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Normal Modes (Free Oscillations) and Anisotropy
April 23, 2008
Subdisciplines of Seismology 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 simplify ing assumptions. Each is therefore relevant to particular types of subsurface investigation. Below is a table summarizing the salient characteristics of the three.
Figure 1: Summary of 3 subdisciplines of seismology.
As the table suggests, the normal modes provide a framework for repre senting global seismic waves. Typically, these modes of free oscillation are of extremely low frequency and are therefore difficult to observe in seismo grams. 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.
Normal Modes (Free Oscillations) Please note that normal modes were only touched on in the 2008 semester course. For further discussion of Normal Modes in the Fourier Domain and Synthetic Seimograms, see 22 April 2005 open courseware materials.
Figure 2: (b) 1D and (c) 3D analogs of free oscillation.
To understand normal modes, which describe the modes of free oscil lation of a sphere, its instructive to consider the 1D analog of a vibrating string fixed at both ends as shown in panel Figure 2b. This is useful be cause the 3D case (Figure 2c), similar to the 1D case, requires that 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:
(1) δ (^2) u = 1 δ (^2) u δx^2 c^2 δt^2
.^. .u(x, t) = Aei(t+x/c)^ + Bei(t−x/c)^ + Cei(t+x/c)^ + Dei(t−x/c)
The boundary conditions require that u(0, t) = u(L, t), which implies that A = −B and C = −D. Hence:
(2) 2 i(Aeit^ − Ce−it^ sin ( ωLc ) = 0 ⇒ sin ( ωLc^ ) = 0
⇒ ωL= (n + 1)π, n = 0, 1 , 2 , ..., ∞ c
So there are infinitely many discrete frequencies, ωn, that satisfy (1), and these are called eigenfrequencies.
Figure 3 depicts several modes or eigenfrequencies that satisfy (2). n= corresponds to the fundamental mode and all n ≥ 1 correspond to higher modes (overtones).
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 k = 2 π^2 πR λ and^ l^ =^ λ where^ λ^ is^ the^ wavelength;^ so^ the^ angular^ order^ is^ like a wave number! l also denotes the spherical harmonic degree.
Normal Mode Nomeclature The wave equation, subject to spherical boundary conditions, gives rise to the so-called spherical harmonics:
u¨ = c^2 �^2 u → (spherical boundary conditions)→ Spherical harmonics.
For example, the gravitational potential can be expressed in spherical harmonics by:
The above equation 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 A^01 , A^11 , B 11. 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 4 and 5).
Figure 4: Spheroidial and Toroidal Motion Diagrams
Figure 5: Surface and Radial patterns of Normal Modes
Figure 6: Some spherical harmonics
Normal Mode Seismology: Density Density is very poorly constrained by body and surface waves. Free Os cillations are constrained by gravity and tell us about density structure: ρu¨ = � · σ + fgravity
Reflectivity Method A Propagator matrix describes a propagating wave field through multiple layers. A displacement field is connected to the adjoining layers by bound ary conditions. The matrix system allows a stepwise relationship between layers called the Reflectivity Method.
Receiver functions are related to studies with interfaces and body waves and are used for inverse scattering problem. (See Chevrot and van der Hilst, EPSL 2000.)
Anisotropy So far we have assumed isotropy, i.e. wavespeeds do not depend on the direction of wave propagation. We solved the wave equation assuming plane waves: (^) � (^) � p = sin^ i^ sin^ j^ λ+2μ^ μ α =^ β where^ vp^ =^ α^ =^ ρ and^ vs^ =^ β^ =^ ρ.
Snell’s Law and Generalized Hook’s law held only for isotropic medium.
Anisotropy, however, cannot be ignored as it is the focus of increasing research in seismology and is inherent in our environment from stratifica tion, sedimentation, anisotropic minerals, etc.
Figure 7: In an isotropic medium, wavefronts are concentric circles with radius dependent on the velocity (vp or vs) as shown above. The raypaths are perpendicular to wavefronts and the slowness vector is perpendicular to the wavefront. The energy goes with group velocity and group velocity is also perpendicular to wavefronts.
Now we look at some basic theory to get insights into the problem and concepts. A full treatment is beyond the scope of this class.
We have used the Generalized Hookes Law:
τij = cijkl�kl τij = τji → cijkl = cjikl �kl = �lk → cijkl = cjilk
And from thermodynamics:
cijkl = cklij
These relationships reduce the number of independent elements from 81 to 21 elements.
In the isotropic case, Generalized Hooke’s Law simplifies to:
cijkl = λδij δkl + μ(δikδjl + δilδjk)
Where only the two independent parameters are the Lam constants, λ and μ.
c 1111 = c 2222 = c 3333 = λ + 2μ c 1122 = c 1133 = c 2233 = λ c 1212 = c 1313 = c 2323 = μ Else = 0
An anisotropic medium is a more complex system and uses symme tries such as: Orthorhombic- (e.g. olivine) with 9 elements Hexagonal- with 5 elements Cubic- (e.g. MgO) with 3 elements
Anisotropy is not due to individual minerals but the whole medium or lattice. Lattice preferred orientation (LPO) is the deformation of olivine by plate motion. Olivine minerals can align along their long axes from high stress deformation. We can use studies of anisotropy as a measure of strain, but we have problems distinguishing history dependent LPO from signals of current strain.
Anisotropic Case: Hexagonal Symmetry Hexagonal symmetry is very useful for seismology where there is rotational symmetry around a symmetry axis (not necessarily vertical but often is).
Figure 9: Anisotropic medium with hexagonal symmetry. This layered sys tem has a vertical symmetry axis which is most useful for Earth. In the transverse direction (ie. direction perpendicular to the symmetry axis) we have isotropy. This is also known as Vertical Transverse Isotropy (VTI).
The advantages of VTI are:
- q SH and q Sv can still be treated separately.
- The velocities vary only with incidence angle and not with azimuth.
If the symmetry axis is horizontal you have azimuthal anisotropy or Horizontal Transverse Isotropy (HTI). For mathematical convenience we introduce a new notation conven tion in order to collapse the 4th order tensor into a 2nd order tensor (ie. cijkl → CIJ ): Let 11 → 1; 22 → 2; 33 → 3; 23 → 4; 13 → 5; 12 →6.
For example, c 1122 = C 12.
