dry spells using dry
spell indices; mean dry day persistence, Maximum number of
consecutive dry days and mean dry spell length. Trends of
these indices were calculated using logistic regression with
logit function as the link function and maximum likelihood
method for parameter estimation36. Although these models
were able to catch up trends of certain dry spell properties they
were not intended for forecasting purposes.
114806-7575 IJBAS-IJENS © December 2011 IJENS
IJENS
International Journal of Basic & Applied Sciences IJBAS-IJENS Vol: 11 No: 06
C. Use of moving average
In all the studies discussed above, the length of dry spells was
obtained on the basis of the precipitation sums in a given
length and the maximum number of consecutive dry days in
the original precipitation series {rt } . An unique study done by
one author37 smoothed the original series
{rt } by taking the
moving average of order n and the length of the dry spell
obtained using smoothed series (without distinguishing
months or years), {St } where
1 i + n −1
(1)
∑ rt .
n t =i
The study used the daily rainfall data in 1957-2006 of 56
locations in Estonian. The use of the moving average series
allowed finding severe dry spells. The estimation of extreme
dry weather conditions on the basis of moving average of
daily precipitation allows determining the most drastic periods
and trends of the precipitation regime in a better way than
using original series. During 1957-2006, the most severe
drought was observed in 2002 with the mean number of 13 dry
days in August-September. The mean number of dry days in
2002 was more than twice as big as the number in 2006, which
was the second driest year over throughout the periods. One of
the drawbacks of using smoothed series is the selection of the
order of moving average. In this study too no attempt was
done to forecast either the length of dry spells or onset of dry
spells.
S tn =
D. Markov Process
A Markov process is useful for analyzing random events
whose likelihood depends on what happened last38. Time
series dry spells depends on what happened in the past, and
Markov process models can be used to study properties of dry
spells. A stochastic process, whose state at time t is Yt (t > 0),
such that the value of
of
Ys (s<t) does not depend on the values
Yu (u<s) then the process is said to be Markov process39.
156
state being wet or dry. The transition matrix of the Markov
model of order 1 for two state (D=0 for the dry state and W=0
for the wet state) is given by
D W 

D p
p01 
(3)
00

W p10 p11 
where
pij = probability of the present day state i (i=0 or 1)
given that previous state is j (j=0 or 1). Thus, using
Geometric distribution the probability of a dry spell lasting
exactly n days will be given by the following formula;
n −1
n −1
(4)
Pn = p 00
p 01 = p 00
(1 − p 00 )
where
p00 = the probability of a dry day occurring after a dry day
p01 = the probability of a wet day occurring after a dry day.
Thus depending on the lag period, Markov model of any order
can be used. However, it should be noted that increasing the
order increases the complexity of formulae and reduces the
applicability.
The Markov chain probability model for the analysis of wet
spells and dry spells was first introduced by Gabriel and
Neumann (1957) using 27 years (1923-1950) of rainfall data
from November to April at Tel Aviv in Israel considering the
threshold of 0.1 mm38. The results were validated using chisquare tests. Since then the Markov process models have been
used extensively by many authors.
Daily rainfall data from 1984 to 1993 in Havana were used to
estimate the year round probabilities for both the dry and wet
seasons using time varying Markov chain40. In this study
probabilities of a dry spell greater than a particular length
based on the condition of the previous days (that is, the
probabilities of dry spell followed by dry season and that of
followed by wet season) were computed. However, more
details were not shown in this paper. Nevertheless, the
predicted probabilities would be more beneficial from the
short-term planning point of view as the lead time is longer.
That is,
P(Yt +h = y / Yt1 = y1,Yt 2 = y2 ........,Ytn = yn ) = P(Yt +h = y / Ytn = yn)
(2)
It indicates that the probability of its having state Y at time
t+h, conditioned on having the particular state Yt + h at time t,
is equal to the conditional probability of its having that same
state Y but conditioned on its value for all previous times
before t. This captures the idea that its future state is
independent of its past states, but depends only on the
immediate past. In dry spell analysis as a day is classified
either ‘wet state’ or ‘dry state’ depending on the rainfall
amount of a day, Markov process can be applied.
