THE E L L I P T I C I T Y CORRECTION TO TRAVEL-TIMES
OF P A N D S EARTHQUAKE WAVES.
K. E. Bullen.
(Received 1937 April 7)
T h e importance of the effect of the Earth's ellipticity on the traveltimes of earthquake waves was recently brought into prominence in a paper
by Gutenberg and Richter,* and Jeffreys -i-has since given a theoretical
analysis of the general effect. I t is proposed in this paper to examine in
detail the effect on P and S waves.
I . Jeffreys 1 defines a standard sphere of reference such that each internal
surface of equal velocity for any pulse is spherical and encloses the same
volume as the corresponding surface of equal velocity within the actual
Earth. Now let E be the epicentre of any given earthquake and 0 a given
observing station, and let their geocentric co-latitudes be a , y respectively.
Let the angular distance between E and 0 be A (computed using geocentric
latitudes). Take r as the distance from the Earth's centre of any point P
on the path of a ray that travels from E to 0, and T - 6 ~as the distance
of the corresponding point from the centre of the standard sphere. Let
c be the velocity of the ray at P,9 the angle subtended at the Earth's centre
by EP, and s the length of path of the ray between E and P. As usual, p
is defined by the equation
d%-pc
z
7'
Then if T' is the actual travel-time of the ray between E and 0, and T the
time between the two corresponding points on the surface of the standard
sphere, Jeffreys shows that to the first order in Srlr
[ I dr
( d 2 I I)
T ' - T = p --6r
+ p 6~ - -+-- d%.
r2 dtl
dfPr Y
Further, from the definition of the standard sphere it follows that to the
same order
6r = €YAY2,
where
s,= - cos2 %',
E being the ellipticity of the surface of equal density through P, and 9' the
co-latitude of P.
2. We shall investigate the case of P waves first, and commence by
considering the second term of equation (2). From ( I ) we have
1: 1;
+
*
Gerl. Beitr. Geophys., 40, 380-389, 1933.
t M.N.R.A.S., Geophys. Stippl., 3, 271-274,
1 LOC.cit.
1935.
Mr. K. E. Bullen,
I44
a.e.
(3)
Differentiating with respect to r and cancelling a factor, we have,
constant along the same ray,
Thus the second term of
(2) may
p being
be put in the form
a.e.
r3 dc
c3 dr
(4)
approximately, where Q is now measured in degrees. This last form is
the one most suited to computation.
2 . 1 . Towards evaluating S,, let
be the azimuth of 0 referred to E.
Then for the typical point P
+
cos 8'
= cos
8 cos a + sin Q sin a cos 4.
+
(5)
Since a and are constant along the particular ray, it is desirable to have
cos 8' expressed as a function of Q (ranging from 0" to 105') for given values
of ( a , 9). For this purpose it is convenient to put (5) in the form
cos2 8'
= ( I - sin2
a sin2+) cos2 ( Q - p),
where
Values of a were taken at 10' intervals from 30° to 150° (values outside
these limits not being included since there are practically no recorded
epicentres or observing stations within 30' of the Earth's poles). T h e
values taken for were oo, 30°, 60°, 90" (values outside this range are not
necessary as S , is an even function of +, and, further, its value is unchanged
when W, # are both replaced by their supplements). T h e expression cos2 8'
was then computed by ( 6 ) correct to three figures for each combination
of values of ( a , #) and a graph constructed for each case giving S2as a
function of 8. Care was taken to ensure two-figure accuracy in values of
S , as read from these graphs.
2.2. T h e factor A8 of (4) was next considered. Ry (3) we have
+
1937 May Ellipticity Correction to Travel-times of P and S JVaves
145
A 3 being here measured in degrees. I n obtaining values of A@, r was taken
at depths of multiples of IOO km. below the Earth's outer surface, Ar being
IOO km. except at the ends and lowest point. T h e formula (7) was used
for all but the lowest range of depth, the value of A0 for this range being
found by subtraction. T h e formula becomes inaccurate near the lowest
point of the path, but the maximum error that could be introduced here
was, from the nature of the variation of the functions subsequently to be
integrated, found to be negligible. This process was carried out with
the different values of p corresponding to each of the various values of A
subsequently used in evaluating (4). T h e values used for c were obtained
from the recent results of Gutenberg and Richter.* Values for p were
taken from the P and S tables of Jeffreys and the author.? T h e following
results are obtained in a typical case (A =60°). Increments At' (in degrees)
are given for the specified ranges of depth down to the lowest point.
Range of Depth
(in km.)
ae
Range of Depth
A0
0- 5 0
50-150
150-250
250-3 5 0
3 50-45 0
450-550
5 50-650
650-750
0.27
0.55
0.57
0.64
0.76
0.89
I '03
1.19
750- 850
850- 950
950-1050
1050-1150
1150-1250
1250-1350
1350-1450
1450-1520
1.35
1.59
I.7j
1.96
2.44
3.12
5.26
6.63
r3 dc
- being independent of the particular
c3 dr'
epicentre and observing station, was computed as a direct function of Y.
