The KL Procedure

In the Karhunen-Loeve procedure, also called the Proper Orthogonal Decomposition, the spatial modes in a time sequence of data are orthogonalized by diagonalizing the spatial correlation matrix.

The first step is to find the mean of the data set and subtract it from each frame so that only the perturbations from the mean are considered. Next a two-point correlation is calculated for every pair of data points in one frame, generating an N by N matrix, where N is the total number of spatial points in one frame of data. This spatial correlation is calculated for each time frame and then all correlation matrices are added to get one time average spatial correlation matrix, K.

This matrix is then diagonalized. The matrix which affects the transformation is the matrix of eigenvectors of K. These eigenvectors or eigenfunctions correspond to the orthogonal spatial modes present in the original data, and form a complete orthonormal basis for the original data. Each eigenvalue represents the total energy present in its corresponding eigenfunction.

Eigenfunction Analysis of Coherent Structures on the Solar Surface
Authoring NASA Official:Dr. Milton Halem, Chief, Earth and Space Data Computing Division
Contact:Marilyn Mack/Code 930 marilyn.j.mack.1@gsfc.nasa.gov