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SEISMOLOGY: N ORMAL MODES (FREE O SCILLATIONS )
D ECOMPOSING S EISMOLOGY I NTO S UBDISCIPLINES
Seismology can be decomposed into three representative subdisciplines: body waves, surface waves, and normal modes of free oscillation. Technically, these domains form a continuum, each pertaining to particular frequency bands, spatial scales, etc. In all cases, these representations satisfy the wave equation, but each is subject to different boundary conditions and simplifying assumptions. Each is therefore relevant to particular types of subsurface investigation. Below is a table summarizing the salient characteristics of the three.
Seismic Domains Type Application Data
Boundary Conditions Body Waves P-SV SH High frequency travel times; waveforms unbounded
Surface Waves Rayleigh Love Lithosphere
dispersion; group c(w) & phase u(w) velocities interfaces
Normal Modes Spheroidal Modes Toroidal Modes Global power spectra
spherical earth
As the table suggests, the normal modes provide a framework for representing global seismic waves. Typically, these modes of free oscillation are of extremely low frequency and are therefore difficult to observe in seismograms. Only the most energetic earthquakes are capable of generating free oscillations that are readily apparent on most seismograms, and then only if the seismograms extend over several days.
N ORMAL M ODES
To understand normal modes, which describe the modes of free oscillation of a sphere, it’s instructive to consider the 1D analog of a vibrating string fixed at both ends as shown in panel Figure 1b. This is useful because the 3D case (Figure 1c), similar to the 1D case, requires that
Figure 1
Figure by MIT OCW.
22 April 2005
standing waves ‘wrap around’ and meet at a null point. The string obeys the 1D wave equation with fixed-end BCs, the general expression and solution to which are:
u xt Aei^^ (^ txc )^ Bei (^ txc )^ Ce i (^ txc )^ Dei (^ t xc )
t
u x c
u
∴ = +^ + − + − + + −^ −
( , ) ω ω ω ω
2
2 2 2
2
The BCs require that u (0, t ) = u ( L , t ) = 0, which implies that A = -B and C = -D. Hence:
sin 0
2 sin 0
n n c
L
c
L
c
L
i Aeit Ce it
So there are infinitely many discrete frequencies, ω n , that satisfy (1), and these are called
eigenfrequencies.
Figure 2 depicts several modes or eigenfrequencies that satisfy (2). n=0 corresponds to the fundamental mode and all n≥1 correspond to higher modes (overtones).
Normal Modes in the Fourier Domain
The normal modes, just like the harmonic solutions we saw for the 1D case in (1) and (2), can be thought of as basis functions spanning the set of all possible waves we expect to encounter in a spherical body. Therefore, we can employ weighted mode summation to reconstruct or represent waveforms occurring in a spherical body. Heuristically, u can be represented by
u ( x , t ) = ∑ An modes( ω n ) (3)
and, more precisely
Figure 2
Aside: We have already seen ω- k plots for surface and body waves and have learned how to interpret and manipulate them. Normal modes are also frequently graphically depicted using ω- l plots, where ω has the normal meaning and where l is the characteristic length or angular order. But note that l =2 πR/ λ, and recall that k =2 π/ λ; so the angular order is like a wave number!
L
n = 2
n = 0
n = 1
1 X 1
X 3
Figure by MIT OCW.
22 April 2005
Synthetic Seismograms
Recall that we have already established a method for creating synthetic seismograms for body waves, using the following expression:
u ( xt ) φ( k (^) xky ω z ) e i (^ ⋅^ −ω^ t ) dkxdkyd ω
∞
−∞
∞
−∞
∞
−∞
= (^) ∫ ∫ ∫ , , , , kx , (5)
This triple infinite integral is computationally prohibitive to calculate, and a number of simplifications have been introduced to circumvent these difficulties. Such techniques include the so-called WKBJ method, the Reflectivity method, and the Cagniard-de Hoop method, none of which will be developed here. The point is that simplifications are imperative. For plane waves, two useful simplifications are:
1. Integrate over a finite frequency band, ω 0 - δω < ω < ω 0 + δω. This is useful if the phase
of interest can be isolated within a practical frequency band.
2. Integrate over a finite ‘wave number band’, k 0 - δ k < k < k 0 + δ k. This is possible because
arrivals at a particular station are only incident over a finite sub-range of all possible directions. The wave vector indicates the direction in which a plane wave travels, so we can reduce the range of integration to something finite, since only particular directions can physically arrive at a station for a given event. The same simplifications are applicable for normal modes. Whereas the preceding simplifications are
applicable for body waves in the ω- k domain, with
normal modes we work in the in the ω- l domain
(recalling that l is the angular order – or basically a normal mode wave number). Figure 4 provides an
explanatory cartoon of this in the ω- l domain. The
frequency band of interest is ∆ω, which of course would
correspond to waves having only particular frequency content. The waves might only arrive at a particular station in certain directions, allowing us to limit the range of integration in l as well. The shaded area therefore depicts the actual range over which modes must be summed to approximate waves having desired properties.
Note that c = ω/ k (body/surface waves) and c = ω/ l
(normal modes), so straight lines in the ω- l domain still
correspond to constant wavespeeds.
And, just as in the ω- k domain, we can identify ‘phases’
in the ω- l domain (see Figure 5), which correspond to the
average wavespeeds of rays interacting with various features at depth.
l
ω
∆ω
Figure 4
l
ω
P
ScS
SKS
PKP
PKiKP
Figure 5
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Normal Mode Nomenclature The wave equation, subject to spherical boundary conditions, gives rise to the so-called spherical harmonics : sphericalharmonics BCs spherical u && = c^2 ∇^2 u ⎯⎯⎯ ⎯→.
For example, the gravitational potential can be expressed in spherical harmonics by:
∑∑^ {^ }^ (^ )
∞
= = − −
0 0 Longitudedependent LegendreLatitude Polynomialdependent
grav cos sin cos
l
l
m
m l
m n
m n
l A m B m P r
U ϕ ϕ θ. (6)
Equation (6) displays a 2 l +1 degeneracy. That is, for each l there exist 2 l +1 modes (solutions). For example, for l = 0 there is only one mode; for l = 1, there are three modes corresponding to 0 A 1 , 1 A 1 , and 1 B 1. n indicates the number of nodes along the radius of the Earth (also called the overtone number ), and l is the angular order, which indicates the number of nodal planes on the surface (see Figures 6 and 7). Spheroidal modes (~P-SV; changes in volume) are denoted by (^) n S l and are sensitive to compressional and shear wavespeed as well as density. Toroidal modes (~SH; rotation or shear; no change in volume) are denoted by (^) n T l and are sensitive only to shear wavespeed. There are more spheroidal than toroidal modes.
There are a number of modes that have been given special names. One such mode is 0 S 0 (see Figure 7, bottom left), which is called the breathing mode because the entire spherical volume periodically expands and contracts. Another is 0 S 2 (see Figure 7, top left), which is called the football mode because the extrema of this free oscillation are shaped like an American football (also because a European football displays this oscillation when it is kicked). Two modes that do
Figure 6
0 T 2
2 nodal planes along surface; l = 2 (^) 1 zero crossing along the radius; n = 1
2 nodal planes along surface; l = 2
3 nodal planes along 1 T 2 0 S^2 0 S^3 surface; l = 3
Figure by MIT OCW.
22 April 2005
Figure 8
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Image removed due to copyright considerations.