T h e values taken for E were those recently obtained by the author,z c and
2.3. T h e remaining factor
dc
~
dr
E-
being taken as in the preceding paragraph.
2.4. It was now possible to evaluate the expression (4) for each value
of ( a , 6)and for each A required. T h e values taken for A were 3oo, 4oo,
50°, 60°, 7oo,Soo, 9oo,I O O O respectively ; as will be seen presently, it was not
necessary to evaluate (4) for A less than 30°. T h e increment of Y involved
in a term of a particular summation was usually greater than roo km., but
was adjusted so as to secure with minimum labour that the entire computation error would be not greater than 0.02 second in any case. I n
most cases it would be less than 0.01second. In the following table values
*
Gerl. Beitr. Geophys., 45,345-352, 1935.
B u r . Centr. Sdism. Intern., fasc. 11, 1935. This paper will be henceforth
referred to as Paper I .
1 M.N.R.A.S., Geophys. S u p p l . , 3, 395-407, 1936.
t
Mr. K. E. Bullen,
146
49 2
of the expression (4)are given to two decimal places. ( T h e second decimal
place has been preserved in order to avoid cumulative errors of computation.)
TABLE
I
Corrections to Travel-times of P W a o e s due to the Ellipticities of Internal S t r a t a
of Equal Velocity ( t o be subtracted f r o m Apparent Travel-times)
30'
T
+ 1.49
SO3
-t1.40
100"
i
+ 0.68
+0.50
+ 0.47
+0.30
+ 0.99
+ 0.42
+ 0.25
t 0.70
+0.14
+0.05
+ 1.27
+ 0.56
+O.IO
+ 0.09
+ 0.43
+ 0.07
+0.14
+0.13
+ 0.03
+ 0.02
+ 0.08
+0.14
+ 0.26
+ 0'45
+ 0.24
+0.08
0.00
0.00
+ 0.07
+0.18
+0.34
=40°
+ 1.16
+ 1.16
+ 0.28
+0.76
+0.46
+O.II
-0.14
+0.45
- 0.26
+I.II
+ 1.03
+ 0.99
+0.80
+ 0.73
+ 0.58
+0.58
+ 0.43
+ 0.90
+ 0.84
+@54
+ 0.42
+ 0.26
- 0-09
-0.01
- 0.05
+0.37
+0.34
+0.22
-0.20
+ 0.58
+ 0.40
+0.36
+ 0.22
+0.09
- 0.01
-0.09
-0.24
- 0.27
-0.28
+ 43-45
+0.33
+ 0.46
+0.56
+ 0.60
+0.53
+0.41
+0.33
+0.19
+0.91
+0.70
+0-50
- 0.02
- 0.08
+ 0.02
+0.8S
+0.85
+ 0.69
+ 1.07
+ I .08
+O.IO
-0.12
-0.15
-0.12
- 0.07
+0.14
+0.03
-0.06
-0.14
+ 0.97
+0.26
+0.89
+o.81
+ 0.73
+ 0.60
+ 0.40
t0.23
+0.14
+ 0.04
+ 1.22
+ 0.41
- 0.34
- 0.38
- 0.34
- 0.26
+O.IO
- 0.07
-0.17
- 0.22
-0.15
+O.OI
-0.12
-0.21
- 0.22
-0.14
- 0.05
0.00
+ 0.22
+0.14
+0.32
- 0.29
- 0.39
- 0.35
- 0.42
- 0.42
- 0.41
- 0.44
- 0.38
- 0.28
-0.21
+ 0.07
-0.11
- 0.02
- 0.22
+O.OI
+O.IS
-0.12
+0.16
+ 0'34
- 0.29
- 0.53
- 0.33
- 0.37
- 0.41
- 0.56
- 0.57
- 0.58
- 0.45
- 0.40
- 0.35
- 0.26
-0.64
-0.62
-0.58
-0.65
-0.62
-0.56
-0.43
=60"
30"
40°
50"
60"
70'
80'
90°
100"
~
+ 0.95
+ 0.76
+0.59
+ 0.29
+0.16
+ 0.06
+ 1.25
+ 1.30
+ 1.28
i
60"
"55
30°
40°
60"
70°
80"
9o'
I
+ 0.65
+ 0.77
+ 0.86
+ 0'79
+0.61
+0.55
-I- 0.41
+ 0.03
+ 0.07
+ 0.07
