ON TAKING MEANS OF DENSITY DISTRIBUTIONS IN THE
EARTH
B. A. Bolt
(Received 1957 May
29)"
Summary
The error is investigated when the mean of two solutions of a non-linear
second-order differential equation which gives the variation of density with
'depth in certain layers of the Earth is used as a solution.
A solution (Model M) of the equation for layer E is given, which has a
density of 9-385 g / m 3 at the top of layer E.
Identifying, at the top of layer E, the mean of two numerically estimated
solutions with a similarly estimated solution, it is shown that the error is a
maximum at the bottom of layer E, where it is of the order of 0.04 per cent.
I . Introduction.-For
a spherically symmetrical Earth, if p denotes the density
at distance Y from the centre, and m the mass inside a sphere of radius r ,
dm
- =47+p.
dr
Also, in a chemically homogeneous layer of the Earth where departures from
adiabatic temperature gradients are negligible,
_ - GPm
dP
(2)
dr
(brz '
where G is the constant of gravitation and #I is a function of r determined from the
values of the P and S seismic velocities. The elimination of m from ( I ) and (2)
gives the second-order differential equation
where R ( r ) and Q ( r ) depend on q5 (r).
In two papers, Bullen (I) calculated a density distribution for the Earth by a
method which involved numerical integration of (3) in layers B, D of the mantle
and layer E of the core. The Earth model corresponding to this density distribution has since been called Model A. Two further distributions of the density
in the Earth's interior were calculated by Bullen (2) using equation (3) in layers
B, D and E. I n one of these, since called Model A' (3), the density varies smoothly
throughout the core and has a central value of 12.3 g/cm3. I n the other, which we
call Model A", there is an arbitrarily assigned density-jump of order 1og/cm3
between the bottom of layer E and the top of layer G, and the central density is
22.3g/cm3. Bullen used values midway between those of Models A' and A"
to compute values of the elastic parameters, pressure, and gravity within the Earth.
He states that, subject to his formal assumptions and the further assumption that
the density distributions of Models A' and A represent extreme cases, these mean
*
Received in on@
form 1957January 29.
G 27
B. A. Bolt
370
density values are “ accurate within about I per cent down to a depth of 2700 km,
and within about 3 per cent below this depth ”.
Bullard (4) has remarked that the procedure of taking simple means “is, in
principle, an unsatisfactory basis ” since equation (3) is non-linear and a linear
combination of two solutions is therefore not itself a solution. He states further
that it is probable that little error is introduced, but further investigation is
desirable ”. The purpose of this paper is to carry out this further investigation.
2. The Earth Model M.-A
comparison of Models A and A“ shows that while
the difference in density at the bottom of layer E is of the order of 7 per cent it is
only of the order of I per cent at the bottom of layer D. . Thus the effects of the
non-linearity of (3) will be most marked in layer E. Table I gives the densities
and their simple means in layer E of Models A and A , to one more decimal place
than in Bullen’s book. In Table I, R is the radius of the central core, so that for
layer E, 0.4 <r/R < I .
The mean density at the top of layer E is given as 9.385 g/cm3. Starting with
this value and using the velocities of Jeffreys (5), equation (3) was integrated numerically throughout layer E by the tangent formula, taking twelve equal intervals
of the argument. The model corresponding to the yielded density values will be
called Model M. In Model M, the density at the bottom of layer E is I 1.507g/cm3,
the computational error being less than 0.001. The density values at other depths
are shown in Table I.
TDLEI
Model A‘
Model A”
(g/mS)
(g/-?
9.720
9‘982
10.227
10‘454
10.663
10.856
I I ‘034
I 1.197
1 I ‘347
I I -484
I I .608
I I -720
9.050
9.290
9’517
9’729
9‘927
10.113
10.288
II
‘820
10’454
10.61I
10.761
10.906
I I ‘047
11.186
Mean
(g/cm7
9‘385
9.636
9.872
10.092
10.295
10.485
10.661
10.826
‘0.979
Model M
11.123
11.125
‘257
11.384
I I ‘503
II
’259
II
.386
‘507
II
(g/cm3)
9‘385
9.636
9’87 1
10.092
10.296
10.485
10.662
10.827
10.981
II
Table I shows that, for layer E, the means of the density values of Models A‘
and A“ correspond closely to the density values of Model M. It is seen that the
densities derived on taking the non-linearity of (3) into account differ from the
means shown by amounts which reach 0.004g/cm3(0.04per cent) at the bottom of
layer E. Calculation shows further that in the mantle the correspondence is
within the accuracy of the numerical method used.
3 . The non-linearity of ( 3 ) in relation to further Earth models.-It remains to
consider the distribution of error when means of models intermediate to A’
and A” are taken as a solution of (3). Because of the non-linearity of (3), it is not
apriori certain in such cases that the maximum error occurs at the bottom of E
and is less than 0.04per cent. A detailed analytical study, summarized below,
showed that this is in fact so.
On taking means of h s i t y distributions in the Earth
371
Let layer E be divided into layers of equal thickness, and let the suffx i refer
to the top of the ith layer. Let ( p i , mi) and (pi", mi")be values given from two
separate solutions of (3) intermediate to the solutions A and A". Let undashed
symbols relate to the further solution of (3) for which p1 = 4(pl' + pl"), and write
p;=pi(I +Xi),
mi'=mi(I +Yi).
Then, to sufficient accuracy,
pill=&.(' - x i ) ,
mi"=mt(l - y i ) ,
and further analysis shows that
2 4 (Ap; + Ap;)
I
- ZApi = Z ~ g i A p,i
L
i
(4)
where Api is the increase in density inside the ith layer. The left side of (4)
expresses the error at the top of a layer in E in using the mean of any two solutions
of (3) as the actual solution, and the right side enables this error to be computed at
any assigned depth.
By (4), taking several pairs of solutions intermediate to those of Models A"
and A", it was found that the non-linearity caused the errors to change sign at a
depth of order 400 km below the top of layer El but that this did not result in the
maximum error occurring elsewhere than at the bottom. The error did not
exceed 0.001per cent in the highest 400 km, and the maximum error did not
exceed 0.04per cent.
Since the Models A' and A represent fairly extreme cases, it is clear that the
errors in taking means of the solutions for reasonable pairs of models are negligible
for such purposes as the computation of the elastic parameters, pressure, and
gravity.
Department of Applied Mathmcltics,
The University of Sydney,
AUStralia:
1957May 17.
References
(I) Bullen, K. E., Bull. Seis. SOC.
nlmer., 30, 235-250, 1940; 32, 19-29,1942.
(2)Bullen, K.E., Introduction to the Theory of Seismology, 218-223, Cambridge, 1947.
(3) Bullen, K.E., M.N. Geophys. Suppl., 6, 384-401, 1952.
(4)Bullard, E.C., The Earth as a Planet, Vol. 11, ed. by G.P. Kuiper, 94-95,Chicago, 1954.
(5) Jeffreys,H.,M.N. Geophys. Suppl., 4,594415, 1939.