Composite Agrometeorological Drought Index Accounting for Seasonality and Autocorrelation

25 Pages·2017·1.13 MB·English

Composite Agrometeorological Drought Index
Accounting for Seasonality and Autocorrelation
Mohammad Mehdi M Bateni, Javad Behmanesh, Carlo de Michele, Javad
Bazrafshan, Hosein Rezaie
To cite this version:
Mohammad Mehdi M Bateni, Javad Behmanesh, Carlo de Michele, Javad Bazrafshan, Hosein Rezaie.
Composite Agrometeorological Drought Index Accounting for Seasonality and Autocorrelation. Jour-
nal of Hydrologic Engineering, In press, 23 (6), ￿10.1061/(ASCE)HE.1943-5584.0001654￿. ￿hal-
01654611v2￿
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Journal of Hydrologic Engineering-HE3881 (Accepted Manuscript)
doi:10.1061/(ASCE)HE.1943-5584.0001654

Composite Agrometeorological Drought Index Accounting for
Seasonality and Autocorrelation
M. M. Bateni1, J. Behmanesh2, C. De Michele3, J. Bazrafshan4, H. Rezaie5
1 Water Engineering Department, University of Urmia, Urmia, IRAN (corresponding
author). E-mail: [email protected]
2 Water Engineering Department, University of Urmia, Urmia, IRAN. E-mail:
[email protected]
3 Department of Civil and Environmental Engineering (DICA), Politecnico di Milano,
Milano, Italy. E-mail: [email protected]
4 Department of Irrigation and Reclamation Engineering, University of Tehran, Karaj,
IRAN. E-mail: [email protected]
5 Water Engineering Department, University of Urmia, Urmia, IRAN. E-mail:
[email protected]

Abstract
Drought indices are statistical tools used for monitoring the departure from normal
conditions of water availability. Recently, the multivariate nature of droughts has been
addressed through composite indices, capable of including different factors contributing to the
occurrence of a drought. However, some issues (like the auto-correlation or the proper
definition of the multivariate index) are still open and need to be addressed to make these
indices applicable in the current practice. Here, a composite agro-meteorological drought index
(AMDI-SA) has been introduced, accounting for meteorological and agricultural droughts,
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considering specifically seasonality and auto-correlation. AMDI-SA combines, through the
copula concept and the Kendall function, two drought indices (namely Multivariate
Standardized Precipitation Index (MSPI) and the Multivariate Standardized Soil moisture
Index (MSSI)) in a statistically consistent (normal-distributed) drought indicator.
Nonparametric distributions have been used for the variables of interest and the calculation of
MSPI and MSSI, while parametric and nonparametric (empirical) copulas to build AMDI-SA.
A pre-whitening procedure has been applied to MSPI and MSSI to remove the autocorrelation.
An application to Urmia lake basin in Iran has been presented, drought indices compared, and
investigated their spatial variability. Results showed that MSPI and MSSI are able to justify
72% and 89% of the variability throughout the year. AMDI-SA reflects the combined effect of
soil moisture and precipitation, and has a behavior in between whitened MSPI and MSSI. In
addition, having no memory and being composite index, the AMDI-SA is able to clearly detect
the temporal variability of recorded droughts to a greater extent than MSPI and MSSI indices.