+ 0.05
- 0.02
- 0.09
- 0.16
- 0.25
- 0.42
- 0.43
- 0.44
- 0.46
-0.50
-0.32
-0.12
+O.OI
+0.14
-0.21
0.00
+0.19
+@34
1937 May Ellt$ticity Correction to Travel-times of P and S Waves
60"
OU
- 0.10
+O.OI
+ 0.06
+ 0.20
+ 0.30
+ 0.24
+ 0.42
+0.58
+0.60
+ 0.55
120°
1500
180"
- 0.79
- 0.70
- 0.34
- 0.28
- 0.63
- 0.75
- 0.80
- 0.62
- 0.71
- 0.24
-
0.62
- 0.75
- 0.72
- 0.22
- 0.65
- 0.70
- 0.49
0.22
- 0.62
-
- 0.24
- 0.48
- 0.09
- 0.43
- 0.38
+O.II
0.58
-
0.62
0.27
+0.50
+0.33
t 0.29
+ 0.24
- 0.30
- 0.59
- 0.60
+ 0.43
+O.IS
- 0.35
-0.61
- 0.35
- 0'45
- 0.27
- 0.65
- 0.83
- 0.87
- 0.83
-0.12
- 0.58
- 0.80
- 0.82
- 0.69
-0.10
+ 0.03
-0.52
- 0.55
- 0.40
+O.IZ
- 0.47
- 0.41
- 0.78
- 0.78
- 0.73
- 0.68
- 0.69
- 0.70
- 0.77
+0.28
-
147
-
0.00
-0.56
- 0.38
- 0.14
+ 0.07
+ 0.23
+ 0.34
!=so"
30°
40°
SO0
60"
70°
80"
90°
I0O0
+ 0.08
+0.38
+ 0-43
+ 0.44
+ 0.39
+0.16
t-0.17
- 0.38
- 0.40
tO.12
-
- 0.7 2
- 0.54
- 0.37
- 0.21
- 0.04
- 0.82
0.44
-0.71
0.60
- 0.20
- 0.49
- 0.05
-
- 0.79
- 0.62
- 0.44
- 0.22
0.00
+0.17
- 0.47
+ 0.04
- 0'43
'0.10
+ 0.30
+ 0.36
- 0.82
- 0.72
- 0.65
- 0.76
- 0.54
- 0.42
- 0.68
- 0.37
- 0.21
- 0.21
- 0.04
10.17
: =90°
30"
40
- 0.65
50:
-0.21
603
703
80"
90°
I0O0
- 0.42
0.00
i0.17
+ 0.29
+ 0.37
+ 0.39
- 0.76
- 0.68
- 0.63
- 0.53
+ 0.05
+O.IO
+ 0.09
- 0.47
- 0.45
- 0.46
- 0.91
- 0.87
- 0.84
- 0.84
- 0.77
- 0.72
- 0.72
- 0.74
0.63
- 0.53
- 0.47
- 0.45
-- 0.46
-
+ 0.05
+O.IO
+ 0.09
0.00
+ 0.29
+ 0.37
+0.39
I
3. T h e first term of the right-hand side of
By (3) it may be put in the form
(2)
was now considered.
where E,, is the ellipticity of the Earth's outer surface. Its values for the
various a, and h were therefore easily computed from the material used
in dealing with the second term. It will be noticed that the results are
different from those deducible from the figures given in the paper of Gutenberg and Richter ; * this is because the standard sphere of reference used
by them differs in radius from the sphere of Jeffreys used here.
4. T h e results of Tables I and I1 may now be combined to give the
total ellipticity correction due to equation ( 2 ) . I t will be noticed that for
each case with A =30° the results in Table I are very nearly - 0.71 times
the corresponding results in Table TI. This agrees with the approximation
+
*
Gerl. Beitr. Geophys., 40,T a b l e
I ,
p. 383, 1933.
Mr. K. E. Bullen,
148
4,2
for short distances mentioned in Jeffreys's paper, and shows that below 30°
it is sufficient for the total correction to calculate the first term in ( 2 ) and
multiply it by 0.29. T h e results for h = 10' and 20' were therefore found
in this way. T h e final tables are given in section 7.
