Geophys. J. R. ustr. SOC.(1973) 32,203-217.
A Correction to the Excitation of the Chandler Wobble
by Earthquakes
F. A. Dahlen
(Received 1972 August 8)
The purpose of this note is to point out that there is a serious and substantial
error in the published numerical results of Dahlen (1971b). That paper, hereafter
called CW, was a theoretical investigation of the hypothesis advanced by Mansinha
& Smylie (1967) that the Chandler wobble of the Earth is excited by earthquake
activity. Elastic dislocation theory was used to compute the change in the products
of inertia of a realistic SNREI Earth model produced by an arbitrary infinitesimal
seismic dislocation, and these results were used together with an empirical earthquake moment-magnitude relationship of Brune (1968) to estimate the total
excitation of the Chandler wobble by all observed large earthquakes since 1904.
Unfortunately, an algebraic error was committed in the evaluation of the re uired
four static deformation scalar radial functions eZ1(r),Y z 1 ( r ) ,4?21(r), and -%,l(r>
(see Appendix A of CW, and the Appendix to this note), and the numerical results in
CW are incorrect. The corrected numerical results presented here allow one to
draw a much stronger conclusion regarding the hypothesis of seismic excitation of the
Chandler wobble than was possible in CW. Very roughly, the computed shifts of
the Earth’s rotation pole produced by large earthquakes are about an order of magnitude less than those presented in CW. These smaller polar shifts force one to the
conclusion that seismic activity cannot be the primary excitation mechanism of the
Earth’s Chandler wobble. The fact that there must be an error in CW has been
independently pointed out by Israel, Ben-Menahem & Singh (1972), who have made
an independent numerical study of the change in the Earth’s inertia tensor produced
by an infinitesimal seismic dislocation. The corrected results presented here are now
in complete agreement with those of Israel et ul. (1973), and confirm their conclusions
with regard to the excitation of the Chandler wobble.
For the purpose of computing the static deformation of a SNREI Earth model
of radius a, an infinitesimal seismic fault (point tangential displacement dislocation)
may be completely described in terms of its depth h = u-r,, co-latitude 8,,
longitude 4,, strike a, dip 6, slip angle A, and its slip-area [A, &,I, or alternatively
its seismic moment M , = po[Ao&,] (here p, = p(ro) is the material rigidity at the
focal depth h; see Fig. 3 of CW for the definition of these parameters). The changes
AC,3 and AC23 in the products of inertia of the Earth model, expressed in terms of the
fault parameters h, O,, 4,, a, 6, A and [A, Go],
are given by equations (50) of CW,
203
204
F. A. Dahlen
rewritten here, with the correction of a slight misprint
AC13 = [A,~,](Tl(lt)[(sin 2u sin6 cos A
++cos 2u sin 26 sin A) sin 28, cos 4,
- 2(+sin 2u sin 26 sin A - cos 2u sin 6 cos A) sin 8, sin b0
+ T,(h) [ -sin 26 sin A sin 28, cos 4,]
+ T,(h) [(sin u cos 26 sin A -cos u cos 6 cos A) cos 28, cos 4,
+ (sin u cos 6 cos A +cos u cos 26 sin A) cos O0 sin 4, [}
(50CW)
ACz3 = [A,dv,]{Tl(h)[sin2u sin6 cosA
+&os 2u sin 26 sin A) sin 200 sin 4o
+2(+sin 2u sin 26 sin A- cos 2u sin 6 cos A) sin O0 cos 4o
+ T,(h) [ -sin 26 sin A sin 28, sin 4,]
+ T,(h) [(sin u cos 26 sin A -cos u cos 6 cos A) cos 28, sin 4,
- (sin u cos 6 cos A -t cos u cos 26 sin A) cos O0 cos $,I}.
1
N
E
0,
r
W
e 1.0 c
k
I
V
0
I
100
I
200
I
I
300
DEPTH
400
h
I
500
I
600
700
(KM.)
FIG. 1. Plot of
(h),
(h), r3(h) for SNREI Earth model 8073AW. This
replaces Fig. 4 of CW. Note that the function plotted is - rz (h).
205
Excitation of the Chandler wobble
The form of equations (50CW) is correct, but because of the error the three canonical
functions of depth r,(h),r z ( h ) , T3(h)in Fig. 4 of CW are incorrect. The algebraic
details necessary for a correct computation of T,(h), rz(h), T3(h)are described very
briefly in the Appendix to this note. The corrected functions themselves for SNREI
Earth model 8073AW (Fig. 2 of CW) are shown plotted in Fig. 1, which replaces
Fig. 4 of CW. The formulae (50CW) and the accompanying curves r,(h),T,(h),
T3(h)in Fig. 1 are useful for those earthquakes for which one has an estimate of the
seismic slip-area [A, Lo].
