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Environ Eng Res > Volume 28(6); 2023 > Article
Gautam, Silwal, Baral, Subedi, Lamichhane, Chapagain, and Adhikari: Influence of the rainfall and temperature oscillation on air quality in Kathmandu valley: The wavelet analysis


In this study, wavelet analysis and traditional cross-correlation analysis (TCA) have been employed to explore the temporal characteristics of the Air Quality Index (AQI) (PM2.5) and its relationships with meteorological factors, air temperature, and rainfall in the capital city of Nepal, Kathmandu, during 2017 – 2020. The temporal variation of monthly averaged (AQI) (PM2.5) revealed that the air quality was unhealthy in winter, improved in summer, and subsequently became unhealthy again in autumn and spring. We also noticed that the lowest AQI (PM2.5) was recorded in 2020, which is due to the COVID-19 pandemic lockdown. By analyzing the time-frequency relationship of (AQI) (PM2.5) with air temperature and rainfall, using Cross Wavelet Transform (XWT), we observed the interannual, seasonal, and monthly common periodicity between them. The coherency of XWT is then tested by using Wavelet Transform Coherence (WTC), which showed a negative correlation of AQI (PM2.5) with rainfall and air temperature, while an anomaly was seen among AQI (PM2.5) and temperature as they were positively associated for a few months in 2017, 2018, and 2020. Moreover, the results of cross-correlation and multiresolution analysis (MRA) are consistent with wavelet analysis, indicating the driving effect of air temperature and rainfall on (AQI) (PM2.5).

1. Introduction

The increasing urbanization and industrialization in Kathmandu valley have led to air pollution becoming a major issue [1]. According to the World Health Organization, almost all low- and middle-income countries, including Nepal, inhale air that exceeds WHO air quality guidelines [2], and every year, 7 million deaths are caused by air pollution [3]. The main sources of air pollution are vehicles, industries, household combustion, and suspended dust particles [4, 5, 6]. Among other pollutants, particulate matter (PM), PM2.5, and PM10 are particles of great significance [5, 7]. The concave shape of the Kathmandu valley traps air inside, leading to pollution [8, 9] that affects public health [10, 11, 12], the ecosystem [13], and climate change [14, 15, 16]. Although the COVID-19 lockdown temporarily reduced PM emissions [17, 18], Kathmandu was still ranked as the 10th most polluted capital city globally in 2020, according to a report by IQAir [19].
In addition to topography, meteorological conditions also play a role in PM emission [2025]. To date, numerous studies have been conducted representing the relationship between PM2.5 and meteorological conditions. For instance, Wang & Ogawa [22] performed a correlation analysis between PM2.5 concentration and meteorological data in Nagasaki of Japan. They showed a negative correlation of PM2.5 with temperature, whereas positive with precipitation. In Wuhan, generalized additive models (GAM) were utilized to study the connection between meteorological parameters and PM2.5 which revealed the reduction of PM concentration by up to 37% during precipitation [26]. Higher precipitation reduces about 17% – 27% of PM2.5 from the air as higher rainfall rates deliver a large number of raindrops, raising the chances of colliding with PM and settling them down [27]. Some studies have also shown the seasonal variation of PM2.5 with meteorological parameters. For example, Yang et al. [24] reported the seasonal variation of PM2.5 with temperature in different cities in China. PM2.5 was positively correlated with temperature in winter whereas negatively during autumn. Similarly, Chen et al. [28] found a variation in correlation between PM2.5 and temperature in different seasons. The temperature was negatively correlated with PM2.5 in summer and autumn, whereas positively in spring and winter. On the other hand, a temperature rise can cause the alteration in the formation of PM as high temperature triggers the photochemical reaction between the PM [22]. However, in Hong Kong, the study done by Zhao et al. [29] from January – December 2013 concluded the negative association of PM2.5 with temperature. In Kathmandu, a study conducted by Becher et al. [30] showed a higher PM2.5 concentration in winter than in the summer season, which implies a negative association of PM2.5 with temperature. According to these studies, the association between PM2.5 and meteorological parameters seems to vary at geographical and seasonal scales. Hence, the comprehensive study of the relationships of PM2.5 with meteorological factors such as air temperature and rainfall in different locations is crucial for modeling their variation on the local and global scales.
In this particular work, we have implemented a wavelet analysis, which has emerged as a useful technique for investigating localized power fluctuations in a time series [31]. This technique has been employed in the study of water quality [32], seismic waves [33], snow depth [34], and even time series of pollutant concentration [21, 28, 35]. This approach can work with the dynamic window; a window works to observe the high-frequency content and be widened to capture the high-frequency content in the signals [36, 37]. Analyzing any signal involves accessing information about the signal’s frequency content and the time at which that frequency occurs. Therefore, it is our interest to study the temporal pattern of particulate matter (PM2.5) and its connections with meteorological parameters through wavelet analysis. A brief overview of wavelet analysis is discussed in this paper. A more extensive explanation of the wavelet analysis can be obtained in previous studies [28, 3841], they have shown the importance of wavelet analysis for the study of PM over a large time scale.
Previous studies [42, 43] have mentioned that addressing air pollution could benefit the lives and huge monetary value of the country, Nepal. Research of air pollution hazards in Kathmandu valley is crucial as it signifies health impacts on 3.5 million people, its ecosystem, cultural heritage, and even climate change phenomena. So, an attempt has been carried out to fulfill the knowledge gap on the seasonal and yearly trend of AQI (PM2.5) of Kathmandu and observe its variation with i) air temperature and ii) precipitation from 2017 to 2020 using wavelet spectrum analysis. The outcome of this study will help in understanding the nature and trends of the emission of PM, to counteract the soaring air pollution levels of Kathmandu valley.

