# Study on the Thermospheric Density Distribution Pattern during Geomagnetic Activity
## 3 Results
### Comparison of EOF Density Fields of Different Orders
This paper explores statistics on the variance contribution rate of the first four order eigenvectors obtained after the EOF decomposition of atmospheric density on 8 September 2017. Table 1 shows the eigenvectors obtained after the first four-order EOF decomposition and their statistical contribution results. From the statistical data in Table 1, it can be seen that the variance contribution rate of the first-order EOF atmospheric density characteristic vector field reached 98.04%, while the variance contribution rate of the second-order EOF atmospheric density characteristic vector field dropped significantly to 1.35%, and from the second order onwards, the variance contribution rate of each order gradually decreases. In terms of cumulative variance contribution rate, the variance contribution rate of the first two EOF atmospheric density characteristic vector fields reached 99.39%, and the variance contribution rate of the first four orders reached 99.94%, which is already very similar to the original measured atmospheric density distribution field. Because the variance contribution rate of the first-order atmospheric density characteristic vector field has already exceeded 95%, it can be considered that the first-order atmospheric density characteristic vector field is sufficient to contain most of the information of the original measured atmospheric density field. Therefore, we assume the first-order EOF atmospheric density characteristic vector field is \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Order Number** & \begin{tabular}{c} **The Eigenvalue** \\ **(\(\times 10^{-20}\))** \\ \end{tabular} & \
\begin{tabular}{c} **Variance** \\ **Contribution Rate** \\ \end{tabular} &
\begin{tabular}{c} **Cumulative Variance** \\ **Contribution Rate** \\ \end{tabular} \\ \hline First order & 0.2408 & 0.9804 & 0.9804 \\ Second order & 0.0033 & 0.0135 & 0.9939 \\ Third order & 0.0010 & 0.0042 & 0.9980 \\ Fourth order & 0.0003 & 0.0014 & 0.9994 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: First-order EOF eigenvectors and their contribution statistics. sufficient to highly fit and restore the measured atmospheric density distribution field. In the subsequent research, in order to filter out some incidental or even white noise features in the measured atmospheric density vector field and to avoid the contingency of research conclusions, the first-order EOF atmospheric density characteristic vector field is adopted as the research object in the data processing process to explore the long-term and large-scale changes of atmospheric density. After the EOF decomposition is obtained, it is necessary to test the data obtained. Because the fitting degree of each order of the eigenvector field to the original measured density field has been analyzed in previous works, there is no composite analysis test here. Instead, the North test examines whether the vector fields of different orders are independent; that is to say, whether the vector fields of each order can be regarded as vector fields with independent characteristics. Sometimes the research data may be mixed with random or false data. After empirical orthogonal decomposition, these invalid and real data can be decomposed into several principal components and eigenvectors. Therefore, the North significance test is necessary. In the North significance test, each order vector field's eigenvalues need to be substituted into the function to obtain each eigenvalue's lower and upper error limits. For example, suppose that the error of two adjacent eigenvalues intersects the lower and upper limits. In that case, the eigenvector field corresponding to the eigenvalue does not pass the significance test. Therefore, the vector field corresponding to the eigenvalue is invalid data and must be eliminated. Table 2 shows the corresponding upper and lower error limits for the first five-order eigenvector fields obtained by EOF decomposition after the North test.