Traveltime sensitivity kernels: Banana-doughnuts or just plain bananas?

Kernels compared

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 $v_0(0)=1.5 \;{\rm km s^{-1}}$, and $\frac{dv_0}{dz}= 0.8 \; s^{-1}$. 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 band-limited wave-paths (Woodward, 1992). 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.

Traveltime sensitivity kernels: Banana-doughnuts or just plain bananas?