TABLE
I1
Corrections t o Traael-times of P Waves due to Departure of Earth's Surface
fyom Standard Sphere (to be subtracted from Apparent Travel-times)
d
A
60"
9oc
120°
1.75
1.64
- I .46
- 1.18
- 0.89
- 0.62
- 0.37
1.34
1.13
- 0.90
- 0.6 j
- 0.41
- 0.27
- 0.23
- 0.99
- 0.25
- 0.28
1.65
-1.75
- 1.71
-1.52
- 1.24
- 0.76
- 0.58
- 0.34
-0.16
-1.30
- 0.58
- 0.30
O0
,\
1500
I 80"
-0.76
-0.49
-0.32
-0.69
-0.21
-0.21
-0.24
-0.40
-0.70
- 1.09
-0.27
1=30
30°
40°
50°
60"
70°
80"
90°
I ooo
I
- 2.25
- 2.1 I
-
2.27
2.18
-~ 1.95
- I .64
I '24
- 0.84
- 0.5 I
- 2.09
-
- 2.01
-
1.71
- 1.42
1.07
- 0.70
- 0.40
-
-
-
-
- 0.73
- 0.50
- 0.30
- 0.22
- 0.25
- 0.37
-0.61
-0.43
-0.27
-0.50
-0.84
- 1.26
=40
40°
-1.84
-1-96
SO0
- 2.01
60"
70"
80"
9oa
- 1.85
- 1.64
- 1.24
-0.84
-0.43
30°
I ooo
-
-0.97
-0.57
- 1.20
-
1.06
- 0.83
- 0.14
- 0.06
10.05
- 0.35
-0.15
+ 0.04
+0.16
+0.19
+0.15
+O.I2
10.22
- 0.42
- 0.05
- 0.57
- 0.98
- 0.84
- 1.26
- 0.05
+ 0.20
-0.23
+0.13
+0.15
1.09
1.28
- 1'35
- 1.27
- 0.28
+ 0.06
+0.18
+0.16
+ 0.05
+O.II
+0.18
f0.12
- 0.07
1'500
30°
40"
50°
60"
70°
80"
9ov
I0OC
1
-
-
- 0.64
-
-
- 0.64
- 0.54
- 0.42
1.30
1.54
169
- 1.64
- 1.52
- 1.24
- 0.84
0.40
-
-
1.09
-0.19
-0.12
+ 0.45
+O.I8
+0.50
10.35
+ 0.48
+0.32
10.60
+0.62
- 0.40
-
+ 0.02
- 0.43
+0.02
+ 0.25
-0.10
10.46
+0.61
-0.49
-0.73
-0.88
-0.93
-0.02
+ 0.47
-0.04
+038
+ 0.70
- 0.79
'0.53
0.00
- 0.84
1.26
=Go"
- 1.26
-
1.35
- 0.84
-0.85
-0.69
-0.37
-0.03
-0.02
+0.05
i-0.19
+0.37
+0.57
+0.76
+ 0.81
+ 0.92
+ 0.99
+ 1.02
+ 1.01
+0.76
+ 0.85
+0.92
+0.90
+ 0.85
+ 0.74
+ 0.57
+ 0.38
10.85
+0.85
+ 0.79
t0.60
+ 0.34
+0.85
+0.81
0.68
- 0.02
- 0.42
- 0.37
-0.71
- 0.84
+
+ 0.37
0.00
-
1-19
1937 May Ellipticity Correction to Travel-times of P and S Waves
O0
30°
60'
+ 0.95
+ 1.37
149
12o0
150"
I 80"
+ 1.13
+ 1.18
+ 1.20
+ 1.13
+ 1.09
+ 1.09
+ 1.02
+ 0.88
+ 0.60
+ 0.29
+ 1.07
+ 0.94
+ 0.68
0.00
- 0.50
= 70'
30°
40°
50°
60"
70°
- 0.06
+0.14
- 0.43
- 0.74
- 0.97
- 1.09
-0.13
- 0.34
- 0.49
- 0.56
80"
- 1.02
- 0.50
+0.56
+ 0.49
+ 0.47
+0.51
+0.60
+ 0.70
90°
-
- 0.28
+0.82
0.84
IOOO
-0.51
30°
40'
50'
60"
70'
80"
90'
looo
+0.45
+O.II
-0.27
-0.58
-0.82
-0.92
-0.84
-0.63
0.00
+0.62
+0.36
+O.II
-0.12
- 0.27
- 0.32
- 0.25
- 0.05
+ 0.98
+ 0.99
+ 0.92
+0.88
+ 0.86
+0.87
+ 0.89
+ 0.97
+ 1.06
+ 1.03
+ 1.13
+ 1.23
+ 1.30
+ 1.36
+ 1.39
+ 1.26
+ 1.33
+ 1.40
+ 1.44
+ 1.52
+ 1.56
+ 1.58
1.59
+ 0.97
+0.32
- 0.07
f0.82
+o.71
- 0.28
- 0.84
- 0.5 I
-
+ 1.28
+ 1.28
+ 1.26
+ 1.20
+ 1.13
+ 1.04
+ 1.17
+ 0.98
+ 1.07
+ 0.77
+0.50
+ 0.49
f0.12
+ 0.97
+0.91
- 0.25
- 0.62
- 0.84
- 0.33
- 0.93
+ 0.99
+ 0.85
+ c.75
+@53
+0.50
+0.18
- 0.21
.
