PART II

-137- 1. Introduction

The ane1asticity of the Earth's interior gives rise to a variety of important phenomena in solid earth geophysics. Examples are: the short period effects such as seismic wave attenuation and tidal dissipation; the long term phenomena such as response of Earth's crust and mantle to loading and unloading on the surface ("isostasy") or the transverse movements of large segments of Earth's upper layer.

("plate tectonics"). The mechanisms responsible for the anelastic behavior of the Earth's interior are largely controlled by defect structure of the crystals making up the rock minerals. A host of such mechanisms, thought applicable to geophysics, are reviewed and listed

in the literature (Gordon and Nelson 1966, Jackson and Anderson 1970).

Further experimentation under high pressure and temperature pertinent to mantle conditions are necessary to decide their relative importance.

Recently, the attenuation of seismic body and surface waves at different frequencies have been measured. Jackson and Anderson (1970) summarize this information up to 1970. This information from seismol- ogy, when carefully interpreted, can serve to put a constraint on the physical mechanisms of ane1asticity operating in the Earth's interior.

It will be shown in the present work that ane1asticity, as in- ferred from the seismic wave attenuation data, also produces a non- negligible shift on the periods of Earth's toroidal oscillations. This result implies that in constructing velocity and density models of the Earth using free oscil1at1on data, one has to invert thelll together w1th ane1astic structure when the free oscillation period measurements have an accuracy limit less than the shift due to anelasticity. Computations

in this work show this limit to be I~T/TI ~ 2.3 x 10-4 for the toroidal oscillations. Dziewonski and Gilbert (1972) recently determined the

free oscillation periods excited by the 1964 Alaskan earthquake. The resolution of the fundamental toroidal modes oTt reported by them varies between 0.031% and 0.262% and is generally about 0.1% for the higher frequency toroidal modes oTt at which the ane1astic shift becomes important. The magnitude of present observational error is about five times bigger than the shift of free oscillation period due to ane1asticity.

-139-

2. Derivation of Equations for Shift in Earth's Toroidal Free Oscillations Due to Ane1asticity

The equation of motion for the elastic-gravitational oscilla- tion of a nonrotating earth with lateral heterogeneity is given by

(II.2-l) and

(II.2-2)

where Po: equilibrium density distribution

~~i: displacement field of free oscillation characterized by the mode numbers k, i , and m

w~i: angular oscillation frequency

Pl

=

-V·{Po~kt)' change in density due to displacement

$ = $0+ $1 ' total gravitational potential

$0 equilibrium gravitational potential

$1 change in gravitational potential due to free oscilla- tion

G : gravitational constant.

The operator H is defined by its components in Cartesian coordinates (Dahlen 1968) by

[H {~,$)Ji = -dj (r ijkt 0kt) + di[poSjdj$oJ - V' (po~) di$o

+ Podi$l -

di[SkdjT~jJ

⁺

d j(SkdkT~j]

^{+ dj[l}

T~ l {diS t- di

^Si)]

(II.2-3)

The mode numbers are suppressed in the above equation. In equation

(I 1. 2-3) where

TO: static stress field tensor

:::

~o

^_

lo - ~ro)ii ~

, is the static stress deviator tensor E elastic stress tensor

..

and the stress-strain relationship (in Cartesian coordinates)

has been used in the equation of motion to obtain the expression for operator H (?,cjl) in equation (11.2-3). r;jkR. in equation (11.2-4) are components of the elastic coefficient tensor. The boundary condi- tions are (Alterman, Jarosch and Pekeris 1959; Backus 1967)

S continuous, except at mantle-core boundary where only ~.? needs to be continuous

cjll continuous

~ • Vcjll + ? • ~ 47TG Po conti nuous

~

.

n • E cont, nuous

(11.2-5)

where

n

^is the outward normal to a undeformed surface of continuity.

Since equation (11.2-2) can be integrated, to express cjll (~) in terms of the displacement field S

-141-

4>1

(~)

^{= -G}

J

^{'V • (po?) g(!.'.:,) dV'}

+ G

J

^{Po (!:') 9}

(~.!.")

^{S (:' ) •}

~'

^{dS' (I I. 2-6)}

The operator H is hermitian when the density Po is laterally homogeneous. It is not hermitian when the density possesses a lateral heterogeneity. This is discussed by Saito (1971) and the adjoint opera- tor H of the operator H has been constructed by him. The eigenvalue equation of the adjoint operator is

(11.2-7)

with

J

^{Po~kR, -m • ~k' m' R,' dV = 0 if k 1 k'. R, 1 R,' or m 1 m'}

~~t is the adjoint solution of ~~R, If H is hermitian. H = H •

m _ m -m ^~

YkR, - wkR, • and ?kR, =?k • where denotes complex conjugation.

