Theory

A generic discrete linear inverse problem may be written as

$${\bf d} = {\bf A \; m}$$ (1)

where ${\bf d}=(d_1 \; d_2 \;...)^T$ is the known data vector, ${\bf m}=(m_1 \; m_2 \;...)^T$is the unknown model vector, and ${\bf A}$ represents the linear relationship between them. A natural question to ask is: which parts of the model influence a given observed data-point? The answer is that the row of matrix, ${\bf A}$,corresponding to the data-point of interest will be non-zero where that point in model space influences the data-value. Rows of ${\bf A}$ may therefore be thought of as sensitivity kernels, describing which points in model space are sensed by a given data-point.

For a generic linearized traveltime tomography problem, traveltime perturbations, ${\bf \delta T}$, are related to slowness perturbations, ${\bf \delta S}$, through a linear system,

$${\bf \delta T} = {\bf A \; \delta S}.$$ (2)

The form of the sensitivity kernels depend on the the modeling operator, ${\bf A}$.

Under the ray-approximation, traveltime for a given ray, T, is calculated by integrating slowness along the ray-path,  

$$T=\int_{\rm ray} s({\bf x}) \;dl.$$ (3)

Assuming that the ray-path is insensitive to a small slowness perturbation, the perturbation in traveltime is given by the path integral of the slowness perturbation along the ray,  

$$\delta T=\int_{\rm ray} \delta s({\bf x}) \;dl.$$ (4)

Since traveltime perturbations given by equation (4) are insensitive to slowness perturbations anywhere off the geometric ray-path, the sensitivity kernel is identically zero everywhere in space, except along the ray-path where it is constant. The implication for ray-based traveltime tomography is that traveltime perturbations should be back-projected purely along the ray-path.

We are interested in more accurate tomographic systems of the form of equation (3), that model the effects of finite-frequency wave-propagation more accurately than simple ray-theory. Once we have such an operator, the first question to ask is: what do the rows look like?