1. Introduction
Droughts are extreme hydrological events and representative of natural hazards, which
impose serious challenges to ecosystems and human societies. Droughts may affect a wide
variety of sectors such as agriculture or hydroelectric power generation, with diverse
geographical and temporal extent. Droughts have multiple aspects and may be classified into
four main types: meteorological, agricultural, hydrological and socioeconomic droughts
(Dracup et al., 1980; Heim, 2002). Meteorological droughts are related to the deficiency of
precipitation over an extended period of time, from which other types of drought originate.
Agricultural droughts relate to insufficient water to meet the need of crop production, or plant
growth (Heim, 2002).
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Drought indices are useful tools to detect, monitor, and evaluate drought events (Zargar et
al., 2011). Several indices have been developed for drought monitoring (Mishra and Singh,
2010). Among these, Standardized Precipitation Index (SPI) is one of the most commonly used,
applied to local, regional, and global scale studies. SPI is widely used, primarily for its
simplicity, standardized nature, and flexibility of use across different time scales (e.g., 1-, 6-,
12-month). On the other hand, SPI has potential limitations. The assumption of one suitable
probability distribution function for precipitation data could be inconsistent under different
window sizes and could not account for seasonal variability. Moreover, the selection of one
single time scale could be too simplistic or misleading in practical applications to water
resource managers, decision-makers, and users. Using a novel approach, Bazrafshan et al.
(2014) proposed the Multivariate Standardized Precipitation Index (MSPI) using the 1st
principal component of SPI aggregates at different time windows. They showed the superiority
of MSPI, when the appropriate time window for a drought study had not been identified in
advance.
However, drought analyses based on a single variable may not be sufficient because
drought phenomena have complex dynamics involving multiple variables (e.g., precipitation,
runoff, and soil moisture) (Hao and AghaKouchak, 2013). Moreover, a single drought index
may not be sufficient to describe all aspects of drought onset, persistence and termination (see
e.g., Dracup, 1980; Kao and Govindaraju, 2010; Hao and Singh, 2015; Waseem et al., 2015;
Hao et al., 2016). For example, indices which are used to monitor meteorological droughts
usually capture the drought onset earlier (Behrangi et al., 2015), while soil moisture index
describes the drought persistence more reliably (Entekhabi et al., 1996; Heim, 2002). The
characterization of droughts with a composite perspective is to go over the inadequacy of
drought characterization using a single variable. This can be done by developing drought
indices combining multiple hydrological variables, or drought indices (Hao and Singh, 2015).
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Recently, several composite drought indices have been developed combining different drought
indicators to improve drought characterizations from multiple aspects (e.g. Waseem et al.,
2015; Rajsekhar et al., 2014; Azmi et al., 2015).
Hao and AghaKouchak (2013, 2014), using theoretical and empirical copula functions,
presented parametric and non-parametric versions of the Multivariate Standardized Drought
Index (MSDI), respectively. MSDI is an agro-meteorological drought index based on the
bivariate distribution of SPI and SSI (standardized soil moisture index). To obtain MSDI, the
joint cumulative probability is transformed with the inverse CDF of a standard normal
distribution. However, it is important to note that since the joint cumulative probability is not
uniformly distributed on [0,1], the transformation will not result in normally-distributed index.
In addition, copula function, to be applied, needs input data to be time-independent, which does
not hold for SPI and SSI in general.
The objective of the present paper is to develop a composite agro-meteorological drought
index, properly defined, copula-based, addressing seasonality and auto-correlation issues.
Furthermore, the proposed framework could be used to assimilate two or more standardized
drought indices, into a single drought index that may be useful for comprehensive decision-
making.

2. Methods
In section 2.1, SPI and SSI indices have been briefly recalled. To avoid lack of information
in drought monitoring, and include all within-year variations, SPI and SSI are computed at
twelve time windows, from 1 to 12 months. In section 2.2, the modified SPI (SPI𝑚) and the
modified SSI (SSI𝑚), subgrouped by the ending month m, have been considered to account the
seasonality of the variables. In section 2.3, SPI𝑚 at 12 time windows (w=1,...,12 months)
𝑤
combined, through the principal component analysis, in the multivariate standardized
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precipitation index, MSPI. Similarly, it has been done for SSI𝑚 in the multivariate standardized
𝑤
soil moisture index, MSSI. These indices are standardized (subtracting the mean and dividing
by the standard deviation) to remove the seasonality. In section 2.4, the autocorrelation within
MSPI and MSSI time series has been removed through a whitening procedure, building
whitened series indicated by MSPIwh and MSSIwh, respectively. In section 2.5, a new agro-
meteorological drought index, denominated AMDI-SA, has been introduced combining
together MSPIwh and MSSIwh using the concept of copula and the Kendall function. In section
2.6, an interpretation of the proposed index has been presented.