2. Data Collection and Methods

2.1. Study Area

For the present study, Kathmandu valley has been selected which comprises three districts; Kathmandu, Bhaktapur, and Lalitpur, covering an area of 685 km2. The average altitudinal range of the valley is 1400 m; more than 3.2 million residents, representing 24% of the Nepalese suburb population, reside here [44, 45]. Kathmandu city is recognized as the city of temples, one of the top tourism areas, which since past few years has been tallied as one of the world’s most polluted capital cities [46]. The government of Nepal has established seven air quality monitoring stations across Kathmandu valley (as per the study period), which provide data on PM2.5 and PM10. The study area map is shown in Fig. 1.

2.2. Dataset

The World Air Quality project [47] provided the AQI (PM2.5) dataset, originally collected by the Government of Nepal, Department of Environment (DoE) [48], from 1st January 2017 to 30th December 2020 upon our request via their official website: https://aqicn.org. The daily average of AQI (PM2.5) of Kathmandu city was taken for this study. AQI is a numerical value for reporting daily air quality [49]. According to US EPA (U.S. Environmental Protection Agency) Standard, AQI < 50 is good, and < 100 is the moderate category. In contrast, AQI > 100 signifies unhealthy condition of air quality [9]. The data relating to meteorology (air temperature and rainfall) was acquired from the Department of Hydrology and Meteorology [50] via: http://www.mfd.gov.np. The missing data gap was filled by applying the appropriate interpolation technique.

2.3. Data Analysis

2.3.1. Cross-correlation analysis

Correlation is described as the dependency between two-time series and the cross-correlation, denoted by ρi,j, is the measure of the correlation between two or more time series [51]. It is used in orienting two-time series where one of which is delayed with respect to the other, and the peak occurs at the lag at which both series are best lined up or best correlated [52]. It is a dimensionless quantity whose values range from −1 to +1. The coefficient values around ±1 signify that the series is highly corrected, whereas the values around zero suggest poor correlation among series [53]. Many researchers have employed cross-correlation analysis as a powerful tool in various research fields [34, 41, 53, 54]. Mathematically, the cross-correlation between xi and xj is illustrated as follows:
where γi,j is covariance, σi2 and σj2 are the respective variance of xi and xj series.

2.3.2. Wavelet analysis

The time-series study on the pollutant’s concentration using wavelet analysis is considered the most effective method over the traditional methods [21]. It can capture many natural phenomena which exhibit self-similarity at different scales [55]. In this approach, the analyzing function should have zero mean value and be confined in both time and frequency to be admissible as a wavelet [56]. Wavelet function generally represents orthogonal or non-orthogonal wavelets, whereas wavelet basis only represents an orthogonal set of functions [57]. This approach has two main forms: Continuous Wavelet Transform (CWT) and Discrete Wavelet Transform (DWT). Continuous wavelet transform (CWT)

