Study area
The climate of Bangladesh is humid, warm, and tropical. The western part of
Bangladesh covers approximately 41 % or 60 165 km2 of the country.
The geographic coordinates of the study area extend between a latitude of
21∘36′–26∘38′ N and longitude of
88∘19′–91∘01′ E. The annual rainfall and average
temperature in the study area vary from 1492 to 2766 mm, with an average of
1925 mm, and 24.18 to 26.17 ∘C, with an average of
25.44 ∘C, respectively (Kamruzzaman et al., 2016a). Bangladesh is
the fourth-largest producer of rice in the world (Scott and Sharma, 2009),
and the livelihood of a majority of the people (approximately 75 %)
(Shahid and Behrawan, 2008; Kamruzzaman et al., 2016b) depends on
agricultural practices. The crop calendar of Bangladesh is related to the
climatic seasons. Rice is grown during three seasons (Aus,
Aman, and Boro) in Bangladesh. Almost 73.94 % of the
cultivable area in the country is used to cultivate Boro rice
(Banglapedia, 2003). The Aus and Aman rice varieties are
mainly rain-fed crops. However, Boro rice is almost completely
groundwater-fed (Ravenscroft et al., 2005) and requires approximately 1 m of
water per square meter in Bangladesh (Harvey et al., 2006; Michael and Voss,
2009).
Methods
In this study, the WBCs were calculated and their trends were identified
using the MK or Modified MK (MMK)
test for evaluating the long-term water
balance of the highly irrigated western part of Bangladesh. The DWT data of
the WBC time series were analyzed for identifying the time period
responsible for the trend in the data. The WBCs were forecasted using the
ARIMA model, whose performance was statistically evaluated. If the
performance of the model was unsatisfactory for forecasting the WBCs,
denoising of the original time series was conducted using DWT techniques to
improve the performance of the model. The descriptions of the methods are
presented in the following sections.
Calculation of the potential evapotranspiration and water
balance components
The potential evapotranspiration (PET) is a key parameter to
estimate the WBCs. In this study, the potential evapotranspiration was
calculated using the Penman–Monteith equation (Allen et al., 1998). The
soil–water balance concept proposed by Thornthwaite and Mather (1955) is one
of the most widely used methods for estimating the WBCs. This method is
suitable for assessing the effectiveness of agricultural water resource
management practices and regional water balance studies because it allows the
actual evapotranspiration (AET), water deficit, and water surplus
to be estimated (Chapman and Brown, 1966; Bakundukize et al., 2011; Karim et
al., 2012; Viaroli et al., 2017). The actual evapotranspiration
(AET) is the amount of water removed from the surface due to
evaporation and transpiration. The amount by which the PET
exceeds the AET is termed as the deficit. The surplus is the
excess rainfall received after the soil has reached its water-holding
capacity (de Jong and Bootsma, 1997). Calculating the field capacity of the
soil is essential for estimating the WBCs. The field capacity of the soil in
the study area was calculated using the soil texture map of Bangladesh
prepared by the Soil Resource Development Institute, Bangladesh (SRDI, 1998),
where the description of the soils was presented by Huq and Shoaib (2013).
The values suggested by Thornthwaite and Mather (1957) for the water-holding
capacity of the soil and rooting depth of the plants were used for estimating
the WBCs in this study. The first step of the calculation involves
subtracting 5 % rainfall from the monthly rainfall data because this
amount of water is lost due to direct runoff (Wolock and McCabe, 1999; Karim
et al., 2012; Kanoua and Merkel, 2015). The remaining rainfall amount is
included in the calculation. The WBCs, such as the AET, surplus,
and deficit, were estimated using the formulas presented in Table 1. The
details of the WBC calculations are available in the Supplement.
Calculations of water balance components (Thornthwaite and Mather,
1957).
Wet months (P-R0>PET)
Dry months P-R0<PET
AET
PET
P-R0+ΔSB
Deficit
0
PET-AET
Surplus
P-R0-PET
0
P is the rainfall (mm), R0 is the direct runoff (mm), PET is
the potential evapotranspiration (mm), AET is the actual
evapotranspiration (mm), and ΔSB is the changes in soil moisture
storage (mm).