The order one (order 1) process assumes that the present state
(wet or dry) depends only on the condition of the previous
The probabilities of the dry spell lengths exceeding 5, 7, 10,
15 and 20 days were computed using Markov model of order
two under the assumption of binomial error structure41. For the
study, rainfall data from 1923-1978 in Anuradhapura, Sri
Lanka were used. The results obtained were compared with
the probabilities calculated using the empirical distribution of
frequencies of dry spells. However, the model has not been
tested for an independent data set and also no forecasting
lengths or its onsets of dry spells was carried out. Rainfall data
from 1930 to 1987 for one semiarid region, Kibwezi and from
1972 to 1991 for one semi-humid region, Kabete were used to
compare Markov model of order one and a random model in
simulating the length of the longest dry spells9. The level of
persistence in the spells in Kabete was found strongly a
Markov model whereas Kibwezi tended to be random in 50%
114806-7575 IJBAS-IJENS © December 2011 IJENS
IJENS
International Journal of Basic & Applied Sciences IJBAS-IJENS Vol: 11 No: 06
cases. The validity of fitted Markov model to Kabete was
illustrated using plots only. It can be concluded that none of
the above studies were able to forecast the length of the
maximum or critical dry spells.
Markov models of orders one to ten were developed for the
duration of dry spells in 35 stations using data from 1951199024. The adjustment of the general distribution of the dry
spells according to their length and the adjustment of the
longest spells, those that exceed one, two or three months
duration, have been considered. Out of these 35 stations, no
significant fit was found for 10 stations. Of the remaining 25
stations different order of Markov models were recommended.
The fitted Markov models for a given location was verified by
using the chi squared test. Finally, they concluded that the
Markov models were incapable of giving acceptable
probability estimates for the duration of dry spells. As the
order of Markov models were spatially varied, it can be
hypothesised that the models could depend on temporal scale
as well. However, Markov models were not verified for
different time periods.
In another study the probabilities of agricultural dry spell
exceeding 10 days in two different locations in East Africa
were computed using the Markov model of order one42. In
their study, an agricultural dry spell was defined by
incorporating water availability using water balance model.
The study found that the probability of dry spells exceeding 10
days varied from 20% to 70% or more depending on onset of
rainy season. Thus one can confirm the instability of Markov
models in explaining the dry spell variability. Nevertheless, it
should be pointed out that the study of agricultural dry spells
using water balance model is a better approach than
meteorological approach to find the impact of dry spells on
crop production, if the necessary parameters of the water
balance model are available for a particular crop.
The trends in rainfall and the behaviour of dry spells in Spain
were analysed using a Markov model of order one25. The
analysis revealed that the mean length of dry spells is not
significantly different among three regions and the mean
length of the dry spell was the highest during July-August
months. However, in this study too no attempt has been made
to forecast dry spell length and also no statistical method was
applied to compare observations and the estimated
frequencies. Frequencies of dry spells of different lengths
obtained using empirical frequency distribution with Markov
model of order one were compared by another study28. The
goodness of fit of the theoretical distribution was assessed and
found that it was difficult to adjust the empirical frequency
distribution using a first order Markov model as the
probability of a dry day after a dry day was very high in a
great part of Argentina.
Above studies confirmed that Markov models have a well
developed literature throughout many years in different
countries. Because of easy application, non parametric nature,
easy interpretation and simplicity in calculation, several
authors have used such models, but there are some drawbacks.
157
In most of the studies it was noted that the Markov models
usually overestimates the very short dry sequences, and
underestimates the very long dry sequences According to
certain studies43,44,45 the Markov models do not reproduce
long term persistence. Therefore, it can be confirmed that
Markov models are not suitable to explain the variability of
the length of dry spells in a given location or in a given period
and so for the use of forecasting length of dry spells.
