On sensitivity kernels for 'wave-equation' transmission tomography

ABSTRACT

We combine seismological scattering theory with the theory of distributions to study some properties of sensitivity kernels for finite frequency seismic delay times. The theory to be used for calculating the kernels depends on the way the measurements are made. For example, the sensitivity to the traveltime proper, that is the time to instantaneous onset of the phase arrival as defined in geometrical ray theory, has a non-zero value on the ray and zero elsewhere, whereas measurements of the smooth later part of the wave excitation can have more complicated kernels. Analysis based upon the Born approximation reveals that the behaviour of such kernels is determined by the regularization of the nth derivative of the Dirac delta supported on the unperturbed source-receiver ray, where n is the spatial dimension. Such regularization induces approximately the desired band-limitation and the associated Fresnel zone averaging. If the regularization is symmetric about the singular support of the delta its odd order derivatives vanish on this support, but the even orders render non-zero values. This explains why 3-D finite frequency kernels for body wave delay times seem—in specific, rather restrictive circumstances—to have zero sensitivity on the ray (as in the so called 'banana-doughnut kernels'). Indeed, a regularization of the kernel can have a zero on the unperturbed ray, in the absence of caustics (that is, the medium must be simple or quasi-homogeneous) and only if the exact source signature (e.g., source time function) is known and used; both conditions can be met in synthetic cases but not—in general—in applications to real data and heterogeneous media. In general, one will not know where the oscillatory kernels have zero values, which has implications for the use of such kernels for the linearization of transmission tomography. We briefly clarify these observations by invoking a multiresolution analysis of sensitivity kernels, connected with a time-frequency analysis of phase arrival times.