[9] The physical model used in the study of the acoustic-to-seismic transfer function consists of elastic layers covering an elastic half space (Figure 1). The parameters associated with the air and substrate are identified by subscripts 0 and ∞; *c*_{0} and ρ_{0} are the speed of sound and the density in the air column; *d*_{j}, *c*_{ℓj}, *c*_{tj}, and ρ_{j} are the thickness, the velocities of the compression and shear waves, and the density in the *j*th elastic layer; and *c*_{ℓ∞}, *c*_{t∞}, and ρ_{∞} are the velocities of the compression and shear waves, and the density in the elastic half space (substrate). The ground parameters are assumed to be constant within the elastic layer. The air column and the elastic half space are assumed to be homogeneous and semi-infinite. Attenuation effects are taken into account in the layers and in the half spaces assuming that the compression and shear waves velocities are complex values _{ℓ,t} = *c*_{ℓ,t} + *i* · η_{ℓ,t}. This requires that the wave numbers be complex: *k* = (ω/*c*_{ℓ,t})(1 − *i* · (η_{ℓ,t}/*c*_{ℓ,t}))/(1 + (η_{ℓ,t}/*c*_{ℓ,t})^{2}), where ω = 2π*f* and *f* is the frequency. In a rectangular coordinate system (*x*, *y*, *z*) the *z* axis is taken to increase positively upward and be normal to the layer plane. We assume that the displacement fields *U* can be written in the terms of the scalar and the vector potentials ϕ and ψ for the waves of vertical (SV waves) and horizontal (SH waves) polarizations:

When taking only waves of vertical polarization (SV waves) into account, the displacement fields are not dependent upon the *y* coordinate and do not contain a *y* component of the displacement *U* = *U*(*U*_{x}, *U*_{y} = 0, *U*_{z}). *U* is expressible as the sum of an irrotational component, which represents a compression wave, and a solenoidal component, which represents a shear wave. The vector potential ψ represents the component of motion with nonzero vorticity. The reference axis of the system can be chosen such that ψ has only one component along the *y* axis; that is to say, ψ = (0, ψ_{y}, 0). Stress and strain relations for a locally isotropic solid in a linear approach to determine the amplitude of strain are given by the generalized Hook's law written in the tensor form:

where σ_{ij} is the stress tensor; λ and μ are the Lamé's parameters; δ_{ij} is Kronecker's symbol; *U*_{γγ} = ∂*U*_{x}/∂*x* + ∂*U*_{z}/∂*z* is the invariant, which characterizes the relative changing of element volume; and *U*_{ij} = (1/2)(∂*U*_{x}/∂*y* + ∂*U*_{y}/∂*x*) is the linearized strain tensor, which characterizes the increment of squared distance between two closely set points. Using equation (1) and the generalized Hook's law (equation (2)), the normal and tangential components of the displacement and the stress tensor in isotropic media can be written in terms of the potentials ϕ and ψ_{y}:

where λ and μ are the Lamé parameters. The velocities *c*_{ℓ} and *c*_{t} of the compression and shear waves are given by the well-known relations *c*_{ℓ} = and *c*_{t} = . The components of the displacement (*U*_{x}, *U*_{z}) and components of the stress tensor (σ_{xz}, σ_{zz}) involved in the boundary conditions are continuous across the elastic/elastic (*z* = *d*_{j}) and the air/elastic (*z* = 0) interfaces:

The potentials ϕ and ψ_{y} are governed by the Helmholtz equations:

where α^{2} = *k*_{ℓ}^{2} − ξ^{2} and β^{2} = *k*_{t}^{2} − ξ^{2} are the squares of the vertical components of the wave numbers for the compression and shear wave velocities, *k* = ω/*c*_{0}, *k*_{ℓ} = ω/*c*_{ℓ}, *k*_{t} = ω/*c*_{t}, ξ = *k* · sinθ_{0} = *k*_{ℓ} · sinθ_{ℓ} = = *k*_{t} · sinθ_{t} are the wave numbers and the propagation constant, and *θ*_{0} is the incidence angle of the plane wave. Assuming a plane wave incidents in the air, the solutions of equation (6) can be written in the terms of the potentials describing the compression and shear waves:

where ϕ^{+}, ϕ^{−}, ψ^{+}, ψ^{−} are the some arbitrary functions that characterize the elastic waves propagating in the positive (with the superscript ^{−}) and negative (with the superscript ^{+}) direction of the *z* axis. For an incident plane wave of unit amplitude, which excites the given elastic layered system, the depth dependence of the potentials in the air and elastic half spaces are given:

where ϕ_{0}^{−} = *R* is the reflection coefficient; ϕ_{0}^{+} = 1 is the amplitude of the plane wave under the angle of incidence θ_{0} in the air half space; and ϕ_{∞}^{+}, ϕ_{∞}^{−}, ψ_{∞}^{+}, ψ_{∞}^{−} are the amplitudes of the compression and shear waves in the elastic half-space, respectively. Here ϕ_{∞}^{+} = *W*_{ℓ} and ψ_{∞}^{+} = *W*_{t} are the refraction indices of the compression and shear waves in the substrate. Taking into account the radiation conditions at infinity, then ϕ_{∞}^{−} = 0 and ψ_{∞}^{−} = 0. Substituting equations (7) and (9) in the boundary condition (5) across the elastic/elastic (*z* = *d*_{j}) interfaces and performing the differentiation at the boundaries, we obtain 4(*n* + 1) equations in 4(*n* + 1) unknowns ϕ_{j}^{+}, ψ_{j}^{−}, ϕ_{j+1}^{+}, ψ_{j+1}^{−}.

[10] We will characterize the displacement field not by the column vector potential (*z*) = (ϕ_{j}^{+}, ϕ_{j}^{−}, ψ_{j}^{+}, ψ_{j}^{−})^{T} (T denotes the operation of transposition), which in a complicated manner transforms across boundaries, but by the column vector (*z*) = (*U*_{x}, *U*_{z}, σ_{xz}, σ_{zz})^{T}. The vector (*z*) by virtue of the boundary conditions is continuous, as opposed to the vector potential (*z*):

Using the constancy of the column vector (*z*) inside the layer and in equation (10), it is easy to obtain the required relation between (*z*_{j}) and (*z*_{j+1}) at two adjoined boundaries *j* and *j* + 1 [*Brekhovskikh and Godin*, 1989]:

Here *A*_{j} and · *A*_{j}^{−1} are the right and invert matrices of the elastic layer, and *L*_{j} is the diagonal matrix for interlayer contacts:

The consecutive application of equation (11) to *j* = 1, …, *n* layers and interlayer contacts permits connection of the column vector between the boundary of first and second layers (*z*_{1}) and the value at the boundary *n*th layer and elastic half space (*z*_{∞}):

where *D* = *A*_{1}^{−1} · *A*_{2} · *L*_{2} · *A*_{2}^{−1} · … · *A*_{n} · *L*_{n} · *A*_{n}^{−1} · *A*_{∞} is matrix propagator characterizing the viscous elastic layered media and elastic half space; and *A*_{∞} is the characteristic matrix of the elastic half space, which looks like matrix *A*_{j} in equation (12), but all material parameters use the subscript ∞. If we know the elements of matrix *D* and incident plane wave perturbation ϕ_{0}^{+} = 1, then for the combined description of the upper air half space and multilayered elastic media, one can write six boundary equations:

where *q*_{km}(*k* = 1, 2, *m* = 1, 2) are the elements of the matrix *Q*^{−1} for the air half space, and *d*_{ij}(*i* = 1, …, 4, *j* = 1, …, 4) are the elements of the matrix propagator *D* for the layered viscoelastic half space. The invert matrices *Q*^{−1} for air half space are written as:

where ω = 2π*f* is the cyclic frequency, ρ_{0} is density of air, α_{0} = is the vertical component of the wave number, and *k* = ω/*c*_{0}, *c*_{0} is the sound speed in air.

[11] For solving the system of linear algebraic equation (15) with respect to the reflection coefficient ϕ_{0}^{−} = *R*, the refraction indices for compression ϕ_{∞}^{+} = *W*_{ℓ} and shear waves ψ_{∞}^{+} = *W*_{t}, one may use the Cramer's rule taking into account the radiation conditions at infinity ϕ_{∞}^{−} = ψ_{∞}^{−} = 0 and absence of the incident and reflected shear waves in the air half-space (ψ_{0}^{+} = ψ_{0}^{−} = 0):

If the complex pressure *P* and the reflection coefficient *R* are known, the normal particle velocity on the boundary with the solid layer *V*_{z} may be determined. The impedance *Z*_{in}(*f*, θ) and the acoustic-to-seismic transfer function *TF*(*f*, θ) may be expressed as:

To increase computational accuracy, matrices of the fourth order should be replaced by the sixth-order matrices. A matrix technique using the sixth-order matrices was implemented as a functioning computer code to obtain *TF*(*f*, θ). Results of test computations of this code are presented in the work of *Fokina and Fokin* [2000].