E. Use of other distributions
In addition to the Markov models, several other distributions
have been applied to study the properties of the dry spells,
particularly to explain the variability of the dry spell length.
The common distributions used were Negative Binomial,
Truncated Negative Binomial, Logarithmic, Weibul and
Eggemberger-Polya.
1) Negative Binomial distribution
The negative binomial distribution, also known as the Pascal
distribution or Pólya distribution, gives the probability of
( r − 1) successes and k failures in ( k + r − 1) trials. The
probability density function is therefore given by
 k + r − 1 r
 p (1 − p ) k
f (k : r , p) = 
(5)
−
r
1


for k = 0,1,2, ..... with 0 < p < 1 and r > 0
The dry spells in 20 stations in Greece were analysed using
data from 1958 –199746. The probabilities that a dry spell
lasts exactly n days (n = 1,2, ,,,,,20) were calculated annually
and seasonally for the 20 stations using Negative Binomial
model and Markov model of second order. The results
indicated that the dry spell frequencies of a dry spell length
less than 10 in Greece can be modeled by the Negative
Binomial on seasonal basis, but it overestimated when dry
spell frequencies were modeled on annual basis. Markov
models of order two modeled such dry spell frequencies on
annual basis. Both models overestimated the frequency for 1120 spells irrespective of temporal scale. Negative binomial
distribution was used to analyse dry spell frequencies in one
station located in the basin of Northern Tunisia using data
from 1968 to 20077.
2) Truncated Negative Binomial distribution
Negative binomial distribution with the zero class truncation is
known the Zero-truncated Negative Binomial distribution. In
Fisher’s notation the negative binomial has the form of
f (k : r , p) =
(k + r + 1)!
pr
,
(r − 1)! k! (1 + p ) k + r
for k = 1,2,... with 0 < p < 1 and r > 0 , P0 =
(6)
1
.
(1 + p ) k
To obtain the corresponding probabilities for the truncated
distribution,
the equation (6) must be divided by
(1 − P0 ) and getting w = 1 /( 1 + p ) , then
114806-7575 IJBAS-IJENS © December 2011 IJENS
IJENS
International Journal of Basic & Applied Sciences IJBAS-IJENS Vol: 11 No: 06
f (k : r , p) = Pr =
(k + r + 1)!
(1 − w) r ,
1 − w (r − 1)! k!
wk
(7)
k
for k = 1,2,.....
The probabilities of the dry spell length greater than 1- 20
days were computed using Markov model of order one and the
Truncated negative binomial model for the six locations in
Brazil 47. Rainfall data for the period from September to
February from 1949 to 1978 were used. As usual the
parameters for each model were estimated using empirical
frequency distribution of the dry spells of different lengths.
Based on Kolmogrov-Smirnov test it was concluded that the
results obtained from both models were not significantly
different. However, Truncated negative binomial model
showed better fit for all six locations than Markov model, but
Markov models are easy to interpret than truncated negative
binomial.
Rainfall data in 25 locations from 1858 to 2000 in Italy were
used to describe dry spell frequencies of occurrences using
Truncated negative binomial model, the Logarithmic model,
and the Markov chain of order one and above20. The Chisquared test for goodness of fit has been used to compare the
models. The Truncated negative binomial and EggenergerPolya models were found to be more efficient than other
models in fitting observed data. By analyzing data for another
four locations in Italy,
another study confirmed that
Truncated negative binomial distribution and EggembergerPolya were better fitted models to explain the dry spell
frequencies48. Of these two distributions, Truncated negative
binomial is not suitable for very long dry spells which were
considered as extreme events, but Eggemberger-Polya
distribution was better fit for the longer dry spells.