The straightforward way of introducing anelasticity into the equation of motion is to assume a constitutive relation which contains dissipation factors. The simplest such constitutive relation results from introducing complex moduli of elasticity into the linear stress- strain relation as expressed by equation (II.2-4). Anderson and

Archambeau (1964). As an illustration. consider the stress-strain rela- tion of a visco-elastic solid.

(IL2-8)

where A,A', ~,~' are real quantities. In the frequency domain (or in response to sinusoidal disturbance) the stress-strain relation (11.2-8) takes the fonn

(I 1. 2-9)

which shows that a pure imaginary part in the elastic moduli would ac- count for this specific ane1astic behavior. Further; a wider class of anelastic behavior can be represented in this manner. In an anisotropic medium, the more general equation corresponding to equation (11.2-9) ;s

(I1.2-10)

where rijk~ * is a tensor with pure imaginary components and having the

same fonn as the elastic tensor rijk~. In an isotropic material, the operator A takes the form

A = -(A*+2~*) V{V· ) + ~*{v x (V x » - (n*) {VJ

-(V~*) • (V + V) (I1.2-11)

-143- _ as x

ell - :Ix '

1 as as

e12 ⁼ 2(a/' +

a!-),

etc. in the dyadic notation.

With this anelastic operator A ,the eigenvalue equation describing the Earth's free oscillations becomes

(11.2-12) Restricting the treatment to an isotropic stress-strain relation, con- sider the anelastic behavior as a perturbation. The perturbation expan- sion for angular frequency and displacement field are

(I1.2-13) Substituting equation (II.2-13) into equation (II.2-12), the zeroth

order equation is

(11.2-14) i.e., the e1asto-gravitationa1 oscillation of a non-rotating Earth. The first order equation is

(11.2-15)

(l)m

~ kt can be expanded in terms of the zeroth order eigenfunctions as

+

L

eel) S(o)m

tift tt' kt' ^(II. 2-16) Now take the dot product of equation (11.2-15) with

~(o):t'

the adjoint solution of

~(o):t'

and integrate over the volume. By definition of the adjoint solution,

{I1.2-17}

and equation (11.2-7). the first order equation reduces to

w(l)m ⁼

J

S(o)m • A s(o)m dV /(2w(O)m

J

^p s(o)m .s(o)m dV) .

kt _ kt _ kt kt 0 _ kt _ kt

I (11.2-18)

_(o)m

Take dot product of equation (11.2-15) and ~ kt' m'fm , and since

(I1.2-19)

equation (11.2-15) becomes

m' m m m'

[(y^(0)k_n)2 - ((0) )2

J J

S(l) • S(o) dV

N W kt Po- kt _ kt

(11.2-20)

-145-

Substituting equation (11.2-16) into equation (Il.2-20) gives a

=

J S(o) • m'

mm' - kR,

m' m'

x J P S(o) • S(o) dV}

0_ kR, - kt ^{(I I.} 2-21) Similarly, expressions for

products of equations with

bkk , and CR,R,' are obtained by taking dot s(o)m, and s(o)m, respectively and

- k R, - kR,

integrating the resulting equation over the entire volume of the Earth.

The second order perturbation equation is then

N t ow a e k t h e ot pro uc d d t f 0 equa lon t · (11.2-22) w,'th -?(o)mko _ ^N and i nte- grate over the entire volume, with

(I I. 2-23)

equation (11.2-22) becomes

=

[(w(l)m )2 + 2w(0)m w(2)m J

f

p S(O)m _S(O)m dV

k1 k1 k1 0- k1 - k1

(I 1. 2-24) or

(I1.2-25)

This general result is now applied to the toroidal oscillations of the Earth. The displacement field of an elastically ^i~tropic and laterally homogeneous Earth model is taken as the zeroth unperturbed eigenfunc- tion. The operator H associated with such an Earth model is hermi- tian. Therefore, H = H -m."m m m

, ~k1 = ~k1 ' and y k1 ⁼ w_{kt ·} The zeroth order displacement field is then (Alterman, Jarosch, and Pekeris 1959)

(11.2-26)

-147-

where em (9,$) is the vector spherical harmonics defined by

-i

"-

(11.2-27)