Teleseismic observations of the long period behaviour of
the seismic signal produced by earthquakes provide more naturally an estimate of
the seismic moment M , = p,[A,&,].
For that reason, and because the seismic
moment is increasingly becoming a routinely measured seismic parameter, it is
convenient to rewrite the formulae (50CW) in terms of the seismic moment M , by
defining
f,(h)= p,-'(a-h) r-,(h)
f,(h) = p , - ' ( ~ - h ) r,(h)
f3(h) = p , - @ - h )
Then
I
r3(h).
f , (h)[(sin 2a sin 6 cos A ++cos 2a sin 26 sin 1)sin 28, cos 4,
-2(+sin 2a sin 26 sin 1-cos 2a sin 6 cos A) sin 8, sin 4,]
+ T,(h) [ -sin 26 sin 1 sin 28, cos 4,]
+ f3(h) [(sin a cos 26 sin A -cos a cos 6 cos A)cos 28, cos 4,
+ (sin a cos 6 cos 1+cos cos 26 sin 1)cos 0, sin 4,]}
A ~ 2 =
3 M,{ f , (h)[(sin 2a sin 6 cos A++cos 2a sin 26 sin 1) sin 28, sin 4,
+2(+sin 2a sin 26 sin A -cos 2a sin 6 cos A) sin 8, cos 4,]
A c , ~ = M,{
c(
+ f , ( h ) [ -sin 26 sin 1 sin 28, sin
+ f 3 ( h )[(sin a cos 26 sin A -cos a cos 6 cos A)cos 28, sin 4,
- (sin a cos 6 cos A + cos a cos 26 sin A) cos 8, cos $,I},
where the functions f,(h), f,(h), f3(h) are shown plotted in Fig. 2. The seismic
slip-areas [ A o 2 u o ]to be used in conjunction with equations (50CW) and Fig. 1 are
to be given in mks units of m3, whereas the seismic moments M , to be used in
conjunction with equations (2) and Fig. 2 are to be given in the conventional units
of dyne-an.
Several features of the functions r,(h),T,(h), T3(h) and f,(h), f,(h), f3(h)
shown in Figs 1 and 2 deserve comment. Examination of the equations (50CW) or
of the equation (2) shows that vertical strike-slip fault models (6=90", ,l=Oo)
depend only on T,(h) and f,(h), whereas vertical dip-slip fault models (6 = 90°,
A = 90°) depend only on T3(h) and f3(h); the functions T,(h) and f,(h) are
required for any fault model which does not have a vertical fault or auxiliary plane
(6 # 0", 90"). Israel et al. (1972) have independently obtained expressions identical
to (50CW)for AC13 and AC23, but have chosen to present their results in terms of
three canonical fault model types. Their three dimensionless functions of depth
F,(h), F,(h), F,(h) are however directly comparable, except for a multiplicative
factor (of dimension kgm'), with the functions -I?,@),
-r3(h),2r2(h). The
agreement with the corrected functions - r,(h),- T3(h), 2r,(h) presented here is
excellent; the small differences between the two sets of curves which do exist are
206
F. A. Dahlen
2.0
I
I
I
I
I
I
I .:
0.:
0
100
200
DEPTH
FIG.2. Plot of
40c)
300
h
500
600
700
(KM )
(h), -f, (h), f, (h) for SNREI Earth model 8073AW.
attributable almost entirely to the finer partition of SNREI Earth model 8073AW
(defined at 36 points in the upper 700 km) compared with Earth model M3 of Israel
et al. (1973) (defined at 12 points in the upper 700 km). As shown by Israel et al.
(1973), the functions Tl(h) and T2(h) have fairly large jump discontinuities at the
level of the Moho discontinuity (h = 28 km), but T3(h) is everywhere continuous.