3) Logarithmic distribution
The logarithmic distribution (also known as the logarithmic
series distribution or the log-series distribution) is a discrete
probability distribution derived from the Maclaurin series
expansion. The probability distribution function of logarithmic
distribution is given by 49
f (k ) =
158
−1
pk
, k ≥ 1 and 0 < p < 1
log(1 − p ) k
.
(8)
Daily rainfall data at Campina Grande from May to July
during the period 1939 -1972 were used to obtain the
frequency distribution of dry spells of durations up to ten days
and these frequencies were compared with those computed
using a logarithmic model and Eggenberger Polya model50.
The chi-square statistics was used to compare the models. The
results indicated that the Eggenberger-polya model provided
good estimates of dry spell frequencies at the station than that
from Logarithmic distribution. As the observed frequencies of
dry spell greater than 10 days were extremely low,
Eggenberger-polya model was not tested for longer dry spells.
The main drawback of those models is that the above models
are not suitable for dry spell frequencies of duration, generally
greater than 10 days.
4) Weibull distribution
The Weibull distribution is one of the most widely used
distribution in many applications, versatile distribution that
can take on the characteristics of other types of distributions.
The probability distribution of the Weibull distribution is
given by
k −1
k x
−( x / λ ) k
(9)
,x≥0
  e
λ λ
where k > 0 is the shape parameter and λ > 0 is the scale
parameter of the distribution. It was shown that Weibull
model fitted well with the empirical frequency distributions of
dry spell lengths irrespective of the duration of the dry spells
for the selected 43 stations across the Iberian Peninsula26. In
another study51 using 50 years daily rainfall data, it has been
shown that the Weibull distribution fitted well for dry spell
frequencies when the dry spell length varied from 1 to 20.
f ( x; λ , k ) =
5) Mixture of log series with Geometric distribution
(MLGD)
Nine types of probability distributions were fitted to dry spells
data for 16 selected rainfall stations in Peninsula Malaysia
using rainfall observations for the period 1975-200452. These
distributions used were Log series distributions, Geometric
distribution, Modified Log
distribution, Compound
Geometric distribution, Truncated Negative Binomial
distribution, Mixed Geometric series with Log series, Mixture
of two Geometric Distributions, Mixture of Geometric and
Poisson Distributions and Mixture of Log series with
Geometric Distribution. The adequacy of the MLGD and the
existing probability models were evaluated using a chi-square
goodness of fit test. All the data sets were found to
successfully fit the new proposed model, the MLGD. New
model was also found to be fitted well for three data sets in
modelling shorter and longer duration of dry spells which
were not able to fit using the existing eight probability models.
IV. CONCLUSION
A major challenge of drought research is to develop suitable
methods and techniques for forecasting the onset and
termination points of droughts. An equally challenging task is
the dissemination of drought research results for practical
usage and wider application. Consequently many authors in
different countries have analyzed long-term rainfall data either
taking daily series for the year or for a selected season
(generally dry season) in different locations. Almost all
studies concentrated on comparing properties of dry spells
spatially using empirical frequency distributions. The length
of dry spells and frequency of dry spells are the two common
variables analyzed. Further, in almost all studies, results were
not compared on temporal scale and thus change of dry spell
pattern had not been related to climate change in the
corresponding countries.
Some authors have attempted to describe the behavior of dry
spells using Markov models of different orders which varied
114806-7575 IJBAS-IJENS © December 2011 IJENS
IJENS
International Journal of Basic & Applied Sciences IJBAS-IJENS Vol: 11 No: 06
from one to ten. The order of Markov models was found to be
location specific. The Chi-squared test has been used as
goodness of fit test to compare the frequencies derived from
the Markov model and the empirical frequency distribution.
