Speeding up the extended Kalman filter approach for denoising XMCD movies of fast magnetization dynamics

The Kalman filter is a well-established approach to get information on the time-dependent state of a system from noisy observations. In former times it has been used for systems with only a few degrees of freedom (typically about 10). The fast magnetization dynamics is often investigated by x-ray magnetic circular dichroism movies (XMCD movies) where the number of components of the state vector is very large (typically about 10⁵). For such systems the Jacobian matrix which is required in the extended Kalman filter approach cannot be calculated numerically (as it is done in former papers) by use of the Landau-Lifschitz-Gilbert equation of motion in a finite time because of the many degrees of freedom of the state vector. In the present paper it is shown that the calculation of the Jacobian matrix can be much speeded up by using good analytical approximations, which are derived by a tensorial Green's function method to solve the linearized Landau-Lifschitz-Gilbert equation of motion. This makes it possible to investigate the dynamics of magnetic vortices and spin waves in circular discs of Permalloy, which are initiated by time-dependent external magnetic fields.