~r' ~9' ~$ are unit vectors in spherical polar coordinates, and

monic function. Wkt(r) is solution to the equation

for a layered Earth model where ~~o) and p~o) are rigidity and density inside the layer of index s , s=1,2,···N. The boundary con- ditions for equation (1~.2-28) are continuity of Wkt(r) and

(~(o) ~r

Wki (r) - Wki(r)) at layer boundaries. The adjoint displace- ment field is given by

(I 1. 2-29)

Assuming a laterally homogeneous ane1asticity in the Earth model,

(0) 2

J ^L

^{(0) ( ) ( )}

(w ki ) .,(0) p Wki r Wki r dV

).i

(11.2-30) and

Substituting equation (11.2-30) into equations (11.2-18), (11.2-21)

and (11.2-25) we get

and

(l)m _

w kR.- ^{

J

^{~* ~} _II ^{p (0) [WkR.(r )J2} dV W (0) kR.}

(2)m _ () 1

W kO - (w ok") ,,-

y

'< N C. irk

( ^W (0))2 ^{iR. 1 (l)m 2}

i

( (0) )2_ ( (0))2 + 2( (0))2 (w kR.)

W iR. W kt W kR.

(2) (1) 2 _

(w~~)) {~+ ~(\kR.)

^}

(I!. 2-31)

(11.2-32)

The attenuation factor (Q~R.) of the toroidal oscillation is given by

(2 )m and the frequency shift is given by ^W kR..

(I!. 2-33)

-149-

3. Numerical Calculations and Results

In actual computation, a laterally homogeneous Earth model con- sisting of seismic velocities Vp' Vs ' and density ^p{o) as a function of radius is fed into a normal mode program to generate the displace- ment function Wkt{r) and angular frequency

w{~l

for kTt' k=O,l;

t=2,3,··· ,99. (m is degenerate for a laterally homogeneous, non- rotating, spherical Earth model). These, together with an intrinsic Q-model as a function of radius, are used to compute the free oscilla- tion dissipation factor {Qk=O,t)-l and the frequency shift due to anelasticity

w{~~O,t

Equations (11.2-31), (11.2-32), and (11.2-33) are employed in these computations. Note from equation (11.2-32)

{2} _ co

y k=O _,., " - ₁₌₁

.1:

(11.3-1) { (O})2 _ {(o) )2

w it w k=O,t

Y(~~O,t

is a summation of positive terms since

to) to)

wit> wk=O,t ^i-l ,2"" {11.3-2}

The actually computed value

( (0) )2

wk=l,i

( w ( ^{0 ) )} 2 _ (w ( 0 ) ) 2 k=l,i k=O,i

(11.3-3)

is the first term of the series (11.3-1) and is the lower limit of the summation. However, as the number of zeros in Wki(r) increases with k , the first term of the series is also the largest.

Two laterally homogeneous Earth models are used in the computa- tion, one characteristic of the Basin and Range mantle province

(CITlll) after Archambeau et al (1969) and the other characteristic of an oceanic structure (Oceanic Series 304702). These models are de- scribed in Tables 11.3-1, 11.3-2, and Figure 11.3-1.

Five intrinsic Q models are employed for the Basin and Range Earth model. Two of them are frequency independent with their low Q zone corresponding to the low velocity channel of the Earth. The only difference between these two Q models, Q(Hl ) and Q(Ll), is in their numerical values inside the low Q zone. This type of frequency-indepen- dent Q-model was adopted in the early work to construct intrinsic

Q-models in the Earth's interior (Anderson and Archambeau 1964). The next two intrinsic Q models, Q(w)l and Q(w)2 are based on a single relaxation mechanism for seismic energy dissipation. They are fre- quency dependent and, except for a scaling factor, are after Jackson's model 10-04 (Jackson 1969). The last intrinsic Q-model for Basin and

-151-

Range Earth model Q(w)3' is constructed after Solomon (1972). In this model, the Q is independent of frequency above the low velocity zone and is the result of two relaxation mechanisms (one of which is attributed to partial melting) in the low velocity zone. It is fre- quency dependent through only one relaxation mechanism below the low velocity zone and above the core-mantle boundary. Only frequency independent Q models, Q(L 2) and Q(H2), are employed for the oceanic Earth model. The Basin and Range Q-models are listed in Table 11.3-3.

The oceanic Q-models are listed in Table 11.3-4. The frequency inde- pendent Q-models are also shown in Figure 11.3-2.