Continuous wavelet transform is a mathematical technique that analyzes a signal as a function of time and scale. It involves decomposing a signal f into wavelets of varying frequencies and scales to identify features or patterns that may be difficult to detect using other methods [58]. The CWT of f can be defined as [58, 59]:
where the mother wavelet is:
Here, ζ̄ is the complex conjugate of ζ, and a and b are the scaling factor (dilation or compression) and translation parameter (time shift) of the wave function, respectively. Moreover, ζL2(ℝ).
The Morlet wavelet, which has finer frequency localization and is analogous to the frequencies obtained in the Fourier spectrum [58], is employed as the generator function for our study. The amplitude of the Morlet Wavelet is regulated by a function corresponding to the standard Gaussian function and has good time-frequency localization capabilities [60]. The expression for wavelet is mentioned below [61]:
Here, ω0 is the non-dimensional central frequency of the wavelet. In addition, π-14 establish the unity energy of the Morlet wavelet, whereas e-ω022 ensures that it satisfies the admissibility condition [62] as discussed below.
The mother wavelet function, Ψ(t), must satisfy the following conditions [62].
  1. The value of the mean be zero,

    which confirms that Ψ behaves to be a wave-like structure and is also called an admissibility condition.
  2. The square of the mother wavelet should be equal to one,

    which ensures the wavelet must have fast decay amplitude (a restriction to a time interval). Discrete wavelet transform (DWT)

In this method, each set in a time series created by the DWT transform is made up of coefficients and describes the time evolution of a given signal in the appropriate frequency range [63]. DWT is non-reductant and only uses as many coefficients as were contained in the primary signal to completely represent the time series x(n). We pick to describe the DWT via the concept of a filter. A set of finite impulse response (FIR) filters is characterized along with L coefficients. We have two filters - a high pass filter and a low pass filter, where both filters are switched on and off at half the sampling frequency. DWT can be characterized by utilizing these filters and implementing them repetitively. The input time series are then treated with filters to get low and high pass components, x1(n) and x2(n), respectively [64].
Here, ck represents the coefficient of the low-pass, and dk represents the coefficient of high-pass filters. Furthermore, the two pairs of filter coefficients are associated through:
The computational advantages are gained in DWT over CWT as it uses a representation that applies a minimum number of possible coefficients. However, DWT lacks the most appealing features of CWT, such as invariance to time and scale shifts [64].

2.3.3. Cross wavelet transform (XWT)

Cross wavelets transform (XWT) is a bivariate wavelet analysis that examines the correlation in the time-frequency domain between two signals to analyze how phase angle represents the mechanism in casual and physical connections between the time series [65]. XWT is widely used in climate, meteorology, and geophysics as it can examine two-time series in a time-frequency domain from multiple time scales [6567]. According to [68], the cross-wavelet power spectrum for two-time series X(t) and Y(t) is as follows:
where WXY(a, b) is the cross-wavelet power spectrum, WX(a, b) denotes the sequence X(t) wavelet transform coefficients, and W*Y(a, b) represents the complex conjugate of the sequence Y(t) wavelet transform coefficients. Moreover, a and b are the scaling parameter and the time translation parameter.

2.3.4. Wavelet transform coherence (WTC)

Wavelet transform coherence is a bivariate framework that studies the relationship over a continuous time and frequency space between different time series and their evolution [69]. The wavelet coherence spectrum is used to measure how coherent the cross-wavelet transform is in time-frequency space [60]. Grinsted et al. [65] have defined cross wavelet coherence spectrum as:
Here, S is the smoothing operator and is given by [65]:
where Sscale and Stime represents smoothing along the wavelet scale axis and smoothing in time. The wavelet coherence value around 1 shows a greater similarity between the time series, whereas coherence near 0 shows no relationship [70].