Trend test
In this study, the trends in the WBCs were detected using the nonparametric
MK test (Mann, 1945; Kendall, 1975) because it exhibits a better performance
than the parametric test (Nalley et al., 2012) for identifying trends in
hydrological variables, such as rainfall (Shahid, 2010), temperature
(Kamruzzaman et al., 2016a), PET (Kumar et al., 2016), soil
moisture (Tabari and Talaee, 2013), runoff (Pathak et al., 2016), groundwater
level (Rahman et al., 2016), and water quality (Lutz et al., 2016). The MK
test cannot be used to accurately calculate the test statistic (Z) if there
exists a significant serial correlation at lag 1 in the time series data (Yue
et al., 2002) because the variance is underestimated (Hamed and Rao, 1998).
The autocorrelation at lag 1 was checked before analyzing the time series
data. If there existed a significant lag-1 autocorrelation at the 5 %
level, the MMK test (Hamed and Rao, 1998) was applied instead of the MK test.
The estimated Z statistic from the MK or MMK test was evaluated for the
direction of the trend (a positive Z statistic indicated an increasing
trend and vice versa). Moreover, the Z statistic indicated the level of
significance of the obtained trend. If the calculated Z statistic is equal
to or higher than the tabulated value of the Z statistic (+1.96), it
indicates a significant positive trend at the 95 % confidence level. If
the calculated Z statistic is equal to or less than -1.96, it indicates a
significant decreasing trend. Moreover, the sequential values of the
u(t) statistic derived from the sequential MK (SMK) test (Sneyers, 1990)
are used for detecting the change point. The u(t) statistic is similar to
the Z statistic (Partal and Küçük, 2006). The magnitude of the
change was calculated using Sen's slope estimator (Sen, 1968). Numerous
studies have already been conducted (notably Nalley et al., 2012) using the
methods described in this section. Further details regarding these methods
can be obtained from Mann (1945), Sen (1968), Kendall (1971), Hamed and
Rao (1998), Sneyers (1990), and Yue et al. (2002).
Wavelet transform (WT) and periodicity
Wavelet analysis has been used in different parts of the world to identify
the periodicity in hydroclimatic time series data (Smith et al., 1998; Azad
et al., 2015; Nalley et al., 2012; Araghi et al., 2015; Pathak et al., 2016).
WT, a multiresolution analytical approach, can be applied to analyze time
series data because it offers flexible window functions that can be changed
over time (Nievergelt, 2001; Percival and Walden, 2000). WT can be applied to
detect the periodicity in hydroclimatic time series data (Smith et al., 1998;
Pišoft et al., 2004; Sang, 2012; Torrence and Compo, 1998; Araghi et al.,
2015; Pathak et al., 2016) and exhibits better a performance than traditional
approaches (Sang, 2013). There exist two main types of WT, namely continuous
WT (CWT) and DWT. Applying the CWT is complex because it produces numerous
coefficients (Torrence and Compo, 1998; Araghi et al., 2015), whereas DWT is
simple and useful for hydroclimatic analysis (Partal and Küçük,
2006; Nalley et al., 2012). The wavelet coefficients of the DWT with a dyadic
format can be calculated as follows (Mallat, 1989):
ψm,nt-τs=so-m/2ψt-nτosomsom,
where ψ is the mother wavelet, m is the wavelet dilation, and n is
the wavelet translation. The specified fixed dilation step (so) is
larger than 1, and τo is the location parameter. For practical
application, the values of so and τo are considered as 2 and 1,
respectively (Partal and Küçük, 2006; Pathak, 2016). After
substituting these values in Eq. (1), the DWT for a time series xi
becomes the following:
Wm,n=2-m/2∑i=0N-1xiψ2-mi-n,
where W indicates the wavelet coefficient at a scale s=2m and
location τ=2mn
In the DWT, details (D) and approximations (A) of the time series can emerge
from the original time series after passing through low-pass and high-pass
filters, respectively. When approximations are the high-scale and
low-frequency components, details are the low-scale and high-frequency
components. Successive iterations are performed to decompose the time series
into its several low-resolution components (Mallat, 1989; Misiti et al.,
1997). In this study, four levels (D1–D4) of decomposition were performed
following the dyadic scales. The decompositions are referred to as D1, D2,
D3, and D4, which correspond to a 2-, 4-, 8-, and
16-year periodicity, respectively. The Daubechies wavelet was used because of
its superior performance in hydrometeorological studies (Nalley et al., 2012,
2013; Ramana et al., 2013; Araghi et al., 2015). To confirm the periodicity
present in the time series, the correlation coefficient (Co) between u(t)
of the original data, u(t) of the decomposition (D) time series data, and
different models (D1 + A … D4 + D3 + A) of the time series
data were calculated and the obtained results were compared (Partal and
Küçük, 2006; Partal, 2009).
