Download Equations - Seismology - Lecture Notes and more Study notes Geology in PDF only on Docsity!
We have introduced the equations:
and
With and
These equations can be solved using 3 different methods:
- D’Alembert’s solution (most ‘physical’ approach)
- Separation of variables
- Fourier Transforms (mathematically, the most powerful method)
There is a whole class of theoretical development in applied maths that uses Fourier
Integral Operators (FIOs)
1.D’Alembert’s Solution:
Take as an example, the wave equation:
And the function: (45)
The first term: f(x – ct) represents propogation in the positive x-direction
The second term: g(x+ct) represents propogation in the negative x-direction
c = the wave speed or phase speed/velocity
Consider the profile of the wave at a time and at some later time
Figure 4:
(x-ct) is known as the phase of the wave.
The phase speed is given by: (46)
Figure 5: Diagram to illustrate the concept of wavefronts:
Wavefront = surface connecting points of equal phase
A wavefront is a line in 2d (or surface in 3d) connecting points of equal phase.
In reality, the wavefronts are circular, but locally they behave as a plane wave.
All points along the wave-front have the same travel-time from the origin.
The relationship between the wavenumber (k) and angular frequency is given by:
(47)
The relationship between wavenumber (k) and wavelength is given by: (48)
So we can re-write the phase in terms of wavenumber: (in one
dimension)
In 3 dimensions, this becomes: or
The harmonic function is a solution to the wave equation:
( ω)
( λ)
( k xx +^ k yy +^ k zz^ −^ ω t ) (^ k x.^ −^ ω t )
k c
ω
1 0
1 0
x x c t t
k
π
λ
x t kx c
ω t
⎜ −^ ⎟→^ −
A lot of imaging is done in the frequency domain by finding solutions to the helmholtz
equation.
- Fourier Transforms:
Fourier transforms allow us to understand the relationship between the space-time (x,t)
and wavenumber-frequency (k, domains.
In one dimension, the forward and reverse fourier transforms between the space-
frequency and space-time domains are given by:
(Note: in seismology, we normally take the exponential as having a positive sign when
we are transforming into the space-frequency domain, however this is simply a
convention)
Similarly, we can use fourier transforms to convert between the space-time and
wavenumber-time domains. In 3d the fourier transforms between the space-time and
wavenumber-time domains are:
Combining these gives the double-fourier transform:
Note that does not appear in this equation. This is because, are related via
the dispersion relation.
Hence, if we have specified the angular frequency, and it follows that has
already been determined.
Numerically, this double fourier transform is very difficult to work with.
Note: Synthetic seismograms:
( 69 )
( 68 )
( 67 )
( 66 )
Since gives the length of the vector representing the P-wave and represents
the length of the vector representing the S-wave, it follows directly that P-waves ‘dive’
less steeply into the medium than S-waves.
Figure 7:
S
P
x
z
The phase ‘speed’ c, is given by: (73) and is a vector in the direction of
propogation
At the surface, we measure: (74) which is the ‘apparent’ velocity/speed
Horizontal slowness:
eter (76)
Vertical slowness:
We know that:
The vertical slowness is given by:^ (^77 )
Combining the vertical and horizontal slowness:
( 78 )
Rearranging this gives:
( 79 )
k β =
k α =^ ω
ds c dt
x
dx c dt
cp (75)
= horizontal slowness = ray param
This follows from Snell’s law
ds dt ⎛^1 ⎞ sin( t ) = = c = c (^) ⎜ ⎟ =
dx dx (^) ⎝ c (^) x ⎠
1 sin( t ) ρ = = c (^) x c
cx = ( cx , cz )
2 2 2 2 2 2
sin t cos t 1
c c
c
So, the vertical slowness does change with depth, because c is a function of depth. The
vertical slowness is zero if (which represents a horizontally propagating
wave)
η is imaginary for evanescent waves. (This is important for understanding the
behaviour of surface waves).
There is a direct relationship between the wave-vector and the slowness components:
(80)
(81)
(82)
Figures from notes of Patricia M Gregg (Feb 2005) Kang Hyeun Ji (Feb 2005)
Notes: Katie Atkinson, Feb 2008
2 2
p c
( ) η =
x x
k p
c
ω = =ω
k c
ω
z z
k c
ω = =ωη
k = ( kx , kz ) = ( ω p , ωη ) =ω ( p , η)