1.06
+0.81
+ 0.22
- 0.27
- 0.05
. =90°
30°
40°
50"
60"
70°
80"
90°
100"
+ 0.85
+0.53
+O.I8
-0.21
+ 0.99
+ 0.75
+0.50
+ 0.23
- 0.53
0.00
- 0.74
-0.15
- 0.23
-0.15
- 0.84
- 0.78
+ 1.24
+ 1.20
+1.17
+ 1.09
+ 1.06
+ 1.04
+ 1.02
+ 1.06
+0.23
- 0.53
0.00
-0.15
- 0.23
-0.15
- 0.74
- 0.84
- 0.78
T h e author had previously prepared similar tables using values of c
taken from Gutenberg's earlier results,* the latter being the most recent
then available to the author. Except in a few places, the two sets of tables
agreed within much less than 0.1 second. Another set of values for c
within the Earth has been calculated by Witte,? and these values have been
confirmed in an investigation by Dahm,f who used similar data to Witte.
T h e differences between these values and those of Gutenberg and Richter
are much less than the differences between the latter and Gutenberg's
1929results. Moreover, the chief source of errors in finding the ellipticity
correction is that due to uncertainties in the variation of c. Therefore it
appears probable that each figure in the final tables found by the author
is correct to well below 0.1second.
5 . T h e ellipticity correction as just computed assumes that epicentral
*
Handbuch der Geophysik, 4, I , 1929.
Nachr. yon des Ges. der Wiss., math-phil. K l., Giittingen, 1932.
1 Bull. Seism. SOC.A m . , 26, 1 - 1 1 , 1936.
t
G I1
Mr. K. E. Bullen,
150
4,2
distances are calculated using geocentric latitudes. I n most seismological
work to date geographic latitudes have been used, and this applies in particular
to the Jeffreys-Bullen travel-times. Therefore in order to find travel-times
for the standard sphere it is still necessary to investigate the difference in
distances as computed with geocentric and geographic latitudes.
Let y and y + dy be the geographic and geocentric co-latitudes of any
point of the Earth's surface. Then
cot ( y + d y ) = ( I - e z ) coty,
where e is the eccentricity of a meridian section of the Earth's surface.
Thus, approximately,
d y = 2e0 cos y sin y .
(9)
With notation as in section
I
we have
cos A =cos a cos y + s i n a sin y cos + 9
where $ is the difference in longitude of E and 0. Thus the correction
needed to derive the distance as calculated with geocentric latitudes from
that with geographic latitudes is dA, where
( -sin A dA+sin
a
cosy d a+co s a sin y dy) sin a sin y
= (cos A - cos a cos y)(cos a sin y d a + sin a cos y dy).
By (9) this becomes
- sin A d a + zeO cos a sin2 a cos y + 2e0 cos a cos y sin2 y
= (COSA - cos
a cos y)(2eo COS' a
+ 2eO C O S ~y ) .
If d A be measured in degrees this reduces to
d A = I 14.6~,{2cos a cos y - cos A(cos2 a + cos2 y)}/sin A.
(10)
T h e results obtained in section 2 . 1 for cos 8' enabled c o s y to be expressed
immediately in terms of a and +, and hence by (10)it was possible to find
d A for the various values of ( a , +, A) considered in the pure ellipticity
correction.
For each value of A the results of the operation just carried out were
dt
multiplied by the corresponding - for P waves. T h e complete process
dA
of this computation was performed so that the maximum errors in the
final results should not be appreciably greater than 0.1 second. As with
the pure ellipticity correction, it was possible to save labour by noticing
that the right-hand member of (10)is an even function of
and is also
not affected when a , are both replaced by their supplements. T h e results
are as shown in Table 111.
