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A generic discrete linear inverse problem may be written as
| |
(1) |

where is the known data vector,
is the unknown model vector, and 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, ,corresponding to the data-point of interest will be non-zero
where that point in model space influences the data-value.
Rows of 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, , are related to slowness
perturbations, , through a linear system,

| |
(2) |

The form of the sensitivity kernels depend on the the modeling
operator, .
Under the ray-approximation, traveltime for a given ray, *T*, is
calculated by integrating slowness along the ray-path,

| |
(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,
| |
(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?

** Next:** Born traveltime sensitivity
** Up:** Rickett: Traveltime sensitivity kernels
** Previous:** Introduction
Stanford Exploration Project

4/27/2000