Frequency domain seismic forward modelling: A tool for waveform ...

the partial dierential equations for numerical modelling. The discretized equations
for the time domain acoustic or elastic wave equations using either a nite dierence
or a nite element approach can be written as
M ~
~u(t)+ K ~
~u(t)= ~ f(t) (1.1)
(see for example Marfurt, 1984a), where ~u(t) is the discretized waveeld (i.e., the
pressure, or the displacement) arranged as a column vector, M is the mass matrix,
~
K is the stiness matrix and ~ f(t) are the source terms, also arranged as a column
~
vector.
Equation (1.1) can be approached in either the time domain or in the frequency
domain.
From this point on, this thesis is concerned with the frequency
domain method for solving these problems. Taking the temporal Fourier transform
of equation (1.1) yields
where
u(!) =
Z 1
,1
K ~
u(!) , ! 2 M ~
u(!) =f(!) (1.2)
~u(t)e ,i!t dt and f(!) =
Z 1
,1
~f(t)e ,i!t dt (1.3)
are Fourier transforms. If viscous damping is included, equation (1.2) becomes
K (!) u(!)+i! C (!) u(!) , ! 2 M (!) u(!) =f(!) (1.4)
~ ~ ~
where C (!) is the damping matrix. Details of the nite-element and nite-dierence
~
approaches can be found in many textbooks (Zienkijevic, 1977; Bathe and Wilson,
1976). In Chapter 3 and Chapter 5 I will give explicit formulas for the matrix coecients
for both the acoustic and elastic wave equations. The mass, stiness and
damping matrices are computed by forming a discrete representation of the underlying
partial dierential equations and the physical parameters (for example, the
seismic velocities, the bulk density and the attenuation parameters). For simplicity
I rewrite equation (1.4) as
S ~
(!) u = f or u = S ~ ,1 (!) f (1.5)
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