6. T h e ellipticity corrections for S waves may be derived to a satisfactory
degree of approximation from those for P waves. T h e ratio of the values
of c for P and S waves at different depths down to the Earth's central core
+
1937 May Ellipticity Correction to Travel-times of P and S Waves
151
TABLE
111
Times to be subtracted f r o m Apparent Travel-times for Distances computed using
Geographic Latitudes, to give Times f o r Distances as computed from Geocentric Latitudes
60"
O0
90°
I zoo
80"
1500
I
+ 0.3
+ 0.3
+ 0.2
+ 0.3
+ 0.2
-0.1
= 30'
10°
20°
30°
40°
SO0
60"
70°
80"
90°
IOOO
+I.I
+I.5
+ 1.9
+ 2.2
+ 2.4
+ 2.2
+ 1-9
+ 1-6
+ 1.3
10'
+0.3
20'
+0.7
30'
+I-I
40'
+1.6
+1.9
+2.2
50'
60"
70°
80"
90'
100'
10°
20°
30°
40°
SO0
60"
7oa
80"
90°
I0O0
I
+ 0-6
+2.2
+2.0
+1.8
+I-6
0.0
+0 - 2
+0.5
+ 1.0
+ 1.4
+ 1.8
+ 1.9
+ 1.9
+ 1.8
+ 1.7
+ 0.6
+ 1.1
+ 1'5
+ 1.8
+ 2.0
+ 2.2
+ 2.0
+ 1.7
+ 1-4
+ 1.0
+ 0.4
+ 0.8
+ 1.2
+ 1.6
+ 1.9
+ 2.0
+ 2.0
+ 1.8
+ 1.5
+ 1.3
+O.I
+ 0.4
+ 0.6
+ 1.0
+ 1.3
+ 1.7
+ 1.7
+ 1.6
+ 1.5
+ 1.4
+0.7
+ 1.1
+ 1.4
+ 1.6
+ 1.7
+ 1.7
+ 1.4
fI.1
+ 0.8
+0.7
+ 1.0
+ 1.1
+ 1.1
+ 1.1
+ 0.9
+0.5
+ 0.2
0.o
+0.5
- 0.2
+0.5
+0.5
+ 0.9
+ 1.2
+ 1.5
+ 1.5
+ 1.6
+ 1.4
+ 1.1
+ 0.9
+ 0.6
+0.3
+ 0.6
+0.8
+ 1.0
+ 1.2
+ 1.2
+ 1.2
+ 1.0
+ 0.9
+0.7
+0.8
+ 0.9
+0.9
+ 0.9
+ 0.7
+ 0.4
+ 0.2
+0.5
+ 0.7
+ 0.6
+ 0.4
+ 0.2
0.0
- 0.5
- 0.4
- 0.8
- 0.2
- 0.9
- 1.2
- 0.4
- 1.1
0.6
- 0.8
- 0.9
- 1.2
- 1.4
- 1'5
- 1.5
- 1.6
-
+ 0.3
+0.4
+ 0.3
-
1.4
- 1.5
+O.I
0.0
- 0.1
- 0.3
-- 0.3
-
0.8
-0.1
-
- 0.2
- 1.2
- 1.4
- 1.5
- 1.6
0.4
- 0.6
-
-
0.8
0.6
- 1.1
- 1.4
-
1.8
1.9
1.9
- 1.8
- 1.7
-
-
- 0.9
- 1.0
- 1.5
+0.4
f0.2
-0.1
+0-5
+O.I
0.4
- 0.7
+0.6
+0.6
- 0.2
- 1.0
-
-
0.4
- 0.6
- 0.8
- 1.5
- 1.6
- 1.7
- 1.9
- 2.2
- 2.2
- 0.9
- 0.9
- 0.9
-
1.7
- 2.0
-0.1
- 0.4
- 0.7
- 1.3
- 1.5
0.0
-0.1
+ 0.6
+0.5
+0.3
+O.I
0.0
-0.1
0.0
-
1.6
-
- 1.5
- 1.4
- 0.3
- 0.7
- 1.1
1.6
1.8
- 1.6
-
=60'
10°
20°
30°
40°
50"
60"
70°
80"
9oa
I0O0
-
- 0.3
- 0.3
-0.1
0.0
+0.2
+0.3
+ 0.8
+1-2
+ 1.4
+ 1.5
+ 1.6
+ 1-6
0.0
+O.I
+ 0.3
+0.5
+ 0.5
+0.8
+ 1.1
+ 1.3
+r.4
+ 1.4
+ 0.9
+ 0.9
+ 1.0
+ 0.9
+ 0.8
+ 1'3
+0.7
+0.7
+ 0.2
+ 0.3
+ 0.4
+ 0.4
+ 0.3
+ 0.3
+0.2
+O.I
0.0
- 0.1
-0.1
- 0.3
- 0.5
- 0.6
- 0.8
- 0.9
-
0.8
-
0.8
- 0.7
-
1.7
- 1.8
- 0.6
- 1.1
- 1.5
- 2.0
- 2.2
-
2.3
1.8
- 2.2
- 1.5
- 1.4
- 1.9
- 1.6
- 1.3
-
- 1.2
MY. K. E. Bullen,
60"
30°
IOO
20°
30'
40'
jo"
60"
70°
-0.8
- 1.1
- 1.1
- 1.1
-0.8
-0.4
- 0.4
0.0
+O.I
-0.5
0.0
i0.2
-0.3
+O.I
+0.2
- 0.2
+0.2
+O.I
+0.2
+ 0.4
+O.I
i 0.4
+o.j
i0.I
+0.7
+ 0.9
+ 1.0
+ 1.1
+0.6
+0.6
i0.6
+0.6
- 0.6
- 0.2
0.0
- 0.8
- 0.8
- 0.7
- 0.6
- 0.2
- 0.2
0.0
-0-2
0.0
-0.1
0.0
-I
0.0
0.0
0.0
0.0
0.0
0.0
+O.I
0.0
180"
- 0.2
- 0.3
-0.5
-0.5
- 0.7
- 0.8
- 0.7
- 0.6
- 0.6
- 0.5
- 0.6
- 1.1
- 1.4
- 1.6
- 1.7
- 1.7
- 1.6
- 1.2
- 1.0
- 0.8
- 0.8
- 0.3
- 0.3
- 0.4
-0.5
-0.5
- 0.6
- 0.5
- 0.4
-0.3
- 0.2
- 0.8
- 0.9
- 1.1
-
- 1'3
- 1.5
- 1.4
- 1'4
1.7
- 1.9
- 1.9
- 1.8
- 1'4
- 1.3
- 1.7
- 2.0
- 2.2
- 2.2
-
1.9
- 1.j
- 1.2
- 0.8
''4
~
80"
i0.3
+ 0.3
+ 0.2
+ 0.3
90'
rooo
+0.6
+0.8
+o.j
+0.3
- 0.9
- 1.3
-1.5
- 1.6
- 1.4
- 1.2
-0.7
-0.3
- 0.8
- 1.0
-0.3
0.0
- 0.3
- 0.8
- 0.9
- 0.3
0.0
- 0.3
- 1.0
-
- 1.2
- 0.3
0.0
- 0.3
- 1.2
-
- 1.2
- 0.4
0.0
- 0.4
- 1.2
-
1.6
- 1.1
- 0.4
0.0
- 0.4
-
1.4
- 0.9
- 0.3
0.0
- 0.3
- 0.6
- 0.2
0.0
- 0.2
- 0.2
-0.1
0.0
-0.1
- 1.1
- 0.9
- 0.6
- 0.2
0.0
0.0
I oo
zoo
30"
40'
500
60"
