Mathematical explanation on the POD applications for wind pressure fields with or without mean value components

The influence mechanism of mean value components, noted as $P_0$, on POD applications for complete random fields $P_C(t)$ and fluctuating random fields $P_F(t)$ are illustrated mathematically. The critical philosophy of the illustration is introduction of a new matrix, defined as the correlation function matrix of $P_0$, which connect the correlation function matrix of $P_C(t)$ and $P_F(t)$, and their POD results. Then, POD analyses for several different wind pressure fields were presented comparatively as validation. It's inevitable mathematically that the first eigenmode of $P_C(t)$ resembles the distribution of $P_0$ and the first eigenvalue of $P_C(t)$ is close to the energy of $P_0$, due to similarity of the correlation function matrixs of $P_C(t)$ and $P_0$. However, the viewpoint is not rigorous mathematically that the first mode represents the mean pressure and the following modes represent the fluctuating pressure when $P_C(t)$ are employed in POD application. When $P_C(t)$ are employed, POD results of all modes would be distorted by the mean value components, and it's impossible to identify $P_0$ and $P_F(t)$ separately. Consequently, characteristics of the fluctuating component, which is always the primary concern in wind pressure field analysis, can only be precisely identified with $P_0$ excluded in POD.