Advancing the Velocity Gradient Technique: Using Gradient Amplitudes and handling thermal broadening

The recent development of the Velocity Gradient Technique allows observers to map magnetic field orientations and magnetization using the direction of velocity gradients. Aside from the directions, amplitudes of velocity gradients also contain valuable information about the underlying properties of magneto-hydrodynamic (MHD) turbulence. In this paper, we explore what physical information is contained in the amplitudes of velocity gradients and discuss how this information can be used to diagnose properties of turbulence in both diffuse and self-gravitating interstellar media. We identify the relations between amplitudes of both intensity and velocity centroid gradients and the sonic Mach number $M_s$ and they are consistent with the theory's predictions. We test the robustness of the method and discuss how to utilize the amplitudes of gradients into self-gravitating media. To extend the velocity gradient technique we also discuss the usage of amplitude method to Position-Position Velocity (PPV) space as a possible way to retrieve the velocity channel maps before the contamination of thermal broadening. We discuss that the Velocity Gradient Technique with these advancements could potentially give a significantly more accurate statistical insight into the properties magnetized turbulence.