It can be seen from equations (1) that a discontinuity in Tl(h), T,(h), or T,(h) may
be either due to a discontinuity in the perhaps more basic functions fl(h), f2(h),
f,(h), or more directly due to a discontinuity in the material rigidity p(a-h). The
functions f2(h) and f3(h) are discontinuous at the Moho, but f l ( h ) is everywhere
continuous; thus the rather large jump discontinuity in Tl(h) at the Moho is directly
attributable to the corresponding jump discontinuity in p(a -h), whereas the
continuity of T3(h)is seen to be somewhat artificial since it arises from a cancellation
of the jump discontinuities in f3(h) and p(a-h). As Israel et aI. (1973) point out,
the changes ACI3 and AC23 produced in a SNREI Earth model by a vertical dip-slip
fault (6 = go", 6 = 90") are almost linearly proportional to the earthquake focal
depth h, for constant orientation and slip-area [ A , z , ] . In particular, the changes
AC13 and AC23 produced by a vertical dip-slip fault acting at the surface (h = 0)
of the Earth model are identically zero (in fact, it is apparent from the form of
Volterra's relation given in equation (37CW) that the entire static deformation field
207
Excitation of the Chandler wobble
Table 1
Fault parameters used for computation
h (W
80
40
a
Event
1960 Chile
8
h
35.5"
270"
25
128-5"
285.5"
170"
1964 Alaska
50
29.0"
213-0"
135"
170"
270"
1964 Alaska (Kanamori)
50
29.0"
213.0"
114"
160"
270"
produced by a vertical dip-slip fault at zero depth is identically zero). Most of these
seemingly peculiar features of the curves in Figs 1 and 2 are artifacts of the mathematical model under consideration. The faulting is modelled by a point tangential
displacement dislocation, and this leads directly to jump discontinuities in f , ( h )
and r , ( h ) across the Moho, as well as to the puzzling result that a vertical dip-slip
surface fault does not deform the Earth model. A real finite vertical dip-slip fault
which intersects the Earth's surface will of course produce a net change (though a
relatively small one) in the Earth's products of inertia; real finite faults in general will
presumably produce changes AC13 and AC2, which may be estimated with fair
accuracy by suitably smoothed (near the Moho) versions of the functions f , ( h ) ,
f z ( h ) , f , ( h ) in Fig. 2.
As an example, the theory developed in CW was used to compute the theoretical
changes in the products of inertia and the consequent theoretical shifts of the Earth's
rotation pole produced by two particularly well-studied earthquakes, the 1960
Chilean earthquake and the 1964 Alaskan earthquake. The fault parameters h,
O,, &,, a,6, I and the slip-areas [A, Go]
for these two events were taken, respectively,
from the summary study of the 1960 Chilean earthquake by Plafker & Savage (1970)
and the similar summary study of the 1964 Alaskan earthquake by Plafker (1969).
These parameters were deduced from a combined investigation of the P and S wave
focal mechanism solutions, the areal distribution of aftershocks, and the measurement and interpretation of local ground deformation data. An independent determination of the seismic source mechanism of the 1964 Alaskan earthquake has been made
by Kanamori (1970b), by means of a particularly thorough investigation of the
radiated long period surface waves. Such a study provides a direct estimate of the
seismic moment M, rather than the slip-area [ A , ~ u , ] ;the seismic moment of
7.5 x loz9dyne-cm obtained for the 1964 Alaskan earthquake by Kanamori
(1970b) is, to my knowledge, the largest seismic moment ever inferred for any
seismic event. The fault parameters used for computation in CW, as well as those
obtained independently by Kanamori (1970b), are tabulated in Table 1 (a misprint
in Table 1 of CW has been changed). The corrected changes ACI3and ACZ3in the
products of inertia of SNREI Earth model 8073AW, and the consequent corrected
theoretical pole shifts corresponding to these assumed fault parameters, are given
in Table 2, which replaces Table 2 of CW. These corrected theoretical polar shifts
Table 2
Changes in the inertia tensor components and associated polar shifts produced by the
Chilean and Alaskan earthquakes
Event
1960 Chile
Mo
(dyne-cm)
-
1964 Alaska
-
1964 Alaska
(Kanamori)
7.5 x
[Aozol
Act3
A&
Polar shift
(m3)
(kg m2)
(kg mf)
(0.013
1.2~10'~- 3 . 1 7 ~ 1 0 ~8.64~10"
~
1.02
1.2~
10l2 - 4 . 1 6 ~ 1 0-0.98x
~~
-
-6.14~loz7-2.44~
Direction
110"E
0.48
193"E
0.73
202"E
208
F. A. Dahlen
produced by the 1960 Chilean and 1964 Alaskan earthquakes may be compared with
the ‘ observed ’ polar shifts which Smylie & Mansinha (1968) determined by a process