2.1 SPI and SSI indices
The most common drought index is the Standardized Precipitation Index (SPI), introduced
by McKee et al. (1993). SPI is calculated with reference to different time scales, and can assess
drought severity. As it is a standardized index, the frequency of extreme drought events at
different locations and time scales are consistent and comparable. Let X and F (x) denote
w Xw
the accumulated precipitation at time scale w and its corresponding probability distribution
function, respectively. F (x) is transformed into the standard normal precipitation index (SPI)
Xw
at time scale w as:
SPI =Φ-1(F (x)), where Φ-1 is the inverse of the standard normal distribution.
w Xw
Sometimes it is hard to discriminate among canonical forms of F (x), or they may not
Xw
provide a good fit to the data (Soláková et al. 2014; Lall et al., 2016). On the other hand, using
different distribution functions could lead to different tail behavior and thus inconsistencies in
characteristics of extremes across space (Farahmand and AghaKouchak, 2015). Therefore, a
nonparametric approach is used to obtain the probability values of X . The Gringorten plotting
w
position is used to calculate the cumulated frequency of non-zero values, F=(i−0.44)/(n+0.12),
i
where n denotes the sample size of non-zero values, and i refers to the rank of the non-zero
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observation x , ordered from the smallest to the largest. Since there are zero values of
(i)
precipitation data, the frequency of zeros has to be added to the plotting position of non-zero
values to estimate F . The method of handling zeros proposed by Stagge et al. (2015) is used,
Xw
which is superior to using the relative frequency for the probability of zero values. Thus, F (x)
Xw
is calculated as:
𝑛0 +(1− 𝑛0 )(𝑖 −0.44), for 𝑥 = 𝑥
(𝑖)
F (𝑥) = {𝑛tot+1 𝑛tot+1 (𝑛+0.12) (1)
X𝑤 𝑛0+1 for 𝑥 = 0
,
2(𝑛tot+1)
where n is the number of zero values, and n =n+n .
0 tot 0
The standardized soil moisture index (SSI) (e.g., AghaKouchak, 2014) can be defined in
a similar way to SPI. Here, SSI has been derived from soil moisture data averaged up to 100
cm depth.
However, SSI, like SPI, has two weaknesses:
1) The index does not take into account the seasonal variability within the annual regime. In
other words, it fits all data (whether it has been observed in wet or dry season) to the same
probability distribution.
2) Increasing the temporal window (w), it increases the temporal overlap between two
successive values of the index, introducing more auto-correlation to the time series of the index,
and bias in the probability distribution fitting.
2.2 Modified SPI and SSI indices
Kao and Govindaraju (2010) proposed the following modification in the calculation of SPI to
account for the seasonal variability of data. The aggregated precipitation X , at a given time
w
window w, is grouped according to the ending month m (m=1 means January, ..., m=12
December). Thus, the series {X } is subdivided into 12 smaller subseries corresponding to 12
w
months of the year, {X m}. X m(y) is the aggregated precipitation over the time window w,
w w
having m as the ending month and relative to the year y. Thus, X 10(y) is the value of the year
1
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y, with a window size 1 having October as ending month, while X 10(y) is the value of the year
5
y, with a window size 5 from May to October. In doing so, observations in each set {X m} will
w
not have overlapping information, when w≤12, and reduce the auto-correlation among the
samples {X m}. In addition, observations in each set {X m} are subject to the same seasonal
w w
effect, and hence, the seasonal variation is accounted for, properly (Kao and Govindaraju,
2010). Then for each of the 12 variables X m, the empirical frequency gives an estimation of
w
the probability distribution, F m(x), and the modified index SPI𝑚 = Φ−1(F (𝑥)) is
Xw 𝑤 X𝑤𝑚
obtained. Similarly, it is possible to calculate the modified index SSI𝑚.
𝑤
2.3 MSPI and MSSI indices
MSPI, recently developed by Bazrafshan et al. (2014), applies the multivariate technique
of Principal Component Analysis (PCA) to a set of SPIs referred to different values of w. This
technique is applied to SPI𝑚 and SSI𝑚.
𝑤 𝑤
Suppose that observations are stored in O, a vector of K variables with covariance matrix
Cv. The Principal Component Analysis (PCA) is a linear combination of K variables: PC =
𝑖
𝐸T𝑂 = ∑𝐾 𝑒 𝑂 with k=1,...,K, where PC is the ith principal component, 𝐸𝑇 is the ith
𝑖 𝑘=1 𝑘𝑖 𝑘 i 𝑖
eigenvector of Cv sorted in descending order of corresponding eigen values and e is the kth