70'
80"
90°
looo
0.0
+0.3
0.0
0.0
0.0
0.0
+ C'4
+ 0.7
0.0
+ 0.2
0.0
0.0
0.0
+O.I
+O.I
0.0
- 1.1
- 0.8
- 0.5
- 0.3
- 1.0
- 0.6
- 0.3
1.3
1.5
- 1.2
- 0.7
- 0.3
0.0
0.0
+ 0.2
+0 . 3
is very nearly constant, ranging from 1.78 to 1.83. T h e ratios of p for
S and P waves for different A are as follows :A
p-ratio
O0
zoo
30'
40°
50'
60"
70°
80"
1.80
1.81
1.79
1.76
1.83
1.86
1.94
2.02
90'
2.02
105'
1.90
T h u s up to 50' the ratio of p c for P and S waves respectively is very close
to unity. Hence by (3) dr/dQ is very nearly the same for all depths down
to about 1400 km. for both P and S waves. For distances up to 50' the
paths of S waves may thus be taken the same as those for P waves, the
fractional error thereby introduced into the ellipticity correction being seen
to be less than the standard fractional errors in Tables I and 11. T h e
1937May Ellipticity Correction to Travel-times of P and S Waves
153
ellipticity corrections for S for this range of distance (assuming distances
calculated using geocentric latitudes) may therefore be satisfactorily obtained
by multiplying the corresponding corrections for P by 1.8,the ratio of the
values of c.
We now investigate how far this procedure may be justified when A
is greater than soo, and examine first the effect on the first term of (2).
By (8) this term may be put in the form
x
=kz/(r2
-p2c2)/c,
where k is independent of p and c, and
Y
is the radius of the Earth.
Then
From ( I I ) it can be shown that the biggest fractional error introduced into
the first term of (2) in the procedure described is I in 40.
T h e second term of (2) does not admit of quite such direct treatment
in this connection, but when the integrand is expressed in terms of p , c
and r it is found that the error introduced should be roughly of the same
order as with the first term. A few special cases were examined, and showed
that the error would generally be well below 10 per cent. unless the final
result was small, in which case a larger percentage error would not matter.
Gutenberg and Richter * have drawn attention to the fact that the velocities
of S waves in the Earth's interior are as yet known much less accurately
than those of P waves. T h u s in the case of S there are liable to be appreciable errors in dcldr, which would be important in a direct calculation of the
ellipticity corrections. T h u s the procedure described earlier in this section,
if applied for distances u p to IO~', will lead to corrections as good as can
be obtained at present. Table V contains the ellipticity corrections as
thus calculated.
T h e additional corrections required when distances have been calculated
using geographic latitudes may be obtained from Table 111 by multiplying
by the appropriate ratio of values of p for S and P waves.
7. Tables IV and V give the ellipticity corrections for P and S up
to 105' in a form suitable for direct application when geocentric latitudes
are used in calculating distances. l ' h e corrections are to be subtracted
from the observed travel-times. a is the co-latitude of the epicentre, and
4 the azimuth of the observing station. For values of a beyond 90' corrections can be obtained from the fact that replacing both a and (b by their
supplements does not alter the value of the correction,
*
Gerl. Beitr. Geophys., 45, 3 5 0 , 1935.