of searching for breaks in the astronomically observed polar motion data of the
BIH. The comparison is made in Fig. 3, which replaces Figs 7 and 8 of CW. There
is for both events a complete lack of agreement between the corrected theoretically
computed polar shift and the ‘ observed ’ polar shift obtained by the data analysis
procedure of Smylie & Mansinha (1968). Although both computed polar shifts are
in approximately the same direction as the ‘ observed ’ polar shifts, the theoretical
amplitude is in both cases about an order of magnitude less than the ‘observed’
amplitude. This lack of agreement is not at all surprising, in view of the re-evaluation
and criticism by Haubrich (1970) of the data analysis procedure of Smylie &
Mansinha (1 968). The corrected theoretical computations for the 1960 Chilean
earthquake and the 1964 Alaskan earthquake support his view (private communication) that the noise contamination of the BIH polar motion data is so severe that it
precludes direct observation of the effect on the polar motion path of even the largest
earthquake. Wells (1972)’ in a study of the BIH data performed since the publication of CW, has provided further support of Haubrich’s views. The average size of
all the ‘ observed ’ polar shifts determined by Smylie & Mansinha (1968) was about
lO(O.01”) (Haubrich 1970). This average ‘Observed’ polar shift is a full order of
magnitude larger than the largest corrected theoretical polar shift. This gives one a
rough feeling for the level of noise contamination in the present BIH polar motion
data; it implies that even the largest earthquake will give rise to polar shifts which
are about an order of magnitude smaller than the noise level. The direct observation
of the effect of a large earthquake on the path of the Earth‘s rotation pole will not
be possible until some way is found to reduce the noise contamination in the
observed polar motion path by at least a factor of ten.
Changes in the inertia tensor of the Earth produced by large ( M o = loz9 to
1030dyne-cm, say) earthquakes do give rise to shifts in the path of the rotation
pole which are large enough that they might someday be detected. The interesting
question to which we now address ourselves is this: is there sufficient cumulative
seismic activity to account for the total level of observed power in the Chandler
wobble, i.e. are earthquakes the primary excitation mechanism? The total theoretical
Chandler wobble power P produced by a series of N earthquakes occurring during a
090’E
-10-
-
+
+
(
7
COMPUTEO+T+AFTER
BEFORE
-20
COMPUTEO+T BEFORE
-30-
-40I t
-20
I
-10
I
10
0
Lor)
I
20
30
TO GREENWICH
CHILEAN EARTHOUAKE 1960
COMPUTED POLE S T E P I .02 1.01”)
-20
I
-10
I
1
0
10
(.Ol”)
I
20
TO GREENWICH
ALASKAN EARTHQUAKE 1964
COMPUTED POLE STEP 0.73 (.Ole)
FIG.3. Comparison of computed polar shifts with polar shifts inferred by Smylie
& Mansinha (1968) from BIH polar motion data for the 1960 Chilean earthquake
and the 1964 Alaskan earthquake. This replaces Figs 7 and 8 of CW.
Excitation of the Chandler wobble
209
time interval TN is given by equation (14CW)
Q (R’),
P =-
(14CW)
WO
where wo is the angular frequency of oscillation, Q is the Chandler quality factor, and
( R ’ ) , given by equation (13CW), is the mean squared polar shift per unit time
associated with the earthquake sequence. Paper CW contains a summary of recent
determinations of the Chandler frequency w o , Chandler Q, and the observed total
Chandler power P; the Q is uncertain by almost a factor of ten (Q = 20-200), the
Chandler power is on the order of 200(0.01”)2. Using the corrected theoretical
computations presented here, it is very difficult to attribute this observed Chandler
power to excitation by seismic activity, as may be seen from the following simple
example. The computed polar shift for the 1964 Alaskan earthquake (using the
Kanamori source parameters) is 0.73(0.01”). If there were ten Alaskan earthquakes
per year (occurring at random times), and if the Chandler Q were really as high as
200, then the observed level of Chandler power P could be maintained; if the Q
were really only 30, then 67 Alaskan earthquakes per year would be required. The
actual average level of seismic activity per year is not in fact equivalent to ten Alaskan
earthquakes per year, and for that reason, seismic activity cannot be the primary
excitation mechanism of the Chandler wobble.
The extent to which the actual observed level of seismicity does affect the polar
motion path may be crudely estimated using a procedure like that described in CW.