3. Results and Discussion

In this section, the findings of data plots, wavelet analysis (continuous, discrete, cross-wavelet, and wavelet coherence), and cross-correlation analysis of AQI (PM2.5), air temperature, and rainfall in Kathmandu from 2017 to 2020 will be interpreted.
Fig. 2 depicts AQI (PM2.5), air temperature, and rainfall measured in Kathmandu City, Nepal, from 2017 to 2020. It is evident that AQI (PM2.5), air temperature, and rainfall time series illustrate the periodic feature after one year. According to the US EPA standard [9], AQI (PM2.5) > 100 is harmful to human health; the plot shows average AQI in Kathmandu exceeds the US EPA standard value for more than half of the period in one year. Based on available data, the monthly average and yearly average AQI (PM2.5) ranged from 46.18 to 158.86 and 91.43 to 115.90, respectively. It infers that AQI (PM2.5) had its minimum yearly average value in 2020 and maximum in 2018. Thus, air pollution was higher in 2018, and it was much improved in 2020 as the COVID-19 pandemic significantly reduced pollutants emission [10, 11].
Similarly, the monthly average air temperature ranges from 5.73 to 24.76° C, and the yearly average ranges from 17.04 to 17.60° C, with the lowest annual average in 2017 and a high annual average in 2019. Although the yearly average air temperature increased gradually from 2017 to 2019, comparable to the temperature rising trend in Kathmandu from 2013 to 2017, as demonstrated by Thapa et al. [71], a slight decrease in the mean value of air temperature (17.19° C) was observed in the year 2020. As we all know, the city’s lockdown in response to the COVID-19 epidemic has resulted in restrictions on Particulate Matter emission [7274]. This could be the reason for the improvement in air temperature in 2020.
Additionally, the rainfall trend indicates that March to September received the most rainfall, while the remaining months received minimal rainfall. Annual rainfall in Kathmandu from 2017 to 2020 ranges from 1253 mm to 1632 mm, with a maximum in 2018 and a minimum in 2017. The Fig. also shows the fact that AQI (PM2.5) and air temperature, as well as AQI (PM2.5) and rainfall, present opposite behavior, i.e., as temperature and rainfall rise, PM2.5 falls, and vice versa., which is in accordance with previous studies of Guo et al. [23], Zhang et al. [35], Qiu et al. [75], and He & Wang [76].
Fig. 3 represents the monthly variation of Kathmandu’s average AQI (PM2.5) and air temperature from 2017 to 2020. It shows that the AQI (PM2.5) in July and August is the lowest, and highest in December and January. This indicates that the pollution is less harmful in summer than in the season winter. This interpretation is equivalent to the previous studies of Yang et al. [24], Zhang et al. [77], and Yan et al., [78] in Chinese regions. The study by Thapa et al. [71] demonstrated similar patterns in air temperature in Kathmandu between 2013 and 2017. It is clearly seen from the Fig. that the AQI (PM2.5) and air temperature obey the pattern of movement in opposite directions; with the increase in temperature, the AQI (PM2.5) lowers, and with the fall of temperature, the AQI (PM2.5) rises. This pattern has also been noticed in pollution investigations in different countries; Thailand, Kathmandu, and Hongkong [30, 40, 79]. As noted in Fig. 3, the study’s key conclusions include comparatively lower average AQI (PM2.5) in 2020 (noticeably from March to June) than in previous years. It indicates that the pollution of Kathmandu temporarily decreased to a moderate zone of US-EPA standard due to the lockdown of the city due to the COVID-19 pandemic (from 24 March 2020 onwards). On the other hand, despite the inner month fluctuations, a comparable seasonal, interannual temperature trend was seen for the whole study period. However, Thapa et al. [71] concluded in their study that over the period 2011–2017, the average annual temperature increase in the Kathmandu valley was 0.06 degrees Celsius.

3.1. Cross-Correlation Analysis

Various researchers performed the correlation analysis of air pollutants (PM2.5 & PM10) with different meteorological variables (e.g., Yang et al., Bhaskar et al.; Huang et al.; Li et al.; Zhao et al.) [24, 49, 8082] in Chinese and Indian regions. This section will establish the correlation of AQI (PM2.5) with air temperature and rainfall in Kathmandu city from 2017 to 2020.
Fig. 4 depicts the cross-correlation coefficients of AQI (PM2.5), air temperature, and rainfall in Kathmandu city from 2017 to 2020. It is noted that the AQI (PM2.5) in 2017 has a strong association with other years’ AQI (PM2.5) at zero-time lags, with a correlation coefficient of greater than 0.8. Additionally, we detect a very high positive correlation between air temperatures of studied years, as indicated by the greater value of the correlation coefficient of 0.99 at zero-time lag.
On the other hand, rainfall in 2017 has a good positive association but is significantly less than the association between air temperature in 2018, 2019, and 2020, as indicated by correlation coefficients of 0.90, 0.85, and 0.76 for time lags of +5, +12, and −9 days, respectively. In general, this implies that the annual variation in AQI (PM2.5), air temperature, and rainfall is relatively similar. Fig. 5 depicts the cross-correlation of AQI (PM2.5) with air temperature and rainfall during 2017 – 2020 in Kathmandu city. As shown in Fig., AQI (PM2.5) correlated negatively with air temperature during the entire study period. A strong positive correlation of AQI (PM2.5) with air temperature can be observed in 2020; however, other years have a slightly lower correlation coefficient. This suggests that the trend of the AQI (PM2.5) and temperature are inversely related which is in accordance with the prior research [24, 26]. Interestingly, it also exhibits a positive correlation with a coefficient value of 0.5 at a time lag in the range of + (100–200) days. The negative relation could be explained by the fact that greater temperature enhances the convection of air, which results in the dilution and dispersion of air pollutants [24, 83]. Thus, high temperature favors the depression of secondary pollutants [84]. Low temperatures might result in increased emissions from domestic heating and power production [24, 85]. Additionally, Yang et al. [24] assert that the phenomenon, known as “scale variation”, plays a significant role in the correlation study of PM2.5 and air temperature. They suggested that seasonal analysis exhibits less correlation than annual analysis. This nature, resulting in a range of correlation coefficient values for long-term series analysis (more than ten years), short-term series analysis (1–2 years), and seasonal analysis, may also be noticed in prior research of Huang et al. [80], Li et al. [81], and Lu et al. [86].
In the relationship between AQI (PM2.5) and rainfall, AQI (PM2.5) also shows a negative correlation with rainfall trends. Similar findings were also obtained by Wang et al. [22], Bhaskar et al. [49], and Lu et al. [86] in different regions; Japan, India, and China. Correlation coefficient values are nearly identical in Fig., indicating a consistent trend over the analyzed years. It reveals that rainfall significantly triggered the depression of air pollutants, and a similar pattern was replicated each year following 2017. Rainfall washes particulate pollutants out of the atmosphere, and wet deposition by precipitation, often known as wet removal, is one of the primary mechanisms for removing particulate pollutants from the atmosphere [49, 87]. However, it is also believed that the intensity of precipitation, the diameter of the raindrop, and particle size of aerosol [88], the increase in total rainfall, the average intensity, and the duration [89, 90, 91] directly influence the removal effect of rainfall on pollutants particles.

