  Derivatives of the displacement tensor with respect to the independent model
parameters of the subsurface, also called Fr\'echet derivatives (or sensitivity
kernels), are a key ingredient for seismic full-waveform inversion with a
local-search optimization algorithm. They provide a quantitative measure of the
expected changes in the seismograms due to perturbations of the subsurface
model parameters for a given survey geometry. Since 2.5-D wavefield modeling
involves a real point source in a 2-D geological model with 3D (spherical) wave
properties, it yields synthetic data much closer to the actual practical field
data than the commonly used 2-D wave simulation does, which uses an unrealistic
line source in which the waves spread cylindrically. Based on our recently
developed general 2.5-D wavefield modeling scheme, we apply the perturbation
method to obtain explicit analytic expressions for the derivatives of the
displacement tensor for 2.5-D/2-D frequency-domain seismic full-waveform
inversion in general viscoelastic anisotropic media. We then demonstrate the
numerical calculations of all these derivatives in two common cases: (i)
viscoelastic isotropic and (ii) viscoelastic tilted transversely isotropic
(TTI) solids. Examples of the differing sensitivity patterns for the various
derivatives are investigated and compared for four different homogeneous models
involving 2-D and 2.5-D modeling.
