The objectives of the present study were to explore the changes in the water
balance components (WBCs) by co-utilizing the discrete wavelet transform
(DWT) and different forms of the Mann–Kendall (MK) test and develop a wavelet
denoise autoregressive integrated moving average (WD-ARIMA) model for
forecasting the WBCs. The results revealed that most of the potential
evapotranspiration (
The water balance model is considerably important for water resource management, irrigation scheduling, and crop pattern designing (Kang et al., 2003; Valipour, 2012). The model can also be used for the reconstruction of catchment hydrology, climate change impact assessment, and streamflow forecasting (Alley, 1985; Arnall, 1992; Xu and Halldin, 1996; Molden and Sakthivadivel, 1999; Boughton, 2004; Anderson et al., 2006; Healy et al., 2007; Moriarty et al., 2007; Karimi et al., 2013). Therefore, accurately forecasting the water balance components (WBCs) and detecting the changes in them is important for achieving sustainable water resource management. However, hydrometeorological time series are contaminated by noises from hydrophysical processes. This affects the accuracy of the analysis, simulation, and forecasting (Sang et al., 2013; Wang et al., 2014). Hence, denoising the time series is essential for improving the accuracy of the obtained results. In this study, the wavelet denoising technique was coupled with the ARIMA (autoregressive integrated moving average) model for forecasting the WBCs after detecting the changes in them by using different forms of the Mann–Kendall (MK) test. Moreover, the time period responsible for the trends in the WBC time series was identified using discrete wavelet transform (DWT) time series data.
Physics-based numerical models are generally used for understanding a
particular hydrological system and forecasting the water balance or water
budget components (Fulton et al., 2015; Leta et al., 2016). To achieve
reliable forecasting using numerical models, a large amount of hydrological
data is required for assigning the physical properties of the grid and model
parameters and calibrating the model simulation. However, numerical models
have numerous limitations, such as the cost, time, and availability of the
data (Yoon et al., 2011; Adamowski and Chan, 2011). Data-based forecasting
models and statistical models are suitable alternatives for overcoming these
limitations. The most common statistical methods for hydrological forecasting
are the ARIMA model and multiple linear regression (Young, 1999; Adamowski,
2007). Many studies have used the ARIMA model to predict water balance input
parameters, such as rainfall (Rahman et al., 2016), temperature (Nury et al.,
2016), and potential evapotranspiration (
The climate of Bangladesh is humid, warm, and tropical. The western part of
Bangladesh covers approximately 41 % or 60 165 km
The national climate database of Bangladesh prepared by the Bangladesh
Agricultural Research Council (BARC) was used for this study. The database is
available for research and can be obtained from the BARC website
(
Study area in the western part of Bangladesh with locations of meteorological stations.
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.
The potential evapotranspiration (
Calculations of water balance components (Thornthwaite and Mather, 1957).
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),
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):
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
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
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 The detail coefficients obtained from the DWT (1 to Detail coefficients from levels 1 to
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).
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):
The mean annual
Distribution of mean annual
The MK or MMK test based on lag-1 autocorrelation was applied to detect the
trend in the
MSE, total mean square error; Co, correlation between original data and DWT models.
Sequential values of the
Most of the trends (73 %) observed in the
Comparison of performance of ARIMA model and WD-ARIMA model.
All the stations except the Bogra station exhibited decreasing trends in the
Distribution of rate of changes of WBCs during the period of 1981–1982 to 2012–2013.
Almost 82 % of the stations exhibited insignificant decreasing trends for
the annual surplus of water. The magnitude of the trends of the original
annual surplus data ranged from
Approximately 73 % of the stations exhibited increasing trends for the
annual deficit of water. The increasing trends were significant for two
stations at the 95 % confidence level (Table S4). However, the Satkhira
station exhibited a significant decreasing trend (Z
The ARIMA model was selected for forecasting the WBC time series. A four-step
analysis was performed during time series modeling. (1) First, the
stationarity of the data was checked using the Augmented Dickey–Fuller (ADF)
test. (2) Then, the autocorrelation function (ACF) was used for selecting the
order of the MA process (Figs. S2–S5). (3) The partial autocorrelation
function (PACF) was then used for selecting the order of the AR process
(Figs. S2–S5). (4) Finally, the appropriate model was selected based on
several trials and model selection criteria, such as Akaike information
criterion (AIC) and Bayesian information criterion (BIC). In addition to the
manual model selection based on the ACF, PACF, AIC, and BIC, the auto ARIMA
function of the “forecast” package (Hyndman et al., 2017) of R (R 3.4.0
language developed by R Core Team, 2016) was used during the trails for model
selection to obtain information regarding the nature of the data for
modeling. The model with the lowest AIC and BIC values and highest
Comparison between actual and wavelet denoise
The WD-ARIMA models were validated to explore their forecasting ability. The
mean percentage error (
Accuracy of WD-ARIMA models of WBCs for validation of the model's predictive ability for the period of 2009–2010 to 2012–2013.
Plot of best WD-ARIMA model first panel represents actual versus
fitted values for the period of 1981–1982 to 2012–2013, the second panel is
normal
This study indicated that a decreasing
The WD-ARIMA model was used in this study for forecasting the WBCs. The
performance of the model indicated the benefit of denoising hydrological time
series data, such as the
In this study, the changes in the WBCs were explored using various forms of
the wavelet-aided MK test. Moreover, a wavelet-aided ARIMA model was used for
forecasting the WBCs. The results obtained from trend analysis indicated that
decreasing trends were dominant in all the WBCs in the western part of
Bangladesh during the period from 1982–1983 to 2012–2013. However, most of
the trends were insignificant at the 95 % confidence level. One
significant positive and one significant negative
Modeling of the study revealed that the WBC time series data was affected by
noises from different hydrophysical interactions. As a result, the classic
ARIMA model exhibited unsatisfactory performance in most of the cases (e.g.,
The national meteorological database of Bangladesh prepared
by the Bangladesh Agricultural Research Council (BARC) was used to accomplish
this study. Data are available for research and can be obtained from the BARC
website (
The supplement related to this article is available online at:
ATMSR designed and wrote the manuscript with input from all co-authors. MSA, MK, MAK, and ATMSR prepared the R code and ATMSR, MK, MAK, and MSA performed the statistical analysis. HMA and ATMSR performed the water balance analysis. QHM and CSJ supervised the whole work.
The authors declare that they have no conflict of interest.
This article is part of the special issue “The changing water cycle of the Indo-Gangetic Plain”. It is not associated with a conference.
We thank the two anonymous reviewers for their constructive comments that greatly improved the manuscript. We would like to thank editor Ana Mijic of the special issue for her comments and support for publication in Hydrology and Earth System Sciences. Edited by: Ana Mijic Reviewed by: two anonymous referees