3.2. Discrete Wavelet Analysis (DWT)

3.2.1. Wavelet decomposition of the temporal distribution characteristics of AQI (PM2.5) (2017 – 2020)

Fig. 6(a) portrays the wavelet decomposition and reconstruction of daily average AQI (PM2.5) in Kathmandu valley during 2017 – 2020. The ‘details’ are represented by the panels on the right side, labeled D1 to D7. The smaller-scale features start appearing on increasing the resolutions (decreasing scales) [93]. The residual of the decomposition process can be regarded as the higher approximation levels, such as A5, A6, and A7, as depicted in Fig. 6(a). At these levels, particularly A7, the average value of the data series is found. The high-frequency component of each of these decompositions is gradually removed, allowing the lower- frequency component to reflect the trend in PM2.5 concentrations quantitatively [94]. The high-frequency coefficient is composed primarily of disturbance noises and random fluctuations of abnormal mutations, which reflect the abrupt changes and disturbances in (AQI) (PM2.5). On the other hand, the low-frequency coefficient is primarily composed of deterministic components, which correspond to the variation characteristics of the (AQI) (PM2.5).
The results in Fig. 6(a) indicate that a similar periodic variation of AQI (PM2.5) in time was observed in the wavelet decomposing signals and the low-frequency reconstructed time series. This signal indicates a possible seasonal variation of the AQI (PM2.5) whose value reached the highest in the winter and the lowest in summer. The AQI decreased from spring to summer, increased from summer to autumn, and dramatically increased from autumn to winter. Similar analysis can also be found in Wang et al. [25] and Zhang et al. [94]. One can notice that the reconstructed low-frequency coefficients in all AQI (PM2.5) layers were relatively stable compared to the similar stages of previous years. This was due to improved air quality following a lockdown during the COVID-19 pandemic [10, 11, 95]. Thus, the hidden information in the AQI (PM2.5) time-series changes was unveiled as a result of wavelet analysis. The findings revealed that the cycles during the four years were essentially identical.

3.2.2. Wavelet decomposition of the temporal distribution characteristics of rainfall (2017 – 2020)

Fig. 6(b) portrays the wavelet decomposition and reconstruction of rainfall in Kathmandu valley during 2017 – 2020. The higher approximation levels (such as A5, A6, and A7) indicated the average value of the rainfall data series. We choose the A7 level to terminate the decomposition since details of higher orders are so smoothed that they are not useful for analyzing rainfall [92]. On the other hand, levels A0, A1, A2, and A3 include the majority of high frequencies, which minimizes correlation and does not significantly improve signal characterization. Consequently, we choose levels A4 – A7 and found that the rainfall characteristics were different each year during the period of 2017 to 2020. However, it could be roughly divided into three stages each year, namely, pre-monsoon (March–May), monsoon (June to September), and post-monsoon (October) stages. This signal indicates the possibility of seasonal variations in rainfall, with the amount of rainfall reaching its maximum level during the monsoon season and minimum during the winter. The result obtained from these details and approximations helps determine the particularities in the original signal [25, 96].