Chi-squared test was not significant in certain cases indicating
that the observed frequencies were significantly different from
the corresponding expected frequencies. As the probabilities
were computed based on Markov models of order one or
above, the results of such studies depend on the climate
condition of the previous day (dry/wet) or on the previous
days depending on the order of the Markov model. However,
the use of higher order models for a given location is not
recommended as it is difficult to interpret. In most of the
cases it was noted that Markov models usually overestimated
the very short dry sequences and underestimated the very long
dry sequences. In none of the studies, the results obtained
were tested for an independent data set and therefore the
results derived from such models may not be valid for future
prediction.
In addition to Markov models, various statistical distributions
such as Negative Binomial, Truncated Negative Binomial,
Weibull, Eggemberg-Polya and mixture models were also
found as best fitted models to explain dry spells for different
locations than Markov models. However, those models are
also location specific. In general Negative Binomial models
performed better than Markov models to explain dry spell
frequencies on the seasonal basis.
Eggemberg-Polya and
Truncated Negative Binomial distributions were more
efficient than the other models in fitting observed data for
longer dry spells on annual and seasonal aggregations.
[2]
[3]
[4]
[5]
[6]
[7]
[8]
[9]
[10]
[11]
[12]
[13]
[14]
[15]
[16]
[17]
In few studies, authors have analysed critical dry spells but
none of the authors considered the time of occurrence of dry
spells. If the selection of critical dry spells were done
according to the time of occurrence rather than the length
irrespective of the time, then it would be much easier to use it
for forecasting purposes. No studies were reported to predict
the starting dates of critical dry spell/s and the lengths of
critical dry spells. Identifying distributions and patterns of
critical dry spells and investigating the possibility of
forecasting the future patterns of dry spells would generate
useful information to various stake holders for their decision
making purposes. Therefore, future research on dry spells
should be geared towards these aspects.
[18]
[19]
[20]
[21]
[22]
[23]
[24]
In addition to the forecasting of the starting time of dry spells
is difficult, forecasting of the gap between two consecutive
critical dry spells is recommended to be explored. It is also
recommended to explore the use of artificial neural network
(ANN) to forecast dry spell properties but such models have to
be statistically validated.
[25]
[26]
[27]
REFERENCES
[1]
Wilhite DA. Glantz MH. (1985). “Understanding the drought
phenomenon; The role of definitions”. Water International. 10:111-120.
159
IPCC (2008). Climate Change, summary for policymakers: A report of
working group of the Intergovernmental Panel on Climate Change,
Montreal, Canada.
2004. “Wildlife and Fish in a Drought”. Wildlife and Fisheries.
http://agnews.tamu.edu/drought./
Jayawardene, H.K.W.I. Sonnadara, D.U.J. and Jayewardene, D.R.
(2005). Trends in rainfall in Sri Lanka over the last century. Sri Lankan
Journal of Physics. 6:7-17.
Sivakumar, M. (1991). "Empirical analysis of dry spells for agricultural
applications in West Africa " .Journal of Climate: 532-539.
Taley SM. And Dalvi VB. 1991. “Dry-spell analysis for studying the
sustainability of rainfed agriculture in India – The case study of the
Vidarbha region of Maharashtra state”. Large Farm Development
Project.
Mathlouthi M, Lebdi F (2008). “Characterization of dry spell events in
a basin in the North of Tunisia”.
Usman, M. and C. Reason (2004). "Dry spell frequencies and their
variability over sourthern Africa." Climate Research 26: 199-21.
Sharma TC. (1996). “ Simulation of the Kenyan longest dry and wet
spells and the largest rain sums using a Markov Model”. Journal of
Hydrology 178 (55-67).
Panu, U. and T. Sharma (2002). "Chalenges in drought research: some
perspectives and future directions." Hydrological Sciences 47(special).
Waidyaratne, K.P. Peiris T.S.G and Samita, S.“Shift in Onset of First
Inter Monsoon Rain in Coconut Growing Areas in Sri Lanka”. Tropical
Agricultural Research. 18:1-12.
Peiris, T. and J. Kularathne (2008). "Assessment of climate variability
for coconut and other crops: A statistical approach." Journal of CORD
24(1): 35-53.
