*2.3. Methods*

This study used a diagnostic method to estimate the long-term soil moisture–temperature coupling over China in different seasons, which is based on two energy balances of ET and PET. The partitioning of the land energy is expressed in Equation (1), where *Rn* refers to the surface net radiation, *G* means the ground heat flux, and λ is the latent heat of vaporization, which can be captured from near-surface air temperature; the ground heat flux is negligible in this study [36,38].

$$R\_{\rm II} - G = \ \lambda E + H\_{\prime} \tag{1}$$

When the annual time series of *E*, *Ep*, *Rn*, and near-surface temperature (*T*) are available, the diagnostic method could be used to estimate the long-term soil moisture–temperature coupling [17], where the metric (Π) can be calculated as:

$$
\Pi = \rho(H, T) - \rho(H\_p, T),
\tag{2}
$$

$$H = R\_n - \lambda E\_\prime \tag{3}$$

$$H\_p = R\_n - \lambda E\_{p\prime} \tag{4}$$

where ρ means Pearson's correlation coefficient, and *H* is the sensible heat flux. Using Π, we can derive an indicator of the long-term soil moisture–temperature coupling, which can be considered as a multi-year average. When considering the σ*T*, σ*H*, and <sup>σ</sup>*Hp* (the standard deviations of *T*, *H*, and *Hp*), Equation (1) could also be expressed in another form as covariances:

$$
\Pi = \frac{1}{\sigma\_T} \left( \frac{\text{cov}(H, T)}{\sigma\_H} - \frac{\text{cov}(H\_{p\_f}, T)}{\sigma\_{H\_p}} \right) \tag{5}
$$

To understand the related heating processes between land and atmosphere during the heatwave events, we used a different diagnostic method based on daily data to derive the soil moisture–temperature coupling at daily scale [8], and the metric (π) is defined as:

$$
\pi\_i = \frac{T\_i - \overline{T}}{\sigma\_T} \left( \frac{H\_i - \overline{H}}{\sigma\_H} - \frac{H\_{p,i} - \overline{H\_p}}{\sigma\_{H\_p}} \right) \tag{6}
$$

where *T*, *H*, and *Hp* indicate the averages of *T*, *H*, and *Hp* over a long term. It can be simplified as:

$$
\pi = T' \times e',\tag{7}
$$

$$
\epsilon' = \left( R\_n - \lambda E \right)' - \left( R\_n - \lambda E\_p \right)' \tag{8}
$$

where *T* represents the anomalies of T, and *e* is equal to *H* − *Hp* and indicates the contribution of soil moisture deficit to sensible heat flux. When there is sufficient soil moisture for the atmospheric demand, this energy term will be zero, and it may increase under arid condition. Only if the potential influence of soil moisture on temperature is accompanied by a large anomalous value of temperature is the local energy balance likely to control air temperature [11].

The Π and the π are two coupling metrics at different time scales; the former includes the long-term record in terms of correlation coefficients to evaluate long-term climatology, while the latter expresses anomalies of one day in terms of standard deviations to evaluate daily extreme. When Π and π values are greater than zero, then the higher the value the stronger soil moisture–temperature coupling. If values are less than or equal to zero, there is no coupling [17].