3.2.3. Wavelet decomposition of the temporal distribution characteristics of air temperature (2017 – 2020)

Fig. 6(c) portrays the wavelet decomposition and reconstruction of air temperature in Kathmandu valley during 2017 – 2020. Using the wavelet decomposition level signals (D4 to D7) and low-frequency reconstruction signals (A4 to A7), it was discovered that the yearly air temperature exhibits the same behavior as the seasonal air temperature, with winter temperatures being lower and summer temperatures being higher. This is owing to the fact that the winter season has the shortest day lengths, whilst the summer season has the longest day lengths. The longer a day lasts, the more time the Earth has to absorb energy from the sun [71, 95, 97]. As a result, longer days’ result in warmer days, whereas shorter days’ result in cooler days, and so on. Although the distribution characteristics showed inter-month fluctuation, as shown by the noise in the chosen decomposition levels, the overall trend of the air temperature is identified with the help of multiresolution analysis. Thus, the decomposition levels have proved sufficient to isolate the periodic variation characteristics.
In addition, when comparing Fig. 6(a), 6(b), and 6(c), it is obvious that there is an anti-correlation in level A7 between the AQI (PM2.5) and rainfall, as well as the AQI (PM2.5) and temperature. This same pattern can be observed at other approximation levels as well. On these long timescales, this anti-correlation demonstrates that the concentration of PM is driven by the air temperature and the amount of rainfall [24, 49, 83, 84, 87].

3.3. Continuous Wavelet Transform Analysis

3.3.1. Periodic variation of AQI (PM2.5) based on wavelet analysis

Fig. 7(a) depicts the wavelet power spectrum of the daily average AQI (PM2.5) in Kathmandu over four years (January 2017 to December 2020). AQI (PM2.5) wavelet power spectra in Fig. 7(a) are derived from daily data collected over a period of four years. As can be seen in this Fig., there are two notable frequencies for AQI (PM2.5) fluctuations: one for the period of 4–8 days and another for 8–16 days, respectively. These oscillations are clearly associated with natural seasonal periodicities, with the highest intensity during the winter months. As expected, these oscillations diminished from spring to summer, increased from summer to autumn, and increased drastically from autumn to winter.
One noteworthy finding is the reduced fluctuations in 2020, indicating the reduced air quality in the Kathmandu valley. Evaluating the time series data of PM2.5 and PM10, Mishra et al. [10] and Baral & Thapa [11] also noted the improvement in air quality following the COVID-19 pandemic lockdown.

3.3.2. Periodic variation of rainfall based on wavelet analysis

Fig. 7(b) portrays the Morlet wavelet power spectrum of daily Rainfall in Kathmandu during 2017–2020. This Fig. shows the dominant frequencies observed in the periods of 4–8 days and 8–16 days every year. However, three dominant frequencies occurred in the period of 16–32 days and 32–64 days in the years 2018, 2019, and 2020. All these oscillations are connected via seasonal patterns that the high power occurs during the monsoon (June to September) and less power occurs in the winter (November to February). The wavelet keeps track of the low-frequency details at the higher scale (periods = 128 – 256 days), i.e., the slowly varying trends in the signal [31, 98]. It suggests that the variation of the rainfall shows inter-month fluctuation as well.

3.3.3. Periodic variation of temperature based on wavelet analysis

Fig. 7(c) portrays the wavelet power spectrum of daily air temperature in Kathmandu during 2017–2020. On increasing the resolutions (decreasing scales), the smaller-and smaller-scale features (high-frequency details) start to appear [93]. The coefficient for high frequencies consists mainly of several disturbance noises and alterations of abnormal mutations which are due to sudden changes and disturbances in the signal. As we can see, dominant frequencies occurred at the lower scale, one in the periods of 4–8 days and others in 8–16 days. It reflects the variation characteristics of the air temperature, with increased fluctuations in the summer rather than the winter. This type of variation illustrates the seasonal air temperature trend, with the lowest winter temperatures and highest summer temperatures. Our result is closely matched with the air temperature trend analysis of Thapa et al. [71] and Shrestha et al. [99].
As depicted by the wavelet power spectrum, the pattern of rainfall and air temperature fluctuation looks exactly opposite to that of AQI (PM2.5), suggesting an anti-correlation between temperature and rainfall AQI (PM2.5). It is also evident from the study of Anusasananan [40] and Liang et al. [100] that the trend of PM2.5 and temperature, PM2.5, and rainfall values move in opposite directions.

