The wavelet coherence of two signals $x(t)$ and $y(t)$, analogous to the correlation in a standard statistical analysis, is used to explore the localized linear relationship between incoming shortwave radiation and air temperature at different scales (frequencies) over the time. It is expressed as:

$R_{xy}^2\left(s\right)=\frac{{\left|\langle s^{-1}W{T}_{xy}\left(\tau ,s\right)\rangle \right|}^2}{\langle s^{-1}{\left|W{T}_x\left(\tau ,s\right)\right|}^2\rangle \langle s^{-1}{\left|WTy\left(\tau ,s\right)\right|}^2\rangle },$ (4)

where $\langle \cdots \rangle$ in Equation (4) is the time and frequency smoothing operator for the Morlet wavelet. $W{T}_{xy}\left(\tau ,s\right)$ the cross-wavelet transform of $x(t)$ and $y(t)$given by:

$W{T}_{xy}\left(\tau ,s\right)=W{T}_x\left(\tau ,s\right)W{T}_y{}^{*}\left(\tau ,s\right),$ (5)

where $W{T}_y{}^{*}\left(\tau ,s\right)$ is the complex conjugate of $W{T}_y\left(\tau ,s\right)$.

$R_{xy}^2$ ranges from 0 to 1; the larger the value, the higher the coherence means greater linear correlation between variables. By transforming incoming shortwave radiation and air temperature data into scales via Equations (1)-(3), the amplitude spectra provides clear spatial variability while the phase indicates their discontinuities.

The cross-wavelet transform also provides the lead and lag relationship in oscillations between two time series. They are known as a phase difference [27] and can be used to extract information about the nature of correlation processes between signals under the assumption that they are causal [30] . The phase difference ${\varnothing }_{xy}$ is given by:

${\varnothing }_{xy}={\tan }^{-1}\left(\frac{\mathrm{Re}\left(W_{xy}\left(\tau ,s\right)\right)}{\mathrm{Im}\left(W_{xy}\left(\tau ,s\right)\right)}\right).$ (6)

where ${\varnothing }_{xy}$ represents phase difference varying in the interval [−π; π]; $\mathrm{Re}\left(W_{xy}\left(\tau ,s\right)\right)$ and $\mathrm{Im}\left(W_{xy}\left(\tau ,s\right)\right)$ are the real and imaginary parts of $W_{xy}\left(\tau ,s\right)$. When ${\varnothing }_{xy}$ ∈ [0, π/2], $x$, $y$move in phase meaning they are positively correlated and $x$ leads $y$; ${\varnothing }_{xy}$ ∈ ]−π/2, 0], $x$, $y$ move in phase and $y$ leads $x$. If ${\varnothing }_{xy}$ ∈ ] π/2; π], the two time series are negatively correlated (anti-phase) and $x$is out of phase and leads $y$ and finally for the last quadrant $y$ is out of phase and leads $x$.

Figure 2 shows the three-year time series data of the incoming shortwave radiation (SWin) and air temperature (Tair), respectively, observed at Dangbo from 01 November 2020 (01 November corresponds to Day of Year (DOY) 1) to 31 October 2023 (DOY 365).

The solar radiation (SWin) reaching the ground at Dangbo exhibited a certain variability which is more pronounced during the wet season (from March to June and September to October). Its evolution also showed a sizeable influence of cloudiness and other factors, limiting the amount of radiation arriving at the surface. In addition, the pattern of SWin reveals the presence of nonlinearities and multiple time scales of variations, which are higher in wet seasons (March to June and September to October) and lower in dry seasons (July to August and November to February respectively).

The distinct SWin and Tair annual patterns in the studied area illustrated in Figure 2 are pieces of evidence that contrast the three years studied but also within the annual cycle. The annual daily average air temperature was about 26.8˚C ± 1.4˚C and the incoming shortwave radiation was 180 ± 50 W·m⁻² (mean ± s.d.) over the studied period.

On average, SWin increases from DOY 100 (around January) to reach its first maxima around DOY 150 (March) but then weakens slowly until July/August where the annual minimum value is observed. The following increase is sharp and lasts until the end of the year (October 31st). The annual evolution of Tair follows a little that of SWin. It is interesting to note that the daily maximum is roughly the same during the three years (~29.0˚C - 30.0˚C), occurring between DOY 100 (during the long dry season) and DOY 150 (March) at the beginning of the wet season. The minimum values (~24.0˚C - 25.0˚C) were however found, depending the year in July/August/September which coincides with the high cloudiness period in West Africa [31] - [33] .

