Evaluation of wet and dry event’s trend and instability based on the meteorological drought index

Study area

The study areas include four meteorological stations (Multan, Bahawalpur, Barkhan and Khanpur) of Pakistan. These selected stations are located in the Southern part of Pakistan, mostly very hot and mildly cold areas. Since our objective is to analyze the trend and weather instabilities with respect to time, therefore, selecting four (stations) time-series data is enough, increasing the number of stations may cause a problem in time series analysis. Monthly quantitative data of precipitation from January 1990 to December 2018 has been obtained from the Karachi Data Processing Center through the Pakistan Meteorological Department, Karachi. The geographical presentation of selected stations is displayed in Fig. 1.

Standardized Precipitation Index at 3-months (SPI-3)

Mckee, Doesken & Kleist (1993) proposed a Standardized Precipitation Index (SPI) for defining and monitoring wet and dry events i.e., beginning, ending and intensity. The SPI is used to measure the precipitation shortage from the long-term historical record of precipitation. In this study, the cumulative precipitation series such as 3 months’ time scale is used to calculate SPI. Because, SPI-3 with a short time scale describes droughts, which is important for the agriculture sector (Caloiero, 2018; Zhang et al., 2012; Ali et al., 2017).

Method for Dry/Wet events trend analysis

Methods for weather instabilities

Mean Marginal Hilbert Spectrum (MMHS)

Hilbert Hung Transformation (HHT) was offered by Huang, Long & Shen (1996). HHT is a very powerful tool and used for dealing with stationary and non-linear series. IMFs are obtained through a decomposition procedure and there is no difficulty in applying the Hilbert Transformation (HT) to obtain instantaneous frequency (IF) and instantaneous amplitude (IA). HT is only applicable for IMFs and not applicable for residual which is a monotonic function. Following (Huang et al., 2003), we can compute the Hilbert transform z(t) of time series y(t) as: (5)${\displaystyle z\left(t\right)=\frac{1}{\pi }P\int \frac{y\left(t^{\prime }\right)}{t-t^{\prime }}d{t}^{\prime }}$ where, P is the Cauchy value and HT be present for all the functions of LP class (Bailey & Titchmarsh, 1938). By definition, the analytical signal R(t) consists of the complex conjugate of and y(t) and z(t) (6)${\displaystyle R\left(t\right)=y\left(t\right)+iz\left(t\right)=a\left(t\right)e^{i\phi \left(t\right)}}$ where, (7)${\displaystyle a\left(t\right)=\sqrt{\left(y^2\left(t\right)+z^2\left(t\right)\right)}}$ (8)${\displaystyle \phi \left(t\right)=arctan\left(\frac{z\left(t\right)}{y\left(t\right)}\right)}$ where, a(t) represents the superlative local fit of amplitude φ(t) and phase varying trigonometric function to y(t) (Huang et al., 2003). IF is obtained by taking the first order derivative of phase t to time t. i.e., (9)${\displaystyle w=\frac{d\theta \left(t\right)}{dt}.}$ The IF is a mono-component function in which only one component represents only one frequency. After completion of HT for each IMF, the following structure of the time series is shown as real part: (10)${\displaystyle y\left(t\right)=RP\left[\sum _{j=1}^n{a}_j\left(t\right)e\left(i\int w_{j}tdt\right)\right]}$ HT is not applicable for the residual term (Non-IMF), because residual is a constant term. With an explanation of the HS, it is also very important to define the Mean Marginal Hilbert Spectrum (MMHS). i.e., (11)${\displaystyle h\left(\omega \right)={\int }_0^{T}H\left(\omega ,t\right)dt.}$

The MMHS presents the contribution of total amplitude which is corresponding to each frequency value.

Non-parametric trend tests

It is very essential to extract the underlying pattern of hydrological and meteorological time series. There are several techniques for identifying patterns and analyzing hydrological meteorological time series trends. The assumptions of parametric test i.e., normality, independence and linearity are not met in the hydrological or meteorological time series. Some non-parametric techniques for dealing with non-normal and non-linear series are available in the literature, such as Mann-Kendall (MK) and Spearman’s Rho (SR) trend test.

Trend analysis of wet and dry events

Later, the SPI-3 is decomposed into IMFs by using the EEMD procedure. The decomposition level is based on the total number of observations. There are 348 total observations as the SPI-3 series consists of monthly observations from the year 1990 to 2018. According to Sang et al. (2016), log₂ (348) provides eight levels of decomposition. The ensemble number is set to 100 and the amplitude of white noise is about 0.2 standard deviation has been added in SPI-3 for construction of IMFs by EEMD (Wu & Huang, 2009). Finally, by applying the EEMD method, SPI-3 of all stations is decomposed into eight IMFs and a residual term, which is depicted in Fig. 2.

Figure 2 shows the IMF-1 and IMF-2 intra-annual fluctuations with the duration (<1) year for all stations, though the IMF-3 and IMF-4 specifies the Intra-decadal fluctuation with periodicity between (1–3) years and (1–6) years in Multan and Khanpur, respectively. Similarly, the IMF-3, IMF-4 and IMF-5 show the intra-decadal fluctuations with the period (1–6) years in Bahawalpur and (1–4) years in Barkhan. The remaining IMFs of all stations signifies the inter-decadal fluctuations with periodicity (>10) years. These fluctuations present the instabilities of wet and dry events corresponding to each IMF. The residuals show a decreasing trend of normal dry from the year 1990 to 1998 and suddenly indicates an increasing trend of normal dry from the year 2010 to 2018 in Fig. 2A. Figure 2B shows the residual stability of a normal wet trend before the year 1998. Figure 2C shows the trend of normal wet is increasing from the year 2007. Figure 2D shows the trend of normal wet is decreasing from the year 1990 to 2007, but the trend is gradually started increasing from 2015 to onwards.

