How many degrees of freedom are evident in a physical process represented by f(s, t)? In some form questions about “degrees of freedom” (d.o.f.) are common in mathematics, physics, statistics, and geophysics. This would mean, for instance, in how many independent directions a weight suspended from the ceiling could move. Dofs are important for three reasons that will become apparent in the remaining chapters. First, dofs are critically important in understanding why natural analogues can (or cannot) be applied as a forecast method in a particular problem (Chapter 7). Secondly, understanding dofs leads to ideas about truncating data sets efficiently, which is very important for just about any empirical prediction method (Chapters 7 and 8). Lastly, the number of dofs retained is one aspect that has a bearing on how nonlinear prediction methods can be (Chapter 10). In view of Chapter 5 one might think that the total number of orthogonal directions required to reproduce a data set is the dof. However, this is impractical as the dimension would increase (to infinity) with ever denser and slightly imperfect observations. Rather we need a measure that takes into account the amount of variance represented by each orthogonal direction, because some directions are more important than others. This allows truncation in EOF space without lowering the “effective” dof very much. We here think schematically of the total atmospheric or oceanic variance about the mean state as being made up by N equal additive variance processes. N can be thought of as the dimension of a phase space in which the atmospheric state at one moment in time is a point. This point moves around over time in the N-dimensional phase space. The climatology is the origin of the phase space. The trajectory of a sequence of atmospheric states is thus a complicated Lissajous figure in N dimensions, where, importantly, the range of the excursions in each of the N dimensions is the same in the long run. The phase space is a hypersphere with an equal probability radius in all N directions.
Complex wavefront sensing based on coherent diffraction imaging using vortex modulation