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velocity perturbation (dv s
0 or dv p
0 and d
ρ
=
0)
density perturbation (dv s
=
0, dv p
=
0 and d
ρ
0)
before scattering
after scattering
before scattering
after scattering
superposition perceived
as traveltime shift
scattered wave
incident wave
scattered wave
incident wave
scattered wave
Fig. 11.3 Scattering characteristics of velocity and density perturbations (e.g. Wu & Aki, 1985; Tarantola, 1986).
Left: An incoming body wave impinges upon a velocity perturbation (black star) with either dv s =
0or dv p =
0and
=
0. The scattered wave, indicated by the double dashed curve, propagates in the same direction as the incoming
wave. The superposition of both waves is perceived as a travel time delay or advance, depending on the sign of the
perturbation (e.g. Dahlen et al. , 2000). Right: The same as to the left, but for a density perturbation with dv s =
0,
dv p =
0. The scattered wave propagates opposite to the direction of the incoming wave. The pulse shape
of the incoming wave remains unperturbed.
0and =
i.e., scattering in the propagation direction of the
incoming wave. While this parametrization in-
troduces body wave sensitivity to density, it also
leads to undesirable trade-offs between the model
parameters, similar to Equation (11.13).
The frequency-dependent travel times of sur-
face waves (dispersion) are mildly sensitive to
density variations. Unfortunately, however, this
sensitivity tends to be oscillatory. A density per-
turbation may therefore interact with both posi-
tive and negative parts of the sensitivity distribu-
tion. The net effect is then nearly zero. Additional
complications arise from the strong trade-offs
between velocity structure and density in the
multi-parameter inverse problem. Perturbations
in seismic velocities and density influence sur-
face wave dispersion simultaneously, and the two
effects cannot easily be distinguished. This led
many authors to either ignore density variations
or to scale v s to ρ using various multiplicative fac-
tors (e.g. Panning & Romanowicz, 2006; Ritsema
et al. , 2011).
At the long-period end of the seismic spec-
trum, around 0.5-3.0mHz, the spheroidal free
oscillations of the Earth are sensitive to the long-
wavelength density distribution in the whole
mantle because of the gravitational restoring
force (Figure 11.4). Lateral heterogeneities lead
to the splitting of normal-mode eigenfrequencies
that can be used for tomography. However, since
normal modes result from the constructive inter-
ference of waves traveling in opposite directions
around the globe, they are primarily sensitive
to even-degree structure, because odd-degree
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