Lecture 16

Earth’s Free Oscillations

Earth’s Free Oscillations

Seismology and the Earth’s Deep Interior Surface Waves and Free Oscillations

Eigenmodes of a stringEigenmodes of a string

Geometry of a string underntension with fixed end points. Motions of the string excited

by any source comprise a weighted sum of the

eigenfunctions (which?).

Normal mode: all particles of the system move sinusoidally with the same frequency

Normal mode: stationary waves that are created from the constructive interference of body and surface waves circling the earth multiple times

Data - seismogramsMw ~9.2 2004 26 December, Sumatra

0 12108642

Time (s)

Time (h)

radial

6S4 30S12

http://icb.u-bourgogne.fr/nano/manapi/saviot/terre/index.en.html#

Park et al., 2005

120 h spectra

Free Oscillations - Spectra contain information about large scale

structure of Earth - Depend on broad averages of the Earth’s structural

parameters - Not affected by limitations of data coverage due to

uneven distribution of earthquakes and stations - Represents a quest to understand Earth’s intrinsic

vibrational spectrum -  Provides a method for calculation of theoretical

seismograms - Intrinsic low pass filters of Earth’s structure

into two peaks in the data spectrum, which is not seenin the theoretical spectrum. Not shown here, butequally important in studying both Earth structureand earthquakes, is the ‘phase spectrum’. Examplesillustrating this are shown in later sections.

Normal mode studies represent the quest to revealand understand the Earth’s intrinsic vibrational spec-trum. However, this is a difficult quest, because it isonly at the very longest periods (!500 s, say) thatthere is the possibility of obtaining data of sufficientduration to make it possible to achieve the necessaryspectral resolution. Essentially, the modes attenuatebefore the many cycles necessary to establish a stand-ing wave pattern have elapsed. Thus, in manyobservational studies, over a wide range of frequen-cies, the normal mode representation has the role,primarily, of providing a method for the calculationof theoretical seismograms. Although observed spec-tra contain spectral peaks, the peaks are broadenedby the effects of attenuation in a path-dependentway. Thus, rather than making direct measurementson observed spectra, the analysis needs to be based oncomparisons between data and synthetic spectra, inorder to derive models able to give improved agree-ment between data and synthetics.

The use of normal mode theory as a method ofsynthesis extends well beyond the realm normallythought of as normal mode studies. For example, ithas become commonplace to calculate global bodywave theoretical seismograms by mode summation ina spherical model, to frequencies higher than100 mHz (10 s period). Typically, such calculationscan be done in seconds on an ordinary workstation,the time, of course, depending strongly upon theupper limit in frequency and on the number of sam-ples in the time series. The advantage of the method

is that all seismic phases are automatically included,with realistic time and amplitude relationships.Although the technique is limited (probably for theforeseeable future) to spherically symmetric models,the comparison of such synthetics with data providesa valuable tool for understanding the nature andpotential of the observations and for making mea-surements such as differences in timing between dataand synthetics, for use in tomography. Thus, theperiod range of applications of the normal moderepresentation extends from several thousand sec-onds to "5 s. In between these ends of the spectrumis an enormous range of applications: studies ofmodes per se, surface wave studies, analysis of over-tones, and long period body waves, each havingrelevance to areas such as source parameter estima-tion and tomography. Figure 2 shows an example ofsynthetic and data traces, illustrating this.

There are a number of excellent sources of infor-mation on normal modes theory and applications.The comprehensive monograph by Dahlen andTromp (1998) provides in-depth coverage of thematerial and an extensive bibliography. An earliermonograph by Lapwood and Usami (1981) containsmuch interesting and useful information, treated froma fundamental point of view, as well as historicalmaterial about early theoretical work and early obser-vations. A review by Takeuchi and Saito (1972) is agood source for the ordinary differential equations forspherical Earth models and methods of solution.Other reviews are by Gilbert (1980), Dziewonskiand Woodhouse (1983), and Woodhouse (1996). Areview of normal mode observations can be found inChapter 1.03. Because of this extensive literature wetend in this chapter to expand on some topics thathave not found their way into earlier reviews but arenevertheless of fundamental interest and utility.

1.02.2 Hamilton’s Principle andthe Equations of Motion

To a good approximation, except in the vicinity of anearthquake or explosion, seismic displacements aregoverned by the equations of elasticity. At long per-iods self-gravitation also plays an important role.Here we show how the equations of motion arisefrom Hamilton’s principle.

