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This section contains images of traveltime kernels computed
numerically for a simple model that may be encountered in a reflection
tomography problem. The source is situated at the surface, and the
receiver (known reflection point) is located at a depth of 1.8 km in
the subsurface. The background velocity model, *v*_{0}(*z*)=1/*s*_{0}(*z*), is
a linear function of depth with , and
. I chose a linear velocity function
since Green's functions can be computed on-the-fly with rapid two-point
ray-tracing.
Figure 1 shows the ray-theoretical traveltime
sensitivity kernel: zero except along the geometric ray-path.

**RayKernel
**

Figure 1 Traveltime sensitivity kernel for
ray-based tomography in a linear *v*(*z*) model.
The kernel is zero everywhere *except* along geometric ray-path.
Right panel shows a cross-section at *X*=1 km.

Figures 2 and 3 show
first Rytov traveltime sensitivity kernels for 30 Hz and 120 Hz
wavelets respectively. The important features of these kernels are
identical to the features of kernels that Marquering et al. (1999)
obtained for teleseismic *S*-*H* wave scattering, and to Woodward's
1992 band-limited wave-paths. They have the
appearance of a hollow banana: that is appearing as a banana if
visualized in the plane of propagation, but as a doughnut on a
cross-section perpendicular to the ray.
Somewhat counter-intuitively, this suggests that traveltimes
have zero sensitivity to small velocity perturbations along the
geometric raypath.
Fortunately, however, as the frequency of the seismic wavelet increases,
the bananas become thinner, and approach the ray-theoretical kernels
in the high-frequency limit.
Parenthetically, it is also worth noticing that the width of the
bananas increases with depth as the velocity (and seismic wavelength)
increases.

**BananaPancake8
**

Figure 2 Rytov traveltime sensitivity
kernel for 30 Hz wavelet in a linear *v*(*z*) model.
The kernel is zero along geometric ray-path.
Right panel shows a cross-section at *X*=1 km.

**BananaPancake2
**

Figure 3 Rytov traveltime sensitivity
kernel for 120 Hz wavelet in a linear *v*(*z*) model.
The kernel is zero along geometric ray-path.
Right panel shows a cross-section at *X*=1 km.

I do not show the first-Born kernels here, since, in appearance, they
are identical to the Rytov kernels shown in
Figures 2 and 3.

** Next:** The Banana-doughnut paradox
** Up:** Rickett: Traveltime sensitivity kernels
** Previous:** Rytov traveltime sensitivity
Stanford Exploration Project

4/27/2000