The 1201 earthquakes of surface wave magnitude M 2 7.0 during the period
1904-1964 in the seismic catalogue of Duda (1965) were in CW classified into three
categories; type 1, shallow focus strike-slip mechanism, type 2, shallow focus low
angle reverse thrust mechanism, type 3, deep focus (h < 100km). The seismic
moment M o of each of these events was estimated by using an empirical momentmagnitude relation. Two such empirical relationships have been proposed; they are
depicted in Fig. 4, which is taken from Aki (1972). The moment-magnitude relation
of Brune (1968) was developed on the basis of a study of the excitation of 100 s Love
and Rayleigh waves; the moment-magnitude relation of Aki (1967 and 1972) was
proposed on the basis of an assumed w’ scaling of the seismic displacement spectrum.
Also shown in Fig. 4 are the measured seismic moments of a few especially well
determined large events, mostly due to Kanamori (1970a, 1970b, and private communication). As discussed in CW, any attempt to infer the seismic moment of a large
(M > 8.0, say) earthquake from the measured 20s surface wave magnitude M is
bound to lead to large uncertainties. This inherent difficulty is reflected in the rather
extreme differences between the two proposed empirical relationships for M > 8.0.
The Brune (1968) relation is supported by the fact that if it is used to compute average
cumulative rates of slip at lithospheric plate boundaries (Davies & Brune 1971), it
gives slip rates which agree well with those obtained by other methods. The Aki co2
relationship, on the other hand, appears from Fig. 4 to more closely predict the welldetermined seismic moments of those few large events so far carefully investigated.
For the purpose of comparison, this study will be performed using both the Brune
(1968) and the Aki co2 empirical relations. In CW, only the Brune relation was
utilized for computation, and it was first converted into a slip-area us. magnitude
relationship, taking p o = 3.3 x 10” dyne cm-’ at all depths (this seems unnecessary
and has not been done here). Equations (2) giving the inertia tensor component
changes ACI3 and AC23 produced by any individual earthquake may be written in
the shorthand notation, similar to equation (56CW),
210
F. A. Dahlen
m
E
Y
0
a
Y
a
W
a
CC
P
J
v)
FIG.4. Seismic moments Mo us. earthquake magnitude, as proposed by Brune
(1968) and Aki (1967, 1972). Taken from Aki (1972).
Average values of the quantity 9" = (9"132 +9"232)' were computed for each of the three
types of earthquakes used in the classification scheme; the corrected results for
SNREI Earth model 8073AW are presented in Table 3. It can be seen from Table 3
that on the average, a shallow focus trench or island arc type earthquake (type 2)
produces a slightly larger polar shift than a shallow focus strike-slip earthquake
(type 1) of equal moment. Before averaging over dip angle 6, the average (9")2 for a
low angle (10' < 6 < 45") dip-slip earthquake is, approximately, (9")2 = 1 . 4 ~
sin 26, i.e. the polar shift produced by such an earthquake is proportional to sin 26.
A simple Monte Carlo technique similar to that described in CW was used to
estimate (R'), the mean squared polar shift per year produced by the recorded seismic
activity during the period 1904-1964. Three separate experiments were performed.
In the first, the seismic moments Mo of the 1201 earthquakes were inferred using the
moment-magnitude relation of Brune (1968), except that for the 21 earthquakes
assigned mantle wave magnitudes M,,, by Brune & Engen (1969), M,,, was used to
Table 3
Average values
Type
1
2
3
( a ) for three types of events.
The units of (g) are such that ACI3
and AC23 are in kg m2 if Mo is in dyne-em.
Description
<k> (kg m2/dyne-cm)
6 = 90°, h = 90°, 0
<h d
20 km
h = 9 0 ° , 10"<6<45", O C h d 6 0 k m
100 < h
< 700 km
0.62 x
1.1ox 10-2
o-wx10-1
21 1
Excitation of the Chandler wobble
Table 4
Chandler power P produced by earthquakes compared with atmospheric excitation.
Method of
Moment release
<RZ>
Chandler power P (0~01")~
moment estimation
Brune
(dyne-cm/yr)
1.5 x 1029
Aki o2
2 . 5 x 1030
Aki o2(modified)
9 - 4x loz9
Atmospheric
-
(0.01")2/yr Q = 30 Q = 100 Q = 200
20
0.20
0.41
0.06
20
110
0.9
4.8
-
2.5
380
760
16
32
8.3
16.6
infer the moment. In the second, the w 2 relationLip of Aki (1 67) was used to infer
the seismic moments, except that those of the 1933 Sanriku earthquake, the 1963
Kuriles earthquake, and the 1964 Alaskan earthquake were taken from Fig. 4.
