Most of the energy from an earthquake arives at a seismic station after the ray theoretical travel time. This is because it takes a sine wave with a period, T, a ramp up time of T/4 for the energy to reach maximum. So, another ray that travels along a similar path to that of the ray theoretical ray path that has an onset time within T/4 will add constructively to the measured pulse. Less similar ray paths leading to travel times that arrive early or late by T/4 < |dT| < T/2 will destructively interfere with the pulse. The constructive and destructive interferences result in positive and negative sensitivities respectively. Because waves from Earthquakes are composed of many frequencies, the sensitivity is dependent on the weighted summation of all the sensitivities to all the frequencies.
Typically, the mid-to-long periods (10-100 s) dominate seismic records because attenuation tends to reduce energy of high-frequency waves faster than low- frequency waves. Additionally, the very-long periods are only excited by the largest earthquakes. So, energy recorded at a seismometer is typically band-limited. Additionally, most seismologists 'clean up' their seismic records by filtering out long-period drift and short period noise. Because of this finite set of frequencies, we can calculate the sensitivities relatively easily.
While finite-frequency kernels are typically used to model sensitivity to 3D variations in seismic velocity, they can also be used to model sensitivity of waves to the topography on a seismic discontinuity. Waves that bounce off of seismic discontinuities, such as S660S which bounces off the underside of the 660-km discontinuity, are highly sensitive to lateral variations in discontinuity topography. Deeper discontinuity topography leads to early arrivals because of the shortened travel path. Shallower undulations lead to delayed arrivals because of the lengthened travel path.
The resulting finite-frequency Frochet kernel for travel time sensitivity to discontinuity topography for an S660S wave looks like like the figure to the left (in map view). The kernel itself depends heavily on the size of the blocks that compose the kernel. Larger blocks average the sensitivities over a larger area, which often channges the shape of the kernel. It is important to grid the Earth into small enough blocks that the kernels do not become greatly distorted. Here the gridding size goes from 0.5 to 5 degree side-lengths. With side-lengths smaller than 1 degree the kernel is only minimally distorted. At side lengths of 5 degrees or more the kernel resembles a cylindrical cap, which is the standard technique for stacking SdS waves.
When you invert stacked finite-frequency kernels and travel-time residuals from stacked waveforms you can recover a structure similar to this
figure. The technique results in higher resolution images of thinning and thickening of the mantle transition zone (WTZ).The
resultant model shows that the transition zone is thick near subduction zones and thin elsewhere.