The computed free oscillation quality factor Qk=O,t for various Earth models and for various intrinsic Q-mode1s is plotted in Figures 11.3-3 and 11.3-4. The attenuation data for free oscillations and surface waves up to 1970 have been collected by Jackson and

Anderson (1970). The toroidal oscillation and Love wave attenuation data reproduced in Figures 11.3-3 and 11.3-4 with the various computed Qk=O,t are from this collection except for the twelve points at the lower right corner, which are from Solomon (1972). The attenuation data included in Figures 11.3-3 and 11.3-4 are Nowroozi (1968; oTt)' Alsop et al (1961; 0\)' MacDonald and Ness (1961; oTt)' Smith (1961;

0\)' Solomon (1972; Love wave), Savarenskii et al (1966; Love wave), Ba th and Lopez-Arroyo (1962; Love wave), Press et a 1 (1961; Love wave),

~

Sato (1958; Love wave), Wilson (1940; Love wave), Gutenberg (1924; Love wave). It appears that the intrinsic Q-models Q(H l ), Q(H2), and

Q(w)l give seismic attenuation which agrees well with the observed

data. However, noting the wide scatter of seismic attenuation data and some recent results of surface wave attenuation in the western United States (Solomon 1972), the low Qk=O,~ values produced by the intrinsic Q models, Q{L l ), Q{L2), and Q{w)2' cannot be ruled out as unreasonable. Values of Qk=O.~ calculated from Q{w)3' the most sophisticated among the Q~odels, matches well both the long period free oscillation attenuation data and short period Love wave attenua- tion data in the western United States. Shifts in free oscillation periods calculated from this intrinsic Q-mode1, Q{w)3' are taken to be representative. The period shift

(1I.3-4)

and the percentage period shift. li=l {OT)k=O,~/\=o,~1 , as calculated from various Earth and intrinsic Q-models are plotted in Figures 11.3-5 and 11.3-6. The maximum values of 10T/TI and the modes at which they occur are listed in Table 11.3-5 for various Earth and intrinsic Q models. When appropriate (~ » 1), the shift in free oscillation

period is also expressed in shift in phase and group velocities of surface waves according to Press (1964). The phase and group veloci- ties of the two Earth models are plotted in Figure 11.3-7. The shift in phase and group velocities for various Earth and intrinsic Q-mode1s are plotted in Figures 11.3-8 and 11.3-9.

-153-

TABLE II-3-l

crT ^{III Basin,}and Range Earth Elastic Model

+UJ_~~_= _____ ~~_s_!.~ _ ~_~J: __ ~A~_G.~ ________________________________________________________________ _ HH , __________ U J _______________ "-q,-"-L~ , _____ J g ______________________________________________________ _

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TABLE II-3-2

Oceanic Earth Elastic Model

MODEL INPUT IN ST'NCARO FD~~.

__ !L!1J_: ____ _ 9_C_g _A_ t:I)_~ __ ~~_"J_~~ _____________________________________________________________________________________ _ _ _ ~!!g ~_ ~ _____ ~9-"-! _"-? _______________ ~9_~~_~~_=_ ______ ~9 _______________________________________________ _

I NO~X ~ ADI US DENSITY VP vs GRAVITY

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4 1.07849E 03 1.24202E 01 1.lb375E 01 0.0 3.80575E 02 5 1.07P49E 03 1.24202E 01 1.10900F. 01 0.0 3.S0575E 02

• 1.1H78~E 03 1.23740E 01 1.09301JE 01 0.0 4.15572E 02 --- -- --T --- ---T: -foY i i;T --,)3---T:2 ~'i5-iE--of-----T.-0 -[ iuM -lfC -- -- -0-.-0--- --- ----- -';:-55-08 W - -0 Z--

8 1 •• Jl01E 03 1.2141"E 01 Q.95Q04E 00 0.0 5.54b2bE 02 --- ---q --- --I :'f2:fozT·-6'-----T.T<jYI'iF--or---q-.-,;;iI~-4F.--,f6------6 :-6 -------. --------. ;-S74f3-~ --02- -

10 2.24102F 0) 1.lh3~lE 01 9.4877~E 00 0.0 7.~606qE 02 11 2.5.202E 01 1. lJOnlE 01 '.161J~~f 00 0.0 8.4981J4E 07 12 2.A8202E OJ l.oul~3E 01 8.7b934E 00 0.0 9.376b2E 02 - --- --I 3--- ---~; -i[fi (fie --^6~----T; If;;"r3Tr--6i---iI-.-~-'~3n--06 - -- ---6-.-0--- --- ---- ---[ ~ -6 t -6 ?o~ --6,,--