3.4. Cross Wavelet Transform (XWT) Analysis

3.4.1. XWT analysis between AQI (PM2.5) and rainfall

Fig. 8(a) portrays XWT between AQI (PM2.5) and rainfall in Kathmandu valley during 2017 – 2020. The XWT finds regions with high common power even in a noisy environment [40, 101]. The wavelet power spectrum of the two-time series (AQI (PM2.5) and Rainfall) clearly exhibits some characteristics in common, such as the strong seasonal variations (from winter to post-monsoon) observed between 2017 and 2020. Both series show strong power in the 2–16-day band; however, the power of the AQI (PM2.5) series is not more than the 5 % significance level in the 16–32-day band for the entire study period. Moreover, the degree of resemblance between the patterns portrayed across this time period is relatively low, making it difficult to determine whether this is a coincidence. This is when the cross-wavelet transform comes in handy [65].
The cross-wavelet power spectra between AQI (PM2.5) and rainfall show significant common power in two significant bands. The first band covers 128–256 days between September 2018 to August 2019 and the second band between 1–16 days around January to June during the entire period of the study. It clearly indicates that both parameters follow a seasonal pattern, although they both exhibit inter-month variation patterns.

3.4.2. XWT analysis between AQI (PM2.5) and temperature

Fig. 8(b) portrays XWT between AQI (PM2.5) and temperature in Kathmandu valley during 2017 – 2020. With the XWT, one can identify regions of high common power and gain additional information regarding the phase relationships. It is reasonable to anticipate a consistent or slowly shifting phase lag between the two series, which may be tested against mechanistic models of the physical process [62].
When the AQI (PM2.5) and temperature are plotted together, the XWT power spectra show significant common power in two significant bands, one at 1–16 days and the other at 256 – 512 days around the pre- and monsoon seasons in the period 2017 –2020. The area of a time-frequency plot is not a reliable measure of causality when the significance level is more than 5%. Even with correctly weighted scales for averaging, two series can be perfectly correlated at a single scale, while the region of significant correlation is significantly less than 5% [65, 102].

3.5. Wavelet Transform Coherence (WTC) Analysis

The wavelet coherence analysis was utilized to investigate the potential relationship between two-time series. The thick black lines show the 5% significance level using Monte Carlo simulations with a phase-randomized surrogate series [62]. The lighter pale colour shows the cone of influence that separates locations with reliable and less accurate estimations. The colour code for power spans from blue (low power) to red (high power). The relative phase relationship is represented by arrows with in-phase pointing right, and anti-phase pointing left [103].

3.5.1. WTC analysis between AQI (PM2.5) and rainfall

Wavelet coherence between AQI (PM2.5) and rainfall is shown in Fig. 8(c). The WTC identifies the regions in time-frequency space where the AQI (PM2.5) and rainfall time series vary. For AQI (PM2.5), a high-power spectrum in the 256–512 days’ band between the entire observation periods was observed with AQI (PM2.5) leading. At the same time, there was a significant region at approximately 16–64 days during the mid-2017 to early 2018 and 2020 winter-autumn seasons, with arrows pointing to the northeast, indicating a delay in rainfall following the AQI (PM2.5) concentration change of approximately six months. The arrows point slightly to the left in every high-power spectrum region, indicating that AQI (PM2.5) and rainfall are out of phase (anti-phase). When the rainfall increases, the AQI (PM2.5) decreases, and vice versa, according to previous studies [40, 62].

3.5.2. WTC analysis between AQI (PM2.5) and temperature

Wavelet coherence between AQI (PM2.5) and temperature is shown in Fig. 8(d). In Fig. 8(d), a high-power spectrum was observed in the 256 – 512 days’ band during the whole observation period. It can be seen that the AQI (PM2.5) and temperature are out of phase, as indicated by the arrows pointing to the left (anti-phase). However, the other three high-power spectra in the 1 – 4 days’ band during the late summer days of 2017, the 4–8 day band during the early 2018 monsoon, and the 32–64 day band during the 2020 pre-monsoon, respectively, were observed. This indicates that the relationship between AQI (PM2.5) and the temperature was positive for approximately 1 – 2 months. It could be attributed to the impact of meteorological variables such as wind, humidity, precipitation, radiation, and atmospheric pressure [28, 94]. Thus, the local correlation, temporal delay, and phase structure of the two-time series in time-frequency space can be investigated using the WTC analysis. While the cross-wavelet analysis reveals high common power regions, WTC discovers localized phase-locked behavior in the same region [40, 65, 102].