ARIMA models
ARIMA models (Box and Jenkins, 1976) are used in hydrological science to
identify the complex patterns in data and project future scenarios (Adamowski
and Chan, 2011; Valipour et al., 2013; Nury et al., 2017; Khalek and Ali,
2016). ARIMA models include (1) an autoregressive process (AR) represented by
order p, (2) nonseasonal differences for nonstationary data termed as
order d, and (3) a moving average (MA) process represented by order q. An
ARIMA model of order (p,d,q) can be written as follows:
∅pL1-LdYt=θ0+θqLUt,
where θ0 is the intercept with a mean of 0, Ut is
the white process with constant variance, ∅pL represents the AR term (1-∅1L-…-∅pLp), and θqL represents the MA term (1-θ1L-…-θpLp).
Wavelet denoising
Wavelet denoising based on the thresholds introduced by Donoho et al. (1995)
has been applied to hydrometeorological analysis (Wang et al., 2005, 2014;
Chou, 2011). In this study, the following three analysis steps were performed
for denoising the time series data.
Decomposing the time series data x(t) into M resolution levels for
obtaining the detail coefficients (Wj,k) and approximation coefficients
using the DWT.
The detail coefficients obtained from the DWT (1 to M levels) were treated
using threshold (Tj) selection. A soft or hard threshold can be used to
deal with detail coefficients and obtain the decomposed coefficient. In this
study, a soft threshold was selected because it performed better than a hard
threshold (Wang et al., 2014; Chou, 2011).Soft threshold processing:W′j,k=sgn(Wj,k)Wj,k-TjWj,k>Tj0Wj,k<Tj.
Detail coefficients from levels 1 to M and approximate coefficients at level
M were reconstructed to obtain denoising time series data.
Selecting the threshold value is essential for denoising the data. In this
study, the universal threshold (UT) method (Donoho and Johnstone, 1994) was
used for estimating the threshold value because it exhibited satisfactory
performance in analyzing hydrometeorological data (Wang et al., 2005; Chou,
2011).
Assessment of model performance
There exist several indicators to assess the performance of the models. The
Nash–Sutcliffe efficiency (NSE) (Nash and Sutcliffe, 1970) coefficient, a
normalized goodness-of-fit statistic, is the most powerful and popular
method for measuring the performance of hydrological models (McCuen et al.,
2006; Moussa, 2010; Ritter and Muñoz-Carpena, 2013). The NSE coefficient
was used in this study to evaluate and compare the ARIMA and WD-ARIMA
models. The NSE is calculated as follows (Nash and Sutcliffe, 1970):
NSE=1-∑i=1N(Oi-Pi)2∑i=1N(Oi-O‾)2=1-RMSESD2,
where N, Oi, Pi, O‾, and SD are the sample size,
number of observations, model estimates, mean, and standard deviation of the
observed values, respectively. The performance of a model can be evaluated
according to its NSE value as very good (NSE≥0.9), good
(NSE = 0.8–0.9), acceptable (NSE≥0.65), and
unsatisfactory (NSE < 0.65) (Ritter and Muñoz-Carpena, 2013).
ERMS is the root-mean-square error and can be calculated as
follows:
ERMS=∑i=1N(Oi-Pi)2N.
The coefficient of determination (R2) is another goodness-of-fit test to
measure the performance of models. The perfect fit of the model draws a line
between the actual values and fitted values, where R2 is 1. If yi
is the observation data, y^i represents the model-forecasted values
of yi and N is the number of data points used. R2 is given as
follows (Sreekanth et al., 2009):
R2=1-∑i=1Nyi-y^i2∑i=1Nyi2-(∑i=1Nyi)2N.
Moreover, the mean percentage error (EMP) and mean error
(EM) were also calculated to evaluate the validation of the model
for forecasting. EMP indicates the percentage of bias (large or
small) between the forecasted and actual data (Khalek and Ali, 2016).
EMP and EM can be calculated as follows:
EMP=1n∑t=1nYtactual-Yt(forecasted)Yt(actual)×100%,EM=1n∑t=1n[Ytactual-Yt(forecasted)]2.