3-D Viscoelastic Model
Full mathematical formulation
Example source code
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 Earth’s
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,  and  are Lame
parameters, and j are vector body force components in an orthogonal system (i.e. j = x, y, or
z):
 2 u  2 v  2 w 
2 u      2 

  x
ydx zx 
x

 2 u  2 v  2 w 
2v    
 2
  y
zy 
xy y

(1)
 2 u  2v  2 w 
 w    

 2   z
xdz ydz z 
2
A vector body force is applied at x  y  0, z  a , 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
x  y  0, z  a [Weertman, 1964]. Equation 2 describes a point body force at both source
and image locations, where F is a vector force.

(2)
(x,y,z) = F (x) (y) (z - a) + F (x) (y) (z + a)
(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) Invert the linear system of equations to isolate the 3-D displacement vector solution for U(k),
V(k), and W(k):
U (k) 


V (k) 

W (k)

 2 2
k 2
 k y  k z 
(   )

(  ) 
 kx k y

4
k (  2 ) 


k x kz




k y kx
k
2
x
 k z2  
k
(  )
k y kz



F 
(e  i2 k z a  e i 2kz a )  x 
 kz ky

Fy 
4 2

Fz 
2
 
 k 
2
2
kx  ky 

(  ) 

 kz kx
2

(3)

where k = (kx ,ky ,kz ) and k  k  k.
2
(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.
Fx 
Fx 
U(k) 
 
 


V (k)   Us (k,z  a)Fy  +Ui (k,z  a)Fy 



W (k)

Fz 

Fz 

(4)
(5) Introduce a layer into the system through an infinite summation of image sources. We utilize
 of images [Weertman, 1964] and superpose multiple image sources [Rybicki,
the method
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 1 and 2,
respectively, are also included.
 Us (z  a  2mH) 
Fx 
Fx  

F 
U(k) 
1  2 m  Ui (z  a  2mH) x  (5)
 
 


 
Fy 
V (k)   Us (k,z  a)Fy  +Ui (k,z  a)Fy   
m11  2   U s (z  a  2mH)





W (k)

Fz 

Fz 

Fz 


 Ui (z  a  2mH)


(6) Integrate the point source Green's function to simulate a fault (Equation 6), where the body
force is applied between the lower depth d1 and the upper depth d2. For a complex dipping
fault, this integration can be done numerically.
However, if the fault is vertical, the
integration can be performed analytically. The displacement or stress (derivatives are also
computed analytically) can be evaluated at any depth z above d2.
d2
 Uk,Z dZ

U (z  d )  U (z  d ) U (z  d )  U (z  d )
'
s
2
'
s
1
'
i
2
'
i
1
d1
 U' (z  d  2mH)  U' (z  d  2mH) 
s
2
s
1


'
'
m 
 
  2   Ui (z  d2  2mH)  Ui (z  d1  2mH)

+  1

'
'
m11  2   U s (z  d 2  2mH)  U s (z  d1  2mH) 


'
'

 Ui (z  d2  2mH)  Ui (z  d1  2mH)

(6)
In Equation 6, U' is the depth-integrated solution. The individual elements of the source and
image tensors are


U x U y U z 


U's (k,Z)  U y Vy Vz  and

U z Vz W z 

U x

U'i (k,Z)  U y

U z
Uy
Vy
Vz
U z 

Vz ,
W z 

(7)
where Z represents all z-dependent terms, including all combinations of z, dn, and 2mH. The

six independent functions of the deformation tensor are

ky2 kx2
D  2  2
k
k


kx ky
U x 
 2 2
 
k

U y 

U z  C  i k x
  2 
k
Vy   
2
ky2
kx
Vz 
D


2
2

k
k
 

W
 z 
ky

 i
k

 D  1
k x2 
2 
k 
k k 
 x 2y 
k 
k 
i x  e Z ,
k   Z 
Ze 
k y2 
 2 
k 
k 
i y 
k 
1 

(8)

where

C
(   )
4(   2)
D
  3

k  kx2  ky2 
1/2
and  = 2 k .
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 2 = 0, which corresponds to the endmember 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 (d1, d2), half-space viscosity (), elastic moduli
(, ), density (), gravity (g), and time (t). We simulate a finite-width single and double force
couple, F, with a displacement discontinuity across the fault imbeded in a finely-spaced grid. The
analytic form of the force couple is the derivative of a Gaussian function, where the half-width of
the Gaussian is equal to the cell spacing. As previously mentioned, our Fourier solution satisfies
the zero-traction surface boundary condition and continuity across the boundary. The x-boundary
condition of constant velocity difference across the fault plane (for deeply defined slip) is
simulated using a cosine transform in the x-direction. The y-boundary condition of uniform
velocity in the far-field is simulated by arranging the fault trace to be cyclic in the y-dimension.
This fault model will be used to rapidly explore the realistic 3-D viscoelastic response of the
Earth throughout the earthquake cycle.