In the third, the Aki w 2relation was used again, except that no earthquake in Duda's
(1965) catalogue was allowed to have a moment M o > 103'dyne-cm (i.e. 18
events with M > 8.4 were assigned M o = 103'dyne-cm). Each experiment consisted of lo00 simulated two-dimensional random walks. The results of these three
Monte Carlo experiments are given in Table 4. The total Chandler power P
generated by an excitation process modelled by each of the three experiments is given,
and these are compared with the level of atmospheric excitation measured by Munk
& Hassan (1961). The theoretical Chandler power estimated using the unmodified
Aki o2 moment-magnitude relation is comparable to the observed level of about
200 (0.01")2, even for a low Chandler Q. It may, however, be seen from a comparison with that estimated using the modified Aki w 2 relation that this large estimate of
the Chandler power is attributable almost entirely to a few (18 out of 1201) very large
earthquakes whose moments M , are estimated by the unmodified Aki relation to
be in excess of lo3' dyne-cm.
Since the largest seismic moment ever measured for any seismic event is
7.5 x
dyne-cm, it is almost certain that the seismic moments of these few very
large earthquakes are considerably overestimated by the use of the Aki w 2momentmagnitude relation. It is the author's opinion that even the theoretical Chandler
power computed using the modified Aki w 2 relation represents an absolute upper
limit to the amount of Chandler wobble activity generated by earthquakes. In
CW, it was argued that for various reasons (aftershock activity, aseismic fault
creep, etc.), the use of the Brune (1968) moment-magnitude relation might systematically underestimate the effective seismic moments of large earthquakes by a factor
of 5-10. It may be seen from Fig. 4 that the use of the modified Aki w 2relation more
than compensates for this factor of 5-10. It is concluded therefore that the Earth's
seismic activity can account for no more than 10 per cent (for Q = loo), and
probably even significantly less, of the observed level of excitation of the Earth's
Chandler wobble.
In conclusion, it appears that although large earthquakes (such as the 1960
Chilean earthquake and the 1964 Alaskan earthquake) do produce polar shifts which
might someday, given improved data, be visible, the cumulative seismic activity of
the Earth is not sufficient to serve as the primary source of excitation of the Chandler
wobble. The mechanism of excitation of the Chandler wobble remains an open
question.
Acknowledgments
I first became aware that there was an error in CW after reading a preprint
(Rice & Chinnery 1972) kindly sent me by the authors. Although the aid of numerous
212
F. A. Dahlen
individuals is acknowledged in CW, responsibility for the error is entirely my own.
This work has been supported by the National Science Foundation under the Grant
GA-21387. A.P. (1711), 525.
Department of Geological and Geophysical Sciences,
Princeton University,
Princeton, New Jersey 08540.
References not cited in CW.
Aki, K., 1972. Earthquake mechanism, Proceeding of the Final UMC Symposium,
ed. by R. Ritsema, Moscow, 1971.
Dahlen, F. A., 1971a. Comments on paper by D. E. Smylie and L. Mansinha,
Geophys. J. R. astr. Soc., 23, 355-358.
Dahlen, F. A., 1971b. The excitation of the Chandler wobble by earthquakes,
Geophys. J. R. astr. Soc., 25, 157-206.
Dahlen, F. A., 1972. Elastic dislocation theory for a self-gravitatingelastic configuration with an initial static stress field, Geophys. J. R. astr. Soc., 28, 357-383.
Gilbert, F. & Backus, G. E., 1966. Propagator matrices in elastic wave and vibration
problems, Geophysics, 31, 326-332.
Israel, M., Ben-Menahem, A. & Singh, S. J., 1973. Residual deformation of real
Earth models with application to the Chandler wobble, Geophys. J. R. astr. Soc.,
32,219-247.
Rice, J. R. & Chinnery, M., 1972. On the calculation of changes in the Earth's
inertia tensor due to faulting, Geophys. J. R. astr. Soc., 29, 79-90.
Smylie, D. E. & Mansinha, L., 1971. The elasticity theory of dislocations in real
Earth models and changes in the inertia tensor, Geophys. J. R. astr. Soc., 23,
329-354.
Wells, F. J., 1972. On the separation of the annual and Chandler spectral components
in astronomic latitude and polar motion data, Ph.D. thesis, Brown University.
