Historically, the first problems for impedances considering soil inertia were solved semianalytically for homogeneous half-space without the internal damping and for circular surface stamps. It turned out that horizontal impedances behaved more or less like pairs of springs and viscous dashpots (as mentioned above, in reality the dissipation of energy had completely different nature; mechanical energy was not converted in heat, as in dashpot, but taken away by elastic waves). One more “good news” was that the impedance matrix appeared to be almost diagonal. Though non-diagonal terms coupling horizontal translation with rocking in the same vertical plane were non-zero ones, their squares were considerably less in module than products of the corresponding diagonal terms of the impedance matrix.

These two facts created a base for using the above mentioned “soil springs and dashpots”.
There are several variants of stiffness and dashpot parameters. You can see below a table 1
from ASCE4-98 [1] for circular base mats.

One more table of the same sort in the same standard [1] is given for rectangular base mats.
However, even for homogeneous half-space it turned out that vertical and angular impedances behaved differently from simple springs and dashpots. This can be seen in Fig.10, where the tabular impedances given by ASCE4-98 are compared to the wave solutions given by codes SASSI and CLASSI (number in the legend denotes number of finite elements along the side).
We understand now that the expressions given in the tables are just approximations for the real values. So, there cannot be “exact” or “true” expressions of this kind – different variants exist.
The expressions from the tables look particularly strange for angular damping, where parameters of the upper structure participate. It is real absurd from the physical point of view: impedances cannot depend on the upper structure; soil just “does not know” what is above rigid stamp. This is not an error, as some colleagues think. This is an attempt to take values from the frequency-dependent curves at certain frequencies. These frequencies estimate the first natural frequencies of rigid structure on flexible soil, so they depend on structural inertial parameters. Surely, such expressions are approximate.
The conclusion is that frequency dependence of the impedances exists even for the homogeneous half-space and spoils the spring/dashpot models in the impedance method.

One more comment should be added here. If a package of horizontal layers is underlain by rigid rock, surface waves behave in a completely different manner than for homogeneous half-space. Instead of two surface waves (Love and Rayleigh ones) there exist an infinite number of surface waves. Each of these waves for low frequencies cannot take energy from the basement – they are “geometrically dissipating” (even without internal soil damping), or “locked”. However, when frequency goes up, each of these waves one by one transforms from a “locked” wave into a “running” wave capable to take energy to the infinity. This behavior depends on soil geometry only, not on structure.
It means that in the frequency domain there exists a certain low-frequency range, where the whole soil foundation is “locked”. All the energy taken from the basement is only due to the internal damping. If there is no internal damping, complex impedances are completely real.
In practice internal soil damping is several percent, so impedances are “almost real”. After the first surface wave transforms into a running one, the soil foundation becomes “unlocked” – wave damping appears. Then one by one other surface waves turn into “running” ones, increasing the integral damping in the soil-structure system. The impedance functions for the same soil, but underlain by rigid rock at depth 26 m, are shown on Fig.11. One can see “locking phenomena” looking at the imaginary parts of the impedances.
The conclusion here is that the frequency dependence of the impedances may be rather sophisticated depending on soil layering. The attempts to “cover” the variety of soils by number of homogeneous half-spaces with different properties (usually this is an approach for “serial design” of structures) may lead to mistakes: there is always a possibility that real layered soil will not be “covered” by a set of half-spaces (e.g. no half-space can reproduce the “locking” effects described above).
The additional problem with springs and dashpots arises when the integral stiffness is distributed over the contact surface Q. Physically in every point of Q there are no distributed angular loads impacting basement from the soil. So, only translational springs and dashpots are usually distributed over Q, and angular impedances are the results of these distributed translational springs. For a surface basement vertical distributed springs are responsible for rocking impedances, and horizontal springs are responsible for torsional impedance. The problem here is that all attempts to find the distribution shape for vertical  springs to represent rocking impedances simultaneously with vertical one have failed. If fact, integral rocking stiffness obtained from distributed vertical springs is always less than actual rocking stiffness; on the contrary, integral rocking damping obtained from distributed vertical dashpots is always greater than the actual one. Physical reason of this mismatch is the interaction between different points through soil. Spring/dashpot model is “local” in nature: the response is determined by motion of this very point, and not neighbors. This is not physically true.
The author found a way to treat both problems at once. The idea is to work in the time domain using a platform model of Fig.5 with conventional springs and dashpots (lumped or distributed). Of course we get some “platform” impedances D(ω), different from “wave” impedances C(ω), but the idea is to tune the platform excitation Vb so to account for the difference between “wave” impedances and “platform” impedances. Six components of the platform excitation may be tuned to reproduce six components of response – e.g., six components of the rigid base mat’s accelerations. Such an approach combines the calculations in the frequency domain (platform seismic input) with calculations in the time domain (final dynamic analysis of the platform model), that is why this method is called “combined”. Besides, this method is “exact” for rigid base mats only: the stiffer is a base mat, the more accurate are the results. That is why this method is also called “asymptotic” – full name is “combined asymptotic method” (CAM) [22].

The last item to discuss in this part is practical tools to obtain impedances and seismic loads (or weightless base mats’ motions) in the frequency domain. At the moment the author uses one of two computer codes. For rigid surface basements on a horizontally-layered soil code CLASSI is the most appropriate. For the embedded basements with possible local breaks in horizontal layering code SASSI is used (SASSI can be used for surface base mats also, but is more sophisticated).
In both cases formula (3) is a basic equation for impedances, and the dynamic stiffness matrix G0 linking set of nodes in the infinite soil is a key issue (the second matrix Gint is absent for surface basement in CLASSI and easily obtained by FEM for the embedded basement in SASSI). To get G0, they first obtain a dynamic flexibility matrix, describing displacements due to the unit forces (this is a Green’s function). Here is a difference between two codes.
Professor J.Luco managed [7] to develop Green’s function analytically for the case of surface load and surface response node in horizontally-layered soil in the frequency domain. Then contact surface Q was covered with number of rectangular elements of different shapes.