14 3.47900E 03 1.01J500E 01 8.04000E 00 0.0 1.08154E 03

·---ys--------f:41',-olfC-6j---S-:5flf61ie--oo-----T.-~-bTf6F--IH---t-.-~ii606F--:fo---f~-6ifi54E--0'C

16 3.6564)F 0) 5.41659E 00 1.3b182E 01 7.Z6~56F OU 1.0573AE 03 17 ).~9643E_03 S.29⁷29E 00 1.13199E 01 7.154A4~ 00 1.03738F 03 18 4.1~643[ U3 5.17729E 00 1.30~S4E 01 7.03751E 00 1.01519E 03

---T<j---4:Yi;,-,;~Toj---i;-.o-<5i.qbE--00------i:Tfie-snn-------b-.-<jZ-.,ne--oo------r:-66-44"-"--0'c

20 4.61643E 03 4.9Z649F. 00 1.25Z11f 01 6.A0733E 00 9.97982E 02 ---2-i -----;;:-Ss(;"';Y Co j ------;;:e-54-34E--66---r :2T3TiC 1ff --- -- -~ -.-~ 84 4-" ^~--0-6 ----<j:-<i,,-Ii 'IT E --02--

22 S.096~3E 03 4.67U6ZE 00 1.lA912E 01 b.5~~80F 00 9.9~2S1E Ol 23 S.33641E 03 4.~3CJ'i8f UO 1.1~Z7qE Ot h.3~712F 00 9.9S34H 02 24 5.S40S0F 0) 4.4000~E 00 1.12U'4F 01 6.17A'7f au q.q713~~ 02

-ir ---- ^{~;"r724l U~ '-.2o-i21f 00 1.10i~5F- cit ~.9725bF 00 9:911-0UOE Ol}

Z6 5.7210~[ O~ 4.20~39E 00 1.U9000E 01 5.80711l 00 9.QQ340E 02 -----Tf -----S-:7nifoE--OY - ---.;. i6-<53'fC-oo-------i. U2boiJr oi ----- -S-;if6Hi Co-o ------o-.-ijlf34-oF 1ft 28 5.76100E 03 4.14.0~E 00 9.99000E 00 5.69142E OU 9.9A427E 02 29 5.~0100t 03 4.078^QhE 00 9.88667E 00 5.56000E 00 9.98398E 02

3~ 5.84100E 0) 4.00374F 00 9.79667E 00 5.41502E 00 9.9822~E 02 ------YC-- ----5-: i;-tmiii"E -oj --- -'f.-qB 'i Sf --('-0- ---q -.-noo ^o~--0-6 ----5:-26-f if4E --0 (f --- ---9;-q t 8 7 i.-C6;(-

32 5.011UOE 03 3.838AOE 00 9.65S00E 00 5.146A1F 00 9.97471E O~

------jl-----5:-<i4Tii(fE--oj---"3~-.,~lf50r-6ii---i:s-,;ifooE-1fo---S-.-0-3-HjE--if6------q:-Qi,-92bE--02--

34 5.956UOE 03 3.71qh6E 00 9.46000E 00 4.97582E 00 9.9659AE 02 35 5.97bOOE 03 3.b7U34E 00 9.46000E 00 4.9U171E 00 9.96110E 02 36 S.~7600E 03 '.67014E 00 9.06700E 00 4.90171: 00 9.9hll0E 02 ---37---- ----5:-q~ /:OOE--oj-------j: 6-;;;-<i\fE---60---If: ifo o-.:Hi E -0-0- -----" -.-8b5 ;;i~--0-6 --- -q: -9~-84i!£ --0 Z--

38 5.~9600E 03 3.h34l7E 00 Q.78JOOF 00 4.82955E 00 9.9557~E 02

--- H ---^-6:-;;06-0;'-1' --6 j - -- -- -)-: ,,-f t-6 6E--06--- ---~-.-n OUOF--OO ---- --

,.- . - ?if"

rw -^{00---- -}-q: -qH ooe--62-- 40 6.031UuE ~3 3.5n003< UO A.63500E 00 4.7UA07~ 00 9.94575E 02 41 b.07100E 03 3.53000E 00 8.54q21~ 00 4.5Q)00E 00 9.9l351E 02 42 h.C840uE 03 3.~1615E 00 d.52000E 00 4.54172F 00 9.92941E 02

---,,-j ---⁶ :Y6 YJo-E' --6 j- -----'f.-,,-o(;t; Re--(f 6---if.-46 3-60 e--:fo--- ---,.