Appendix
This Appendix will briefly present the algebraic details of the computation of the
inertia tensor component changes AC,, and ACzJ;it is intended to replace Appendix
A of CW, which contains a serious error.
The most straightforward method of determining the static deformation of a
SNREI Earth model produced by a point tangential displacement dislocation is to
utilize the concept of the equivalent body force introduced by Burridge & Knopoff
(1964) and extended by Dahlen (1972). The displacement field v(r) produced by the
dislocation is identical to that produced by an equivalent double-couple body force
distribution e(r) given by (Dahlen 1972)
e(r) = -Mo(6, &,+So 6,).VG(r-r0),
zO]
(4)
where M o = p, [A,
is the seismic moment of the dislocation, ro is its location,
and 6, and &, are the unit normals to the fault and auxiliary plane, respectively. The
linearized equations relating the elastic-gravitational response v(r) to the equivalent
2 13
Excitation of the Chandler wobble
applied body force e(r) are equations (17CW) and (18CW), rewritten here
-PoV$,-P1 V$o-V(v*poV$o)+V.E+e
V2$, = 4nGp1
=0
i
PI = -V.(PoV)
E = I(V.V)I+~C([VV+
(VV)~].
(5)
In a SNREI Earth model, these equations may be converted into a set of scalar
equations, using the methods developed by Backus (1967). Both vCr) and e(r) may
be decomposed in the form
v(r) = P%!(r)+V1
-Y-(r)-PxV, YV(r)
(6)
e(r) = Pd(r) V, %?(r)- P x V, %'(r),
)
+
where
I
2 c
q ( r ) = I = O m = - I q r ( r ) y(3)
m
I
In (6) and (7), V, is the surface gradient operator (38CW), and the surface spherical
harmonics r;l(P) are assumed to be fully normalized (39CW and 40CW). Writing
Bo = Pnr+n,
6, = Per+es,
and using Gauss' theorem together with the orthogonality relations for vector
spherical harmonics, the expansion coefficients d ; t ( r ) , .Cdr(r) can be shown to be
(the toroidal coefficients % r ( r ) are not required for this purpose)
1
d ~ ( r=) ~0 (yZs(r-ro)[(n,es+erns).~l
+(~0)-6nrer
*($0)1
F
214
F. A. Dahlen
Introducing the three functions
[AP(fO, do, %)I1, [4"'(foy
fro, QO)IZ, and
of the fault location and orientation defined by
[h'"(fo,
a,,
the formulae (8) may be reduced to
Now following Backus (1967), and as in CW, denote the coupled homogeneous
system of six scalar poloidal elastic-gravitational equations
Excitation of the Chandler wobble
215
d
-@(r) = A(r) @(r),
dr
(13)
by the shorthand notation
where
In equations (12)
In terms of this notation, the non-homogeneous system of equations ( 5 ) may be
converted into the non-homogeneous scalar system
d
dr
-@ r ( r ) =
r )@ y ( r ) - d ~ ( r ) Z p - 9 ~ ( r ) Z Q
where
Now let OP(r),QQ(r),@ / ( r ) , OQ'(r)denote, respectively, the solutions satisfying the
free surface boundary conditions to the non-homogeneous linear systems
d
1
-OP(r) = A(r)@,,(r)-- 6(r-ro)Zp
dr
r3
d
1
1
-mQ(r)= A(r) QQ(r)- --6(r -ro)Z ,
dr
1(1+1) r 3
d
-@/(r) = A(r)@/(r)+
dr
(18)
Then, from (ll), the required solution to the system (16) is given by
=
[@Q(r)l[h?ll
+ [-iF@P(r)+~@P'(r)+aI(I+1) @Q(r)l[Ai"12
+ [@P(r)-3@Q(r)+@o'(r)l[A1"13.
(19)
Numerical solutions to the non-homogeneous systems of equations (18) are readily
obtained using the propagator matrix formalism introduced by Gilbert & Backus
(1970). For computational purposes, it is convenient to avoid the use of complex
quantities by writing, as in CW, Y,"' = A,"'+iB,'" and,redefining @;l(r) and 8,l"(r)
by using Ap(Po) and B?(Po), respectively, in place of Y;nt(fo) in equations (10).
216
F. A. Dahlen
To compute AC13 and AC23, one needs only @,'(r), Y , ' ( r ) , @,'(r) and f 2 ' ( r )
in the mantle and (+l(r)),', (&,(r)),' in the Adams-Williamson fluid core. Written
out in full, the expressions [A2'I1, [A,'],, [A,'],, [A2'Il, [x2'],, [A,'], are
exactly the bracketed expressions in (2), i.e.
