(1) |

Secondly, we assume *s*(*x*,*y*,*t*)
is white in space and time, or equivalently, , where the
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,

(2) |

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 () spectrum as the original data: or equivalently, multi-dimensional spectral factorization.

- Kolmogorov review
- Multi-dimensional factorization
- Theoretical comparison between time-distance functions
- On the assumption of translational invariance

5/27/2001