Ariyabandu, M. and Hulangamuwa, P. (2002). Corporate Social
Responsibility and Natural Disaster Reduction in Sri Lanka, ITDG South Asia.
Wijayapala R. (2011).
Importance of drought management policy
highlighted. Sunday Observer.
Dieterichs, H. (1955). "Frequency of dry and wet spells in san salvador."
Pure and Applied Geophysics 33(1): 267-272.
Moon SE. Ryoo SB. And Kwon JG. (1994). “ A Markov chain model
for daily precioitation occurrence in South Korea”. International Journal
of Climatology. 14:1009-1016.
Chaudry, Q., M. Sheikh, et al. (2001) “History's Worst Drought
Conditions Prevailed over Pakistan”. Volume, DOI:
Douguedroit A. (1987). “The variations of dry spells in Marseilles from
1865 to 1984.” International Journal of Climatology 7(6): 541-551.
Lázaro R. Rodrigo FS. Gutirrez L. Domingo F. and Puigdefáfragas J.
(2001). “Analysis of 30-year rainfall record in semi-arid SE Spain for
implications on vegetation”. Journal of Arid Environments. 48:373-395.
Epifani C. Esposito S. and Vento D. (2004). “Persistence of wet and dry
spells in Italy. First results in Milano from 1858 to 2000”. Proceedings
from the 14th International conference on clouds and precipitation 2004.
Bolobna; 18-24.
Tilya FF. and Mhita MS. (2007). “Frequency of wet and dry spells in
Tanzania”. Climate and Land Degradation. Springelink 197-204.
Harrington J. and Flannignan M.(1993). “A model for the frequency of
long periods of drought at forested stations in Canada”. Journal of
Applied Meteorology. 32:1708-1716
Perzyna G.(1994). “Spatial and temporal charachteristics of maximum
dry spells in southern Norway”. International Journal of Climatology.
14:895-909.
Martin-Vide, J. and L. Gomez (1999). "Regionalization of Peninsular
Spain based on the length of dry spells." International Journal of
Climatology 19(5): 537-555.
Ceballos, A., J. Martinez-Fernandez, et al. (2003). "Analysis of rainfall
trends and dry periods on a pluviometric gradient representative of
Meditreeanean climate in the Duero basin, Spain." Journal of Arid
Envioronments, 58(2): 215-233.
Lana X. Martinez M D. Burgueno A. and Serra C. (2005). “Statistical
distributions and sampling trategies for the analysis of dry spells in
Catolina(NE Spain)”. Journal of Hydrology. 324(1-4):99-114.
Vincente-Serrano, S. and S. Begreria-Portugues (2003). "Estimating
extreme dry spell risk in the middle Ebro valley (NE Spain): A
comparative analysis pf partial duration series with a general pareto
distribution and annual maxima with a Gumble distribution." Journal of
Climatology 23: 1103-1118.
114806-7575 IJBAS-IJENS © December 2011 IJENS
IJENS
International Journal of Basic & Applied Sciences IJBAS-IJENS Vol: 11 No: 06
160
[28] Penalba O. Liano MP. (2006). “Temporal variability in the length of norain spells in Argentina”. Proceedings of the International Conference on
Sourthern Hemisphere Meterology and Oceanography: 333-339.
[29] Joseph, E.S. (1970). “Probability distribution of annual droughts”.
Journal of Irrigation and Drainage. Asce96(IR4), 461-473
[30] Gupta, V.K. and Duckstein, L (1975). “A stochastic analysis of
extreme droughts ” . Water Resources. 27(5) 797-807.
[31] Clausen, B. and Pearson, C.P. (1995). “Regional frequency analysis of
annual maximum streamflow drought”. Journal of Hydrology. 137, 111130.
