Our full deformation model consists of a
body-force couple (fault) embedded in an elastic plate overlying a viscoelastic
half-space (Figure 1). We begin by solving for the
displacement vector **u***(x,y,z)* due to a point vector body force at depth. Arbitrarily complex curved and
discontinuous faults are generated in a fin grid of force vectors. We ignore the effects of EarthÕs
sphericity and assume a Poisson material while maintaining constant layer shear
moduli (rigidity) with depth. An overview of this solution follows:

(1) Derive differential equations relating a three-dimensional
(3-D) vector body force to a 3-D vector displacement. We apply a simple force balance in a homogeneous, isotropic
medium and after a series of substitutions for stress, strain, and displacement,
we arrive at Equation 1, where *u, v, *and*
w* are vector displacement components
in *x, y*, and *z, **l** *and *m* are Lame parameters, and *r** _{j}*
are vector body force components in an orthogonal system (i.e.

(1)

A vector body force is applied at , where we assume a sign convention such that *z < 0* is considered downward from the surface and *a* is a negative number. To partially satisfy the boundary condition of zero shear
traction at the surface, an image source, identical to the initial source yet
mirrored in location above the *z*-axis,
is also applied at [*Weertman,
1964*]. Equation 2 describes a point body force at both source and
image locations, where **F **is a
vector force.

(2)

(2) Take the 3-D Fourier transform of Equations 1 and 2 to
reduce the partial differential equations to a set of linear algebraic
equations.

(3)
(3)

Invert the linear system
of equations to isolate the 3-D displacement vector solution for

*U*(**k**), *V*(**k**), and *W*(**k**):

(4) Perform the inverse Fourier transform in the *z*-direction (depth) by repeated application of the
Cauchy Residue Theorem. In the
following equation, **U***(***k***,z)* represents the deformation tensor, while subscripts *s* and *i*
refer to source and image contributions.

(4)

(5) Introduce
a layer into the system through an infinite summation of image sources. We
utilize the method of images [*Weertman*, 1964] and superpose multiple image sources [*Rybicki*, 1971], reflected both above and below the horizontal
axis, to account for both the source vector and the elastic layer, defined by
thickness *H* (Appendix B). The development of this solution
requires an infinite number of image sources to satisfy the stress-free surface
and layer boundary conditions (Equation 5). Contrasting layer and half-space rigidities *m*_{1} and *m*_{2},
respectively, are also included.

(5)

(6) Integrate the point source Green's function
to simulate a fault (Equation 6), where the body force is applied between the
lower depth *d _{1}* and the
upper depth

(6)

In Equation 6, is the
depth-integrated solution. The
individual elements of the source and image tensors are

(7)

,

where ** Z** represents all z-dependent terms, including all
combinations of

(8)

,

The solutions of Equation 8 are identical
to those of *Smith and Sandwell*
[2003] but have been simplified here for further manipulation of the
exponential terms. In particular,
we analytically sum the infinite series for the case of *m** _{2}*
= 0, which corresponds to the end-member case of an elastic plate over a fluid
half-space.

(7) Analytically solve
for Maxwell viscoelastic time-dependence using the Correspondence Principle.

The numerical components of this entire
approach involve generating a grid of force couples that simulate complex fault
geometry, taking the 2-D horizontal Fourier transform of the grid, multiplying
by the appropriate transfer functions and time-dependent relaxation
coefficient, and finally inverse Fourier transforming to obtain the desired
results. Our layered fault model
consists of an elastic plate overlying a Maxwell viscoelastic half-space (Figure 1)
that includes parameters of plate thickness (*H*), locking depths (*d _{1}*,