- . - ,.if,.

^{b iiE --('-6 ----9 :-q} B iflie--6 Z-- 44 6.12100E 03 3.4825AF 00 8.42000E IJO 4.44453F 00 9.91757E 02 -- -- --';S--------6: Ti,T1fJF--OY ------i; -46'i HE --00----- --li:)-b foife -00--- -.. -: 4-0 Ye ~ F--0-6 -- - --q -.-qn i W--62--

46 6.17100E 03 3.4S000E 00 8.3Z010E 00 4.35000E UO 0.901561 02 47 6.11lUOE U3 3.450UOE 00 7.7000UF 00 4.35000E 00 9.90156E 02 48 6.22100E 03 3.434'.2E 00 1.7uOOOE 00 4.31394E 00 9.~86131 02 ---,;ij---,;-;B YcifCo"j---"3:"-jffi9E--OO---r:To60ol:--0-6------.;:-3)-iiHe--oo------Q:-stTi6T-oz--

50 6.27600E 03 3.42994F 00 7.7UOOUE 00 4.37016E 00 9.87041E 02

----'-i -----6: iii b OO-E --01----- -'f:"-Z',-<j4 C oo --- ---'(:7000 ii E--0-6------4-.-3<iTifH~--ij -6 ----q :-lii. -ii-';e --6 2-- 52 6.'1450E 01 3.43U26E 00 7.70UOOE UO 4.48~52F. 00 9.8hO~3F 02

---n -----"-.32 fMF--6j - -3: 4-jUZ8i'--00--- f.T6iiYor-c,-o-^-----i;-.-5Z~f,;F.Oii 9~-1i5-lii;;E --02 54 b.323uUE U' J.~3U28l 00 R.20100F. UO ~.525i4E UO 9.8583~E 02 55 h.)~30~F~' 3.427~~ OU 8.2U01UE 00 4.7561RE 00 9.~4903E 02 56 h.~6400F UJ 3.~27~5E 00 8.lUJ10E O~ 4.76330F. 00 9.~4880E 02 ---5T------6:3-bi,-oii-Co~------3-: ,,-it-"Sf--OO---T.TfooOn,O---- --';:-7;'-13010 --06 -----ij • Jfi;,flic'-"-02- ______ ~_~ _______ !>_,.?"~!>_~~_E __ s>.~ ______ }_._~_2.!-.s_~~ __ !1_q ______ ~~_'!~_~9_q~ __ ~s>. ______ '!!}.?}_!~~ __ ~!l _______ '_!.a_'!!1_~~_~ __ q? __

59 6.36600E 03 1.03JOUE OU 1.S1000E 00 0.0 9.84836E 02 60 6.17100E 03 1.0l000E 00 1.~1000E 00 0.0 9.84836E 02

Depth,z (km) 0-28 28-45 45-60 60-80 80-140

140-146 146-170 170-180 180-200 200-250 250-300 300-350 350-375 375-398 398-400 400-450 450-500 500-550 550-600 600-630 630-645 645-660 660-680 680-700 700-720 720-740 740-760 760-780 780-800 800-849 840-860 860-880 880-900 900-940 940-980 980-1054 1054-2892 2892-6371

-155-

TABLE U-3-3

Basin & Range Intrinsic Q Models

200 120 35 20 10 15 315 615 655 755 860 965 1078 1190 1443 1645 1695 1745 1795 1835 1858 1973 2090 2110 2130 2150 2170 2190 2210 2240 2270 2290 2310 2340 2370 2390 2400

o

200 130 55 43 36 38 325 615 655 755 860 965 1078 1190 1443 1645 1695 1745 1795 1835 1858 1973 2090 2110 2130 2150 2170 2190 2210 2240 2270 2290 2310 2340 2370 2390 2400

o

Q (w) 2 _{Q (w)} 3

-1 -1 1000

Q (w)=Q (W;A,TO,E*,V*) Q-1=0 1~ /(1~ .. 2 2)+

=AuYT / (1+w2T2) . L(l)1'12 t ^oUJ T 1 T=Toexp[(E*+PV*)/RT] WT2/(1~ T2~

T-0.4 sec -4 t

E*-lO kcal/mole T2-4xlO Tl V*=4 cm3 /mole

Pressure Mode1:Bu11en -1 2 2 Model B (Bullen 1965) Q =0.1(l)1'3/(1+w T3) Temperature Model: T3=2 sec

MacDonald Model 19 (MacDonald 1959)