Consider a material which is initially in equili-brium under self-gravitation. Each particle of thematerial is labeled by Cartesian coordinates xi

(i¼ 1, 2, 3), representing its initial position. The

1.0 1.1 1.2 1.3 1.4 1.5 1.6 1.7Frequency (mHz)

0S6

3S21S4

0S72S3

1S5

2S4

4S1

0S8

3S3

2S5

1S6

0S9

1S0

1S7

2S6

5S1

4S2

0S10

Figure 1 Data (solid line) and PREM synthetic spectrum(dashed line) computed using normal mode summation, forthe vertical component recording at station ANMO followingthe great Sumatra event of 26 December 2004.

32 Earth’s Free Oscillations

Woodhouse & Deuss, 2007

Spherical Harmonics

-  Angular solution to Laplace’s equation in spherical coordinates

- Wave propagation on the surface of a sphere -  Can also be thought of as Fourier transformation

on a sphere -  is a spherical harmonic function of degree l and

order m - is associated Legendre polynomial

Nu =Qd

kDT

⇠ 10

Pr =nk⇠ •

—2

f = 0

Y

m

l

(q,f) = Ne

imfP

m

l

(cosq)

5

Nu =Qd

kDT

⇠ 10

Pr =nk⇠ •

—2

f = 0

Y

m

l

(q,f) = Ne

imfP

m

l

(cosq)

5

Nu =Qd

kDT

⇠ 10

Pr =nk⇠ •

—2f = 0

5

Nu =Qd

kDT

⇠ 10

Pr =nk⇠ •

—2

f = 0

Y

m

l

(q,f) = Ne

imfP

m

l

(cosq)

5

Associated Legendre Polynomials

Nu =Qd

kDT

⇠ 10

Pr =nk⇠ •

—2

f = 0

Y

m

l

(q,f) = Ne

imfP

m

l

(cosq)

P

m

l

(x) =(�1)m

2

l

l!

(1� x

2)m/2

d

l+m

dx

l+m

(x2 �1)l

5

https://mysite.du.edu/~jcalvert/math/harmonic/harmonic.htm

m=0 m=/l m=l

48 CHAPTER 2. THE EARTH’S GRAVITATIONAL FIELD

Figure 2.14: Some spherical harmonics.

1. with increasing distance the lower order harmonics become increasingly dominant since thesignal from small-scale structure (large ’s and ’s) decays more rapidly. Recall that theperturbations of satellite orbits constrain the lower orders very accurately. The fine detailsat depth are difficult to discern at the surface of the Earth (or further out in space) owing tothis spatial attenuation.

2. Conversely, this complicates the downward continuation of from the Earth’s sur-face to a smaller radius, since this process introduces higher degree components in the so-lutions that are not constrained by data at the surface. This problem is important in thedownward continuation of the magnetic field to the core-mantle-boundary.

l = 16 m=0 l = 16 m=8

l = 16 m=16 http://geodynamics.usc.edu/~becker/teaching-sh.html

degree=6

−100−80−60−40−20020406080100

degree=50

−120−100−80−60−40−20020406080100120

l = 6 l = 50

Observed geoid

degree=6

−100−80−60−40−20020406080100

degree=50

−120−100−80−60−40−20020406080100120

l = 6

l = 50

Observed geoid

degree=7−31

−30

−20

−10

0

10

20

30l > 6

0"

500"

1000"

1500"

2000"

2500"

3000"

1" 3" 5" 7" 9" 11" 13" 15" 17" 19" 21" 23" 25" 27" 29" 31"

Series2"

Series1"

Degree

Pow

er

Observed geoid

Hager & Richards, 1989

316 B. H. HAGER AND M. A. RICHARDS

mantle and the molten core; globally averaged seismic models (Dziewonski & Anderson I981) show a region of decreased velocity gradient in D". D" is almost certainly a region of increased temperature gradient, i.e. a hot thermal boundary layer, and hence is expected to have a low viscosity. More-detailed regional models (Lay & Helmberger I983a, b; Young & Lay 1987) show a sharp boundary, so D" may well be distinct in composition as well. Second, there is now great interest in constraining the properties of the CMB. The predicted dynamic topography at the CMB is a strong function of the structure of D".