The EEMD procedures decomposed the SPI-3 on the sequence from the largest periodic component to the smallest periodic component. The first component contains the largest fluctuations of wet and dry events and gradually decreases in the fluctuations of wet and dry events on the last component. Besides, the last component (residual) indicates long term weather variability.

The significance of IMFs is tested by correlation approach (Peng, Tse & Chu, 2005). Initially, the correlation is computed between the IMFs and SPI-3 for all stations, then the threshold is applied. The correlation approach indicates that all IMFs by EEMD are statistically significant. The reconstruction of SPI-3 is based on the aggregate of all IMFs and a residual. All IMFs are utilized for reconstruction of SPI-3. Therefore, the MSE is computed between SPI-3 and the reconstructed SPI-3. It is found that the MSE of reconstructed SPI-3 has been found in Multan, Bahawalpur, Barkhan and Khanpur is 0.0024, 0.0052, 0.0034 and 0.0020, respectively.

The MK and SR trend tests are applied on SPI-3 of all stations to verify the reliability of residual trends. Initially, SPI-3 is divided into eight periods. So that the detailed information is obtained about the increasing and decreasing trends. Then the most commonly used 5% significance level is used to identify the trend. Hence, the statistical trend results of MK and SR is described in Table 2.

It is shown in Table 2 that the MK and SR indicate similar results of trend identification in all periods of Barkhan station. Whereas, MK and SR do not indicate similarity of trend identification in 2014 for Multan, in 1994 for Bahawalpur and 1998 period for Khanpur at the 5% level of significance. The positive sign is an indication of the increasing trend, while the negative sign is an indication of the decreasing trend. Moreover, the trend identification of different periods by MK and SR test is compared with the residuals of EEMD (Fig. 2) for all stations. It turns out that the SR test indicates the consistency with residuals of EEMD in mostly periods. For instance, the SR test verifies that the residuals of Multan indicate an increasing trend in 2014 while the trend does not verify by the MK test (Fig. 2A). Likewise, the residuals of Khanpur indicates a decreasing trend in 1998 which is verified by the SR test, while not verified by the MK test (Fig. 2D). SR and MK provide similar trend identification of periods for all stations except the above-mentioned stations i.e., Multan and Khanpur.

Evaluation of weather instabilities

After this, there is no difficulty to apply an HT to obtain the IF and IA. We aim to describe the detailed information of wet and dry events in the time-frequency domain corresponding to each IMF. Therefore, HT is performed on decomposed IMFs of EEMD for all stations. Generally, HT uses the IMFs to present IF and IA corresponding to each IMF. Finally, the minimum, maximum, mean and standard deviation of IA and IF are related to each IMF is presented in Table 3. Table 3 showed that the highest variation (based on the standard deviation of IF) was observed in IMF1 of all stations. It is an indication of non-stationarity of the SPI-3 for all stations.

The MMHS provides an aggregated amount of IA in the time-frequency domain. So MMHS has been applied on the IA and IF of IMFs for all stations. The IA and IF are related to EEMD because this was achieved after HT is applied on the decomposed IMFs of SPI-3. The MMHS plots of four stations are shown in Fig. 3.

In Figs. 3A, 3B and 3D indicate the low IF of wet and dry events at 0.1 cycles per months (about log2(348) months) that is corresponding to mean IF of IMF-2 (see Table 3), while the IFs away from 0.1 are indicating comparatively low IA. Figure 3C shows the low IF of wet and dry events at 0.2 cycles per month (about <1 months) that is corresponding to the mean IF of IMF-1 (see Table 3). Besides, the IFs away from 0.2 indicate comparatively low IA. It is described that Barkhan station underwent several seasonal instabilities over a short period of 5 months. Whereas, Multan, Bahawalpur and Khanpur stations underwent weather instabilities over a short period of 10 months.

The CWPS is performed on SPI-3 of all stations for the analysis of spectral features at the different time scales. The spectral features of SPI-3 can be seen in Fig. 4.

Figure 4 shows the high energy wet and dry events with a periodicity of 32 months are identified during the years (1991–2017) for all stations. Figures 4A and 4B show a spectrum of similar features of wet and dry events with a period of 16 months in Multan and Bahawalpur, respectively. During the years (1991–2010), the high energy wet and dry events are identified with a period of 16 months in Khanpur (Fig. 4D). While the high energy wet and dry events with the periodicity of 16 months are identified during the years (2002–2011) in Barkhan station (Fig. 4C). Besides, some small wavelet spectrums are indicating high energy wet and dry events with 4 to 10 months of periodicity for all stations. But the high strength of wet and dry events is found in Multan, Bahawalpur, Barkhan and Khanpur are 1.49, 1.03, 1.42 and 1.18, respectively. All these high energy wetness and dryness has been identified at the 5% level of significance.