A=0.032 A=0.060

tT1 =20ex pj-(500E*+584V*)/1239+(500E*+4zV*)/T]sec; E*=57kca1/mole,

V~=1.0cm /mole; Temperature Model 200029 after Minster&Archambeau (1971)

TABLE II-3-4

Oceanic Frequency-independent Intrinsic Q Models

Depth (km) 0-5 5-48 48-200 200-395 395-446 446-825 825-1155 1155-2892 2892 -63 71

o

200 10 80 100 300 800 1600

o

o

200 35 80 100 300 800 1600

o

-157-

TABLE II-3-5

Maximum Percentage Shift of Toroidal Free Oscillation Period Calcu- lated from Various Earth Elastic and Intrinsic Q Models

Earth Earth Free Oscil- Calculated Maxlmum Value of

Elastic Intrinsic 1ation Mode Free Oscil- Percent~fe Shift

Model Q Model at Which 1ation xlO

Maximum Oc- Period

curs (sec)

Oceanic Q(L

2 ) OT92 92.89 -13.104

Oceanic Q(H

2) OT99 86.63 -1.0936

Basin ^& Q(L1) OT62 140.43 -5.4140 Range

Basin & Q(H

1) OT68 129.29 -0.7396

Range

Basin & Q (m\ OT30 261. 68 -0.3815 Range

Basin & Q (m) 2 OT30 261.68 -1.3412 Range

Basin & Q (m) 3 OT60 144.59 -2.3022 Range

"'E

~ E

0>

>-.15 r-

u Q) (/)

...

~ E

~ r- u 0 -.J W 0

>

6000 5000

- - OCEANIC MODEL

--- BASIN AND RANGE MODEL

v

p

4000 3000 2000 1000 RADIUS, km

Figure 11-3-1. Laterally homogeneous Earth elastic models used in the perturbation calculations.

10 -I '"""! IT 1"

I . 5^X10-^{2 :}

I

I .

" a

I

!

L..,

~ i i i i..._\

\

\

'-.. ,

10-^{4 I}

6000

-159-

T I I I

-

- MODEL Q (L₁)

--- MODEL Q (H ) -

1

MODEL Q (L_{2 )} ... MODEL Q (H

2) -

\ ^-

\._._._._._._._._.

-

I I I I

5000 4000 3000 2000 1000 RADIUS, km

Figure II-3-2. The frequency-independent Earth intrinsic Q models used in the perturbation calculations. Numerical values of these Q models are listed in Tables II-3-3 and II-3-4.

a

400

300

200

100

•

\

\

\

\

\

\

•

\

\

\

INTRINSIC EARTH MODEL Q ^MODEL

BASIN AND RANGE Q (L,) --- BASIN AND RANGE Q (H,) _ ._ . OCEANIC Q (L₂)

... OCEANIC Q (H

2)

.. _ .. - BASIN AND RANGE Q (w),

.--._- BASIN AND RANGE Q {W)2

• TOROINAL OSCILLATION DATA

• LOVE WAVE DATA •

\ \

.

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,

\

, ,

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.

\

, ..

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,

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•

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\\.

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..

_• ^\ _{'.. • .1}

^..

_•••

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^/

^.

... .. _ .. X , ' .' i"

, ^~

... '7.-

•

• •

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^/ •

'" ^{'" /}

_{.. -} .

.--.~:..::,..... _•••

.

- . _--'-'-

O~

__

^{L-~ _ _ _ _ ~~ _ _} L - = -_ _ _ _ ^~=-~ __ ^~ ____ ^~ 10⁴ 5xl0³

Id

^{5xl02 5x10' 10'}

PERIOD, sec

Figure 11-3-3. Free oscillation quality factor (fundamental toroidal modes) calculated from various Earth elastic and in-

trinsic Q models as compared wi"th the observational data. The observational data are from Gutenberg(l924), Wilson(1940), Sat8(l958), Alsop et al. (1961), Press et ul.(IQ6') , MAcDonald and Ness(1961) , Smith(1961), Blth and Lopez-Arroyo( 19h2), Nowroozi( 19h8), and Solomon (1972),

-161-

400

• •

•

300

•

---

1--

• •

^•

.>£

•

--- _a

^•

200

•

• _{• •} •

• • •

100 ^{••• •}

• •

• ^•

• •

_ .