4. Conclusions

Employing wavelet analysis, this study investigated the variation of the AQI (PM2.5) and its relationship with the meteorological parameters (air temperature and rainfall) from 2017–2020 concluding the following points:
  • The AQI (PM2.5) was higher in the winter and lower in the summer. The AQI (PM2.5) revealed a negative correlation with air temperature and rainfall on a seasonal and yearly basis within the study period. Additionally, our study identified 2020 as the year with the lowest AQI (PM2.5) which is likely the effect of the COVID-19 lockdown, and 2018 as the highest AQI (PM2.5).

  • The cross-correlation analysis found a strong positive relationship between the AQI (PM2.5) of 2017 and the following years (2018–2020). Additionally, a strong negative association were identified between averaged AQI (PM2.5) and air temperature and rainfall in the Kathmandu region in each year of the study period.

  • CWT results demonstrate that the Morlet wavelet has been proven effective for detecting different fluctuation periods of AQI (PM2.5), rainfall, and temperature over a long period of time. The amount of pollution in the air in the Kathmandu valley varied significantly over time, including across seasons and months. Moreover, high-power wavelet regions in XWT with common periodicity in AQI (PM2.5) vs rainfall, and temperature give an accurate background for understanding the variation pattern of meteorological influences on PM2.5 concentrations.

WTC analysis showed a negative association of AQI (PM2.5) with rainfall and air temperature, while an anomaly was seen among AQI (PM2.5) and temperature as they were positively associated in 2020 for a few months in 2017, 2018, and 2020. Although the study demonstrated an interesting reduction in Kathmandu’s air pollution as a result of combating the global epidemic of COVID-19, it was only a short-term result. To better understand the effects of different meteorological factors on air quality, it is crucially important that we investigate the effects of anthropogenic activities on atmospheric pollution during the urbanization process. The permanent control of air pollution in Kathmandu can only be achieved when the responsible authority takes an initiative in addressing the situation and designing a policy to deal with the city’s escalating levels of air pollution. We believe that this study would serve as a reference for future studies and pollution control policies aimed at improving the air quality of Kathmandu.


We want to acknowledge the World Air Quality Project (aqicn.org), the Department of Environment (Government of Nepal), and the Department of Hydrology and Meteorology (Government of Nepal) for providing data of the Air Quality Index (PM2.5), surface air temperature and rainfall for this study. Some of our software contains code that was originally written by C. Torrence and G. Compo. This research did not receive any specific grant from funding agencies in the public, commercial, or not-for-profit sectors.



The authors declare that they have no conflict of interest.

Author's Contributions

S.P.G (Ph.D. student) and A.S (Ph.D. student) performed conceptualization, methodology implication, software, data curation, writing-original draft preparation, visualization, investigation, and validation; B.B (M.Sc. student), N.L (M.Sc. student) and S.S (M.Sc. graduate) worked on introduction, methodology, and final review, N. P. C. (Ph.D.), B. A. (Ph.D.) performed final editing and review.


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Fig. 1
Map of research area displaying three districts: Kathmandu, Bhaktapur, and Lalitpur of the Kathmandu valley
Fig. 2
Daily average AQI (PM2.5) (upper), air temperature (middle), and rainfall (lower) trend in Kathmandu city from the year 2017 to 2020.
Fig. 3
Temporal variation of AQI (PM2.5) (upper) and air temperature (lower) from 2017 to 2020 in Kathmandu city.
Fig. 4
Cross-correlation of AQI (PM2.5) (upper left), air temperature (upper right), and rainfall (lower left) in the year 2017 with 2018 (blue line), 2017 with 2019 (red line), and 2017 with 2020 (yellow line) in the Kathmandu city.
Fig. 5
Cross-correlation of AQI (PM2.5) with air temperature (blue line) and AQI (PM2.5) with rainfall (red line) in the year 2017 (upper left), 2018 (upper right), 2019 (lower left), and 2020 (lower right) in the Kathmandu city.
Fig. 6
Decomposition waveform signals of (a) AQI (PM2.5), (b) rainfall, (c) air temperature data at Kathmandu valley, where the upper panel shows the original AQI signal; D1 means the first decomposition signals and D7 means the seventh decomposition signals, respectively; and A1 A7 mean the seventh low-frequency reconstruction signals.
Fig. 7
The wavelet power spectrum of (a) AQI (PM2.5), (b) rainfall, (c) air temperature.
Fig. 8
(a) XWT between AQI (PM2.5) and rainfall, (b) XWT between AQI (PM2.5) and temperature, (c) WTC between AQI (PM2.5) and rainfall, and (d) WTC between AQI (PM2.5) and temperature time series.
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