Biorthogonal decomposition

The Biorthogonal Decomposition (BOD, also known as Proper Orthogonal Decomposition, POD) applies to the analysis of multipoint measurements

$Y(i,j)\,$

where i=1,...,N is a temporal index and j=1,...,M a spatial index (typically). The time traces Y(i,j) for fixed j are usually sampled at a fixed rate; however the measurement locations x(j) need not be ordered in any specific way.

The BOD decomposes the data matrix as follows:

$Y(i,j)=\sum _{k}\lambda _{k}\psi _{k}(i)\phi _{k}(j),\,$

where ψ_k is a 'chrono' (a temporal function) and φ_k a 'topo' (a spatial or detector-dependent function), such that the chronos and topos satisfy the following orthogonality relation

$\sum _{i}{\psi _{k}(i)\psi _{l}(i)}=\sum _{j}{\phi _{k}(j)\phi _{l}(j)}=\delta _{kl}.\,$

The combination chrono/topo at a given k, ψ_k(i) φ_k(j), is called a spatio-temporal 'mode' of the fluctuating system, and is constructed from the data matrix without any prejudice regarding the mode shape. The λ_k are the eigenvalues (sorted in decreasing order), where k=1,...,min(N,M). This decomposition is achieved using a standard Singular value decomposition of the data matrix Y(i,j). Thus, the oscillations of the spatiotemporal fluctuating field are represented by means of a very small number of spatio-temporal modes that are constructed from the data themselves, without prejudice regarding the mode shape.

A limitation of the technique is that it assumes space-time separability. This is not always the most appropriate assumption: e.g., travelling waves have a structure such as cos(kx-ωt); however, most propagating waves can still be recognised clearly by their distinct footprint in the biorthogonal modes (provided there are not too many): a travelling wave will produce a pair of modes with similar amplitude and a 90° phase difference.