ki
element of the ith eigenvector of Cv. These components are: firstly, extracted in such a way
that the first one (PC ) justifies the greatest percentage of variance of K original variables
1
mutually uncorrelated; secondly, this linear combination is mutually uncorrelated the
components can be at most as many as the original variables; and thirdly, the components are
extracted in such a way that the first one (PC ) justifies the greatest percentage of variance of
1
K original variables (Wilks, 2011). The PCA can be useful if the correlation among the original
variables is high.
Here, the PCA technique was applied to each of two sets of variables SPI𝑚 and SSI𝑚 with
𝑤 𝑤
w=1,...,12 months. However, it can be applied to any arbitrary set and may include other time
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scales depending on the research needs. Thus, the first component of SPI𝑚 is indicated as P𝑚
𝑤 1
and is equal to
P𝑚 = ∑12 𝑒𝑝 SPI𝑚 (2)
1 𝑤=1 𝑤1 𝑤
where 𝑒𝑝 is the wth element of the first eigenvector of covariance matrix of SPIs. Similarly,
𝑤1
the first component of SSI𝑚 is indicated as S𝑚 and equal to
𝑤 1
S𝑚 = ∑12 𝑒𝑠 SSI𝑚 (3)
1 𝑤=1 𝑤1 𝑤
where 𝑒𝑠 is the wth element of the first eigenvector of covariance matrix of SSIs. P𝑚 and
𝑤1 1
S𝑚are characterized by seasonality and are not comparable among different months or places.
1
Therefore, normalized variables P̂ and Ŝ are introduced, respectively for P𝑚and S𝑚,
1 1 1 1
subtracting the mean and dividing by the standard deviation:
MSPI = P1𝑚−𝜇𝑃1𝑚 ≈ P1𝑚 (4)
𝜎𝑃1𝑚 𝜎𝑃1𝑚
MSSI = S1𝑚−𝜇𝑆1𝑚 ≈ S1𝑚 (5)
𝜎𝑆1𝑚 𝜎𝑆1𝑚
where 𝜇 and 𝜎 are respectively the mean and standard deviation of P𝑚, while 𝜇 and
𝑃𝑚 𝑃𝑚 1 𝑆𝑚
1 1 1
𝜎 of S𝑚. 𝜇 and 𝜇 are close to zero, and so can be ignored in the numerator of Eq.s (4-
𝑆𝑚 1 𝑃𝑚 𝑆𝑚
1 1 1
5) (Keyantash and Dracup, 2004).
MSPI and MSSI are normally distributed with zero mean and unit variance. These indices
summarize several information avoiding problems connected to the selection of the appropriate
time scales of SPI and SSI. Similar to SPI or SSI, negative values of MSPI or MSSI indicate
drought conditions, while positive values to wet conditions. Normal conditions are associated
to values of MSPI or MSSI close to zero, see Table 1.
2.4 Whitening MSPI and MSSI
Since MSPI or MSSI are auto-correlated and input variables of copula function must be
free of auto-correlation (statistically “white”), the temporal dependence has been filtered out.
A classical whitening procedure (Box and Jenkins, 1970) has been applied to MSPI and MSSI,
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assuming that these can be described by autoregressive moving-average (ARMA) models.
Whitened residuals of MSPI and MSSI are indicated as MSPIwh and MSSIwh. Details about
ARMA models can be found in Box and Jenkins (1970) and Hipel and McLeod (1996). The
Ljung–Box test has been used to assess the absence of auto-correlation in MSPIwh and MSSIwh
time series, at a significance level of 0.05 (Ljung and Box, 1978; Hipel and McLeod, 1996).
(Wang et al., 2012).
2.5 Composite Drought Index AMDI-SA
To have a comprehensive description of droughts, a drought index has been considered
combining whitened residuals MSPIwh and MSSIwh together within the copula framework. With
reference to the bivariate case, the copula C(u,v) is a cumulative distribution function of
uniform marginals in the unitary interval, u,v[0,1] (Joe, 1997; Salvadori et al., 2007; Nelsen,
2013). Thanks to the Sklar’s theorem (Sklar, 1959), the joint cumulative distribution function
of MSPI𝑤ℎ and MSSI𝑤ℎ, F , can be written in terms of copula as:
MSPI𝑤ℎ,MSSI𝑤ℎ
F (𝑝,𝑠) = C(F (𝑝),F (𝑠)) (6)
MSSI𝑤ℎ,MSPI𝑤ℎ MSPI𝑤ℎ MSSI𝑤ℎ
where F and F are the marginals and C is the copula.
MSPI𝑤ℎ MSSI𝑤ℎ
The Kendall distribution function (K (t)), also called Kendall’s measure, is the probability
C
measure of the set {(F , F )∈[0,1]2: C(F , F )⩽t}, with t [0,1]. It is
MSPI𝑤ℎ MSSI𝑤ℎ MSPI𝑤ℎ MSSI𝑤ℎ
defined as:
K (t)=Pr[C(F ,F )⩽t] (7)
C MSPI𝑤ℎ MSSI𝑤ℎ
where K (t) is a univariate probability distribution. For some copula families, like
C
Archimedean ones, K (t) has an analytical form; and for others, like elliptical copulas, it has
C
not a closed-form, and thus, it is calculated numerically. For more details on the Kendall
distribution function, see Nelsen et al. (2003); Salvadori et al. (2007); Salvadori and De
Michele (2010). Every copula C(𝑢,𝑣) satisfies the relation W C M on [0,1]2, with W
(counter-monotonicity copula) and M (co-monotonicity copula), also referred as the upper and
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