Kernels for several models that allow us to explain most of the variance in the long- wavelength geoid are shown in figures 2-4. For these models, the first letter refers to the style of convection: 'W' for whole mantle and 'C' for a mantle chemically stratified at 670 km depth. The second letter refers to D": 'L' for low viscosity and 'C' for chemically distinct (also with low viscosity). Only relative viscosity variations affect the geoid. The distribution of relative viscosity y with depth is shown in the figures. For example, for our preferred model WL, the distribution is as follows: 0-100 km depth, ) = 1; 100-400 km, 1 = j; 400-670 km,

I = 1; 670-2600 km depth, /= 10; 2600 km depth- CMB, y = i.1

1000 / 4 / iSOOr , ,/

0 1 ? 1

1500 / lo

2000 '

2500-f - , 1 '

"r off f FIGURE 2. Normalized kernels for model WL, showing the effects of a surface mass cr placed at a given depth in the

mantle for I = 2, 4, and 8. The masses displaced by surface deformation and by deformation of the CMB are normalized by dividing by (-oro). A value of unity implies perfect dynamic compensation. The geoid kernel, which includes the effects of the dynamic topography as well as cr, is normalized by the potential due to a mass 0r at the surface of the Earth. The solid line is for degree two, short dashes for degree four, and long dashes for degree eight.

0 1

500- ' / 1 - -- a"/ I" 0 ID G

E 1000 A/ 'O I , ,

2500

^ 1500 T~' " -

' \

- 0 0.5 1.0 0 0.5 1.0-0.5 0 0.5 o- cr o'

FIGURE 3. As in figure 2, but for model WC, which differs from model WL by having a chemically distinct D" layer with a low-viscosity layer above it in the depth range 2300-2600 km to simulate the effects of ann dditional hot thermal boundary layer.

15hatv-icst lyraoei nte et ag 30200 km , " 'iuaeteefcso nadtoa

[ 26 ]

316 B. H. HAGER AND M. A. RICHARDS

mantle and the molten core; globally averaged seismic models (Dziewonski & Anderson I981) show a region of decreased velocity gradient in D". D" is almost certainly a region of increased temperature gradient, i.e. a hot thermal boundary layer, and hence is expected to have a low viscosity. More-detailed regional models (Lay & Helmberger I983a, b; Young & Lay 1987) show a sharp boundary, so D" may well be distinct in composition as well. Second, there is now great interest in constraining the properties of the CMB. The predicted dynamic topography at the CMB is a strong function of the structure of D".

Kernels for several models that allow us to explain most of the variance in the long- wavelength geoid are shown in figures 2-4. For these models, the first letter refers to the style of convection: 'W' for whole mantle and 'C' for a mantle chemically stratified at 670 km depth. The second letter refers to D": 'L' for low viscosity and 'C' for chemically distinct (also with low viscosity). Only relative viscosity variations affect the geoid. The distribution of relative viscosity y with depth is shown in the figures. For example, for our preferred model WL, the distribution is as follows: 0-100 km depth, ) = 1; 100-400 km, 1 = j; 400-670 km,

I = 1; 670-2600 km depth, /= 10; 2600 km depth- CMB, y = i.1

1000 / 4 / iSOOr , ,/

0 1 ? 1

1500 / lo

2000 '

2500-f - , 1 '

"r off f FIGURE 2. Normalized kernels for model WL, showing the effects of a surface mass cr placed at a given depth in the

mantle for I = 2, 4, and 8. The masses displaced by surface deformation and by deformation of the CMB are normalized by dividing by (-oro). A value of unity implies perfect dynamic compensation. The geoid kernel, which includes the effects of the dynamic topography as well as cr, is normalized by the potential due to a mass 0r at the surface of the Earth. The solid line is for degree two, short dashes for degree four, and long dashes for degree eight.

0 1

500- ' / 1 - -- a"/ I" 0 ID G

E 1000 A/ 'O I , ,

2500

^ 1500 T~' " -

' \

- 0 0.5 1.0 0 0.5 1.0-0.5 0 0.5 o- cr o'

FIGURE 3. As in figure 2, but for model WC, which differs from model WL by having a chemically distinct D" layer with a low-viscosity layer above it in the depth range 2300-2600 km to simulate the effects of ann dditional hot thermal boundary layer.

15hatv-icst lyraoei nte et ag 30200 km , " 'iuaeteefcso nadtoa

[ 26 ]