Figure 3 presents the daily wavelet power spectrum of the incoming shortwave radiation (SWin) and air temperature (Tair) at the Dangbo site. Herein, we quantified the variability of these signals with the wavelet coefficient at different time scales and periods. Two major peaks of variability (2 - 8 and 64 - 128 bands) were emphasized with the wavelet power spectrum (WPS) of SWin. The highest value of the WPS was obtained within the 2 - 8 band period and was more concentrated suggesting that there is a higher variance in the signal from daily to almost decadal time period. The second peak is less higher, with some abrupt and localized changes in the variance of SWin revealing rather the difference between months (thus seasons) corresponding to the seasonal variability of SWin. This evidences also the presence of nonlinearities at these scales already emphasized in previous section ( Figure 2 ). When looking at Figure 3 , annual pattern of SWin is clearly depicted with higher power during wet seasons (March to June and September to October) and slight to almost inexistent power during dry season especially between January and February and within the 2 - 8 band period.

The highest wavelet power in SWin coincides almost with that of Tair indicating that these two variables are interconnected especially in the 2 - 8 band period. Beyond this band, there is relatively continuous and lower power of variability compared to the 2 - 8 band. Wavelet coefficients acutely vary form the highest to the lowest suggesting a potential unstable variability of Tair between 16 and 256 band period.

In respect of each band of time, and period, color gradation indicates the level of co-oscillation (weak or strong). The arrow directions indicate however the causation (leading of lagging) and the nature of the relationship. The degree of co-oscillation was reflected through the plots in the time frequency domain. The color ranged from light blue to red, each representing low and high degree of correlation. The red region indicates a stronger co-oscillation while the blue area indicated a weaker co-oscillation. We can see from ( Figure 4 ) that there is a significant co-movement as expected between SWin and Tair across frequencies (periods).

There are some irregular and discontinuous high and low power oscillations for the period below 4 days. For the three years investigated, there is a quasi-inexistence of significant correlation between SWin and Tair during the long dry season for the period lower than 8 days suggesting that there is another possible mechanism that happens at this site and at daily time scale. [23] showed for example that at this site, a lower RH is not accompanied by a high air temperature at daily time scale, suggesting that air mass has changed, a drier air mass with the same temperature has arrived over the site. During other periods of the year, both signals depicted some similarity toward a large significant band of variability, sometimes discontinuous, and mostly positive correlation in 2022 and 2023. Based on the daily data analyzed, we found that SWin and Tair move in phase meaning there are positively correlated (arrows directed to the right on the Figure 4 ), except between December 2022 and February 2023 (long dry season), where arrows are oriented into the left suggesting rather a negative correlation. A clear distinction of the nature of co-oscillation during the long dry season is therefore obtained.

The mean values of coherence ( $R_{xy}^2$), phase ( ${\varnothing }_{xy})$ (where $x$ refers to SWin and $y$ to Tair) and time lags per band period and for the all three years are given in ( Table 1 ) to quantify nature of the co-oscillation within this unified time-frequency domain. On average for the all band periods, 0 < ${\varnothing }_{xy}$ < π/2, except within the 32 - 64 and 128 - 256 band period where ${\varnothing }_{xy}$ equaled to 333.84˚ and 347˚ respectively. The former indicates that SWin leads Tair by ~23.5˚ when (0 < ${\varnothing }_{xy}$ < π/2) whereas when (−π/2 < ${\varnothing }_{xy}$ < 0), Tair leads SWin.

These results suggest at least that at the short time scale (i.e. periods ≤ 32 days), Tair increases with an increasing SWin. The lag of these two signals ranges between 0.09 and 2.30 days thus suggesting that the air column heats up almost instantaneously as radiation increases. However, when looking at the interdependence of radiation and air temperature at a larger temporal scale (>32 days), Tair lags SWin meaning that an increase in SWin may not directly imply an increasing Tair. The lags between SWin and Tair within 32 - 64 and 128 - 256 band period are negative and were −3.37 and −6.26 days respectively. Surprisingly, in the 64-128 band period, ( $R_{xy}^2$) was the smallest (0.76) and ${\varnothing }_{xy}$ = 30 with a lag of 7.72 days between SWin and Tair. This suggests that Tair follows the dynamics of SWin but with a higher delay compared to that of the period ≤ 32 days. All these results together reveals complex dynamics of radiation and air temperature within the annual cycle.