••• •

10³ 5xl0² 10² 5x10' 10'

PERIOD. sec

Figure 11-3-4. Free oscillation quality factor (fundamental toroidal modes) calculated from the Earth Basin and Range elastic model and the frequency-dependent intrinsic Q model, Q(W)3' as compared with the same observational data as those in figure 11-3-3. The fit is much better than those in figure 11-3-3.

u

(l) Ul

r-

o

-IO°.---.---.---.---.---.--~

-5x10 -I

-10-¹ -5x10 -2

BASIN AND RANGE Q(w)3

-

./

; / ...

;t' ••••

/ .,' /(o~.,

• y'

0'-:" ","

. / .-

/ / '

/ /

/

./ ./

/ /' /'

---'- ' - ' -- . _ . _ . -- ^---

. -

..........

....

... ...

...

-

...

--. , --

-- ,. -- '

---

EARTH MODEL INTRINSIC Q MODEL BASIN AND RANGE Q (L I)

--- BASIN~"-^{RANGE Q (HI)}

._ .- OCEANIC Q (L₂)

..... OCEANIC Q (H₂)

.. _ .. - BASIN AND RANGE Q (W)I

.--.-- BAS IN AND RANGE Q (W)2

5xl0² 10³ PERIOD, sec

Figure 11-3-5. Shift in toroidal free oscillation (fundamental modes) periods, DT, due to anelasticity as calculated from various Earth elastic and intrinsic Q models.

~

I-

o

-163-

INTRINSIC EARTH MODEL Q MODEL BASIN AND RANGE Q (L I) BASIN AND RANGE Q (HI) OCEANIC Q (L₂⁾ OCEANIC Q (H_{2 )} BASIN AND RANGE Q (W)I BASIN AND RANGE Q (W)2

- ' -'--'"

•

•

--

..• ~:~ • .".,r: .... .

,

"'

^..

' . ...

' "

^'-

"'-"

,

• nQe L ^ea Calculated by DAHLEN

•

'.

-10-⁵L - - _ . . I -_ _ _ _ _ _ - - - 1 . _ - ' - _ l -_ _ _ ·· , _ ..

,-=--__

..l...-_ _ - ' - _ - '

10² 5xl0² 10³ 10⁴

PERIOD, sec

Figure 11-3-6. Percentage shift in fundamental toroidal free oscil- lation periods, DT/T, due to anelasticity as calcu- lated from various Earth elastic and intrinsic Q models. The values ae 1 ^E calculated by Dahlen (1968) are the percentage ~hif~ of toroidal oscillation periods due to ellipticity alone.

7 . o r - - - . , - - - - . . . , . . . - - - - , - - - . - - - . , - - - ,

~ u 6.0

E

.:£

;?

>rL 5.0

EARTH MODEL - - BASIN AND RANGE ..... OCEANIC

.,...:.:..:. ... ~---.-....

-

^{... ~ ...}

~

^{............... .}

......

.......

.........

.............

4.0~~I~OO~----~20~0~---3~0~0~---4~0-0---5~0-0---~600

PERIOD. sec

Figure 11-3-7. Love wave phase velocity, Vp , and group velocity, V G, as calculated from the two Earth elastic models used in the perturbation calculations.

-165-

1~2r---.---r---r---.---r---~

- '- '- '- '- - .- . _. _-~-.--

'- '- ' -' - -'- '- ' - -

--~:---

---

- ' -'

100 200

................

. - "- -'- " _ .' ....:::-:-.=..::-.=-~.--..---=.~---= . .:..-

EARTH MODEL INTRINSIC Q MODEL - - BAS IN AND RANGE

--- BASIN AND RANGE OCEANIC

OCEANIC

BASIN AND RANGE BASIN AND RANGE

300 400

PERIOD. sec

Q (L₁⁾ Q (H I) Q (L₂)

Q (H₂)

Q (W}I Q (W}2

500 600

Figure 11-3-8. Shift in Love wave phase velocity, DVp, due to anelasticity calculated from various Earth Elastic and intrinsic Q models.

1~2.---.---.---.---r---r---,

_ '_ '- _ .- ,_ 0- _ ___ _ _ . _ _ _ _ . _ _ _ _ _ _ _ _ _

- " -'

EARTH MODEL INTRINSIC Q MODEL

.I / i

100

/ '

200

- - BASIN AND RANGE Q (L ,) BASIN AND RANGE Q (H,) - -.- OCEANIC Q (L₂)

....... OCEANIC Q (H

2)

BASIN, AND RANGE Q (W), ---- BASIN AND RANGE Q (W)2

300 400 500

PERIOD. sec

Figure 11-3-9. Shift in Love wave group velocity, DV , due to anelasticity calculated from various ~arth elastic and intrinsic Q models.

600