=
J(2)
M,[(sin 2a sin 6 cos A + ~ C O S 2a sin 26 sin A) sin 28,
87c
cos+,
- 2(+sin 2a sin 26 sin A -cos 2a sin 6 cos A) sin 0, sin +,I
[A1,]'
=
[A12],
=
J(2)
sin^ sin~sin20,cos+,]
87c
(g)
M , [(sin CI cos 26 sin 2 - cos CI cos S cos A) cos 20, cos 4,
+ (sin a cos 6 cos 2 + cos a cos 26 sin A) cos 8, sin +,I
[A '21
=
J (2)
M , [(sin 2a sin 6 cos ,I+$cos 2a sin 26 sin>I,
87c
sin 28, sin 4,
+2(fsin 2a sin 26 sin A -cos 2a sin 6 cos A) sin 0 cos +,I
[A,'], =
J(")
M,[-sin26
8n
[A,'],
J (g)M , [(sin a cos 26 sin A- cos a cos 6 cos A) cos 20, sin 4,
=
sin]. sin28, sin+,]
- (sin a cos 6 cos A +cos a cos 26 sin A) cos 0, cos +,I
The changes AC13 and AC23 in the inertia tensor components due to the deformation
of the mantle may be expressed, as in CW, in terms of an integration of the Eulerian
formulation over the deformed mantle volume (see equations (44CW) and (48CW)).
It is preferable, however, since it avoids numerical differentiation of the density
p,(r), to follow Smylie & Mansinha (1971) and Israel et al. (1973) in performing an
integration of the Lagrangian formulation over the undeformed mantle volume.
The mantle contribution may be shown to be
where r = c is the core-mantle boundary. The contribution due to the deformation
of the fluid core may not be computed in this way, since the Lagrangian particle
displacement is indeterminate in the Adams-Williamson fluid core (see Appendix
B of CW and Dahlen (1971a)). The density perturbations (pl);" and (p"J;" may
however be determined (see Longman (1963) and CW)
Excitation of the Chandler wobble
217
and the core contribution determined from an integration over the deformed core
volume
It is clear from (20) that when the expressions (21) and (23) are evaluated, the final
result will be of the form (2); it does not seem necessary to write out explicitly the
expressions for the three functions f , (h), f, (h), f 3(h).
Smylie & Mansinha (1971) have utilized a very similar but not identical approach
in their computation of AC13 and ACz3 for an Earth model similar to SNREI model
8073AW. As Israel et al. (1973) point out, the final numerical results of Smylie &
Mansinha (1971) are in significant disagreement with those presented here. The
major difference between the treatment of Smylie & Mansinha (1971) and that
presented here, apart from a different algebraic development, lies in the treatment
of the fluid core. It turns out that, although the philosophical differences are profound (see Dahlen (1971a) and reply), these give rise ultimately to only one relatively
minor difference in the mathematical formulation. The mantle contributions to
AC13 and AC23 are identical; only the core contributions differ. Smylie & Mansinha
(1971) maintain that there is no density change throughout the core volume
((pi);" = (pl);" = 0), but that there is an additional contribution to AC13 and
ACZ3due to a jump discontinuity in %,'(r) and
at the deformed core-mantle
boundary (compare their equations (44) with (22) above). As a check, computations
incorporating this alteration have been performed, in an attempt to ascertain whether
or not the lack of numerical agreement may be attributed to the difference in treatment. As one might suspect, small changes involving only the relatively small core
of the Earth model do not change the results appreciably, on the order of 10 or 20
per cent. This is not sufficient to explain the considerably larger disagreement.
Other miscellaneous corrections to CW include the following: Figs 5 and 6 are
incorrect, but since they are not really particularly useful, they will not be corrected
here. Fig. 5 is incorrect only in scale, since it depends only on rl(h),
but Fig. 6
depends on all of r1(h),T,(h), r3(h). In equation (IOCW), the second 6 ( t - t j )
should read H ( t - tj). As pointed out in the note added in proof in CW, the derivation
of the Volterra relation (34CW) or (37CW) is not rigorous because of a faulty
boundary condition. A rigorous derivation has been given by Dahlen (1972), and it
has been shown there that the relation itself is valid, and that the equivalent body
force distribution is the familiar double-couple utilized above.
eZ1(r)