Model of stochastic oscillations

Under an assumption of translational invariance, we can model the acoustic oscillation recorded in solar dopplergrams as a stochastic source function convolved with a Green's function impulse response. After a three-dimensional Fourier transform, the convolution becomes a simple multiplication:  

$$D(k_x,k_y,\omega)=S(k_x,k_y,\omega) G(k_x,k_y,\omega),$$ (1)

where D is the observed data, S is the source function, and G is the impulse response. We are interested in the three-dimensional time-space acoustic impulse response, g(x,y,t). As it stands, however, equation () has many more unknowns than knowns, so additional assumptions are required before we can estimate G.

Secondly, we assume s(x,y,t) is white in space and time, or equivalently, $S {\bar S}=1$, where the ${\bar S}$ denotes the complex conjugate of S. If this is not true in practice, spectral color from the source function will leak into the derived impulse response.

Under this assumption, equation () reduces to the statement that the power spectrum of the impulse response equals the power spectrum of the data,

$$\left\vert D \right\vert^2 = \left\vert G \right\vert^2.$$ (2)

While defining the amplitude spectrum of G, this equation places no constraints on its phase, and so we need an additional assumption to ensure a unique solution.

We will assume that G is a minimum-phase function, where a minimum-phase function is defined as a causal function with a causal convolutional inverse. It turns out that many physical systems (from electronics to acoustics) have this property. For example, consider the function that maps stress to strain and vice versa: if you specify the stress in a solid, the strain will react accordingly, leading to a causal function relating stress to strain. However, you could also specify the strain in the solid, and then the observed-stress would be a causal function of the strain. Since these two functions are inverse processes, they clearly satisfy the minimum-phase definition.

If this model holds true, then estimating the impulse response reduces to estimating a minimum-phase function with the same ($\omega,k_x,k_y$) spectrum as the original data: or equivalently, multi-dimensional spectral factorization.