[32] Kumar, V. and Panu , U.S. (1997). “Predictive assessment of severity of
agricultural droughts based on agro-climatic factors”. Journal of
American Water Resources Association 33(6),1255-1264.
[33] Gong, D., P. Shi, et al. (2004). "Daily precipitaion changes in the semi
arid region over nothern China." Journal of Arid Envioronments 59(4):
771-784.
[34] Bonsal, B.R. Chakravarti, A.K. and Lawford, R.G. (1993). ”
Teleconnections between North Pacific SST anomalies and growing
season extended dry spells on the canadian prairies”. International
Journal of Climatology 13(8): 865-878.
[35] Nasri, M. and Modarres R. (2009). Dry spell tren analysis of Isfahan
Province, Iran. International Journal of Climatology. 29: 1430-1438.
[36] Schmidli, J. and Feri, C.(2005). Trends of heavy precipitation and wet
and dry spells in Switezerland during the 20th century. International
Journal of Climatology. 25:753-771.
[37] Tammets, T. (2007). Distribution of extreme wet and dry days in Estonia
in last 50 years. Estonian Academic Science Engineering. 13(3):252259.
[38] Gabriel, K. R. amd J. Neumann (1957). “On a distribution of weather
cycles by length”. Quarterly Journal of the Royal Meteorological
Society 83: 375-380.
[39] Medhi, J. (2009). “Stochastic Processes”. New Age Science.
[40] Roque, D.R. (1996) “Rainfall in Little Havana”. Cuba in Transition 8:
142-149.
[41] Abeysekera S. Senevirahtne KE. Leaker A. and Stern RD.(1983).
“Analysis of rainfall data for agricultural purposes”.11(2):165-183.
[42] Barron J, (2004). “Dry spell mitigation to upgrade semi arid rain fed
agriculture: Water harvesting and soil nutrient management for
smallholders maize cultivation in Machakos Kenya”. Doctoral Thesis in
Natural Resource Management, Stockholm University Sweden.
[43] Chang TJ. Kavvas ML. and Delleur JW. (1984). “Modelling of sequence
of wet and dry days by binary discrete autoregressive moving average
processes”. Journal of Climate and Applied Meteorology. 23:1367-1378.
[44] Foufoula-Georgiou E. and Georgakakos KP. (1988). ”Recent advances
in space-time precipitation modeling and forecasting”. Recent advances
in the Modelling of hydrological Systems, NATO ASI Ser.
[45] Wantuch ID. Mika J and Szeidi L.(2000).”Modelling wet and dry spells
with mixture distributions”. Meteorology and atmospheric physics.73(34):1436:5065.
[46] Anagnostopolou C, Maheras P, Karacostas T, Vafiadas M. (2003).
“Spatial and temporal analysis of dry spells in Greece. Theoretical and
Applied Climatology.
[47] De Arruda HV, Pinto HS (1979). “An alternate model for dry spell
probability analysis”. American Meterological Society 1980 volume
108:823-825
[48] Di. Giuseppe E. Epifani C. Esposito S. and Vento D. (2005). “Analysis
of dry and wet spells from 1870 to 2000 in four Italian sites”.
Geophysical Research Abstracts, 7.
[49] Williams CB. (1951).”Sequence of wet and dry days considered in
relation to the logarithmic series”. The Quarterly Journal of the Royal
Meteorological Society. 78(335):91-96.
[50] Kamar K, and Rao TV. (2004). “Dry and wet spells at Campina
Granade”. Revista Brasileira de Meteorologia 20(1) 71-74.
[51] Aghajani G, (2007). “Agronomical analysis of the characteristics of the
precipitation (Case study: Sabzever, Iran)”. Pakistan Journal of
Biological Sciences 10 (8):1354-1359.
[52] Deni SM, Jemain AA. (2009).”Mixed log seriew geometric distribution
for sequences of dry days”. Atmospheric Research 92, 236-243.
114806-7575 IJBAS-IJENS © December 2011 IJENS
IJENS