*2.2. Methods*

All methods used in this paper were coded and computed in the programming language Python.

#### 2.2.1. Empirical Orthogonal Function Decomposition

The empirical orthogonal function (EOF) is applied to the meteorological variable that changes with time, and the meteorological variable is decomposed into two parts, namely, the function of time and the function of space.

Assuming that the sample size is *n* and the meteorological variable *X* contains *p* spatial points (variables), the anomaly value of any spatial point *i* at any time point *j* can be regarded as the linear combination of *p* spatial functions *v ik* and *p* time functions *y ki* (*k* = 1, 2, 3 . . . . . . and *p*). The decomposition is expressed as a matrix form of *X* = *VY*.

The space vector *V* is a matrix of n row and n column, which are orthogonal to each other: *VT* × *V* = *I* (*I* is a unit matrix)

The time vectors *Y* is an n-row and m-column matrix, and *Y* are also orthogonal:

*Y* × *YT* = Λ (Λ is a diagonal matrix)

Defining the matrix *A* as *A* = *X* × *XT*, and then we have: *A* = *V* × Λ × *VT*

*V* is also the eigenvector of *A*, Λ's principal diagonal is the eigenvalue of *A* and the rest are all 0. *Y* can be obtained as *Y* = *VT* × *X*.

This method is used to study the spatio-temporal characteristics of 200 hPa zonal wind. More information about EOF can be found in [23].

#### 2.2.2. Singular Value Decomposition

The singular value decomposition (SVD) method is performed on the covariance matrix of two variables. The anomalies fields of the variables and the normalized variables are commonly used. The decomposition result reveals the spatial correlation of two variable fields within a certain time range to a great extent. The heterogeneous correlation diagrams of the left and right fields explain the correlation between the two variables, and the SVD results are tested by using the Monte-Carlo method to avoid false correlation. The detailed descriptions and application of SVD is given in [24].This approach is used to test the relationships between upper level jet stream (200 hPa zonal wind) and surface pollutants (PMs and O3) over East Asia.

## 2.2.3. Pearson Correlation Coefficient

The Pearson correlation coefficient is a statistic that measures the linear correlation between two variables. It is usually represented by r, and its value ranges between −1 and 1. The calculation formula of the correlation coefficient between variables *x*1, *x*2, *x*3*...xn* and variables *y*1, *y*2, *y*3*...yn* is as follows:

$$\mathbf{r} = \frac{\sum\_{i=1}^{n} (\mathbf{x}\_{i} - \overline{\mathbf{x}})(y\_{i} - \overline{y})}{\sqrt{\sum\_{i=1}^{n} (\mathbf{x}\_{i} - \overline{\mathbf{x}})\sum\_{i=1}^{n} (y\_{i} - \overline{y})}} \tag{1}$$

The correlation coefficient in this study is tested by using the Monte Carlo method. That is, the two variables are considered to obey the normal distribution. The H0 hypothesis is when the correlation coefficient is r, the two variables are not correlated. Given the confidence level α, the corresponding critical value can be determined according to the degree of freedom so that the probability distribution function conforms to P (|r| > r1-α) = α. If |r| > r1-α, the hypothesis H0 is rejected, and the correlation between the two variables is significant. Otherwise, the two variables are not correlated. The specific approaches are as follows.

First, a pair of arrays that conform to the normal distribution with sample sizes of *n* are randomly generated, and the Pearson correlation coefficient between them is calculated.

Second, the first step is repeated 15,000 times, and the obtained correlation coefficients are sorted in descending order. The 5000th correlation coefficient (1-α) is found and marked as r1-α.

Third, the actual correlation coefficient |r| and the r1-<sup>α</sup> are compared. If |r| > r1-α, the two variables are correlated. Otherwise, they are not correlated.

We used this method to analyze the relationships between jet stream (200 hPa zonal wind) and surface meteorological elements (humidity, temperature, meridional wind and zonal wind), as well as the relationships between surface pollutants (PMs and O3) and surface meteorological variables (humidity, temperature, meridional wind and zonal wind). The Monte Carlo method is also used to test whether the correlation is significant [25].

#### *2.3. Relevant Definitions of the East Asian Upper-Level Jet*

In this study, the area with the westerly wind speed greater than 30 m·s−<sup>1</sup> at 200 hPa in East Asia (70–140◦ W, 15–55◦ N) is defined as the East Asian upper-level jet. The position of the East Asian upper-level jet is defined as the latitude of the maximum westerly wind speed at 200 hPa in East Asia. The intensity of the East Asian upper-level jet is defined as the average wind speed on the jet stream axis. Figure 1 shows the average climate state of the 200 hPa jet stream axis in summer from 1989 to 2018. The position of the 200 hPa jet stream in summer is around 40◦ N with relatively large interannual fluctuations.

**Figure 1.** The average climate state of the jet stream axis at 200 hPa in summer from 1989 to 2018 (scatter points and error bars indicate the average position and the variabilities of the jet stream axis, respectively).

## **3. Results**