[TABLE
IV.
Mr. K. E. Bullen,
'54
TABLE
IV
Ellipticity Corrections to Trncel-times of P W u v e s
-
~
;\!
-
30°
60"
180"
IZOO
= 30'
10°
-0.1
-0.1
-0.1
-0.1
-0.1
-0.1
-0.1
20°
- 0.5
- 0.5
- 0.4
- 0.5
- 0.6
- 0.5
- 0.4
- 0.3
- 0.4
-
0.3
- 0.3
- 0.3
- 0-3
- 0.2
- 0.2
- 0.2
- 0.2
- 0.2
-0.1
- 0.2
-0.1
-0-1
-0.1
-0.1
- 0.2
-0.3
- 0.3
- 0.3
- 0.2
- 0.1
- 0.2
- 0.2
- 0-3
- 0.5
0.6
30°
- 0.7
-
40°
- 0.8
SO0
-
- 0.7
- 0.7
- 0.6
- 0.6
- 0.5
60"
0.8
- 0.7
80"
90°
- 0.7
- 0.5
- 0.3
IOOO
- 0.1
70°
- 0.4
- 0.4
- 0.3
- 0.2
- 0.2
- 0.2
- 0.2
- 0.2
- 0.3
- 0.3
- 0.4
-
0.6
- 0.8
- 0.7
- 0.9
= 40
I oo
-0.1
-0.1
-0.1
-0.1
-0.1
-0.1
zoo
- 0.4
- 0.4
- 0.2
-0.1
-0.1
30°
-
0.0'
0.0
60"
-
- 0.6
- 0.6
- 0.5
-0.1
SO0
- 0.7
- 0.7
- 0.2
- 0.2
- 0.1
-0.1
40°
- 0.3
- 0.3
- 0.4
- 0.3
- 0.2
- 0.5
- 0.2
- 0.4
-0.1
-0.1
+O.I
0-2
+O.I
+O.I
0.6
70°
0.6
- 0.6
80"
90°
- 0.5
- 0.3
I ooo
0.0
- 0.5
f0.I
+
0.0
0.0
-0.1
-0.1
0.0
0.0
0.0
0.0
-0.1
-0.1
+O.I
0.0
-0.1
- 0.2
+O.I
0.0
- 0.3
- 0.4
- 0.7
- 0.9
0.0
-0.1
- 0.5
- 0.3
- 0.8
. = 500
I oo
- 0.I
-0.1
20°
- 0.2
30°
- 0.4
4oa
-
- 0.2
- 0.3
- 0.4
- 0.5
- 0.4
70°
0.6
- 0.6
- 0.6
- 0.6
80"
-0-5
0.4
- 0.3
90°
- 0.3
-0.1
SO0
60'
I0O0
1
0.0
-
+O.I
0.0
0.0
-0.1
0.0
0.0
- 0.2
0.0
0.0
0.0
- 0.2
- 0.2
-0.1
0.0
i0.I
+0.2
+ 0.4
+o.r
+0.2
+ 0.3
0.0
0-0
+O.l
+O.I
+O.I
+O.I
+O.I
+O.I
+O.I
+O.I
+o.r
+O.I
+0.2
+0.2
0.0
-0.1
+0.2
0.0
-t 0.4
+0.2
- 0.2
- 0.2
- 0.4
+0.3
+0.3
- 0.4
-
-0.1
- 0.6
- 0.9
0.0
0.7
=60"
I oo
0.0
0.0
0.0
zoo
-0.1
-0.1
0.0
30°
40°
- 0.2
- 0.4
- 0.2
0.0
0.0
SO0
- 0.5
60"
70°
- 0.5
- 0.3
- 0.3
- 0.3
- 0.3
- 0.3
80"
0.6
- 0.6
9oa
- 0.3
I ooo
-
0.0
0.0
+ 0.2
tO.1
+O.l
+0.2
+ 0.3
+ 0-4
+0.5
0.0
+O.I
+ 0.2
+ 0.2
+ 0.3
+ 0-4
+0.5
+ 0.6
+ 0-6
+0 - 5
0.0
f0.I
+0.2
0.0
+ 0.2
+ 0.2
0.0
+0.2
+ 0.2
+0.3
+0.3
+0.3
+0.2
+0.2
+0.2
+O.I
-0.1
+0.3
-0.1
+0.2
- 0.4
- 0.6
+ 0.4
+O.I
+O.I
0.0
- 0.2
- 0.4
- 0.7
- 0.9
1937 May Ellipticity Correction to Travel-times of P and S Waves
O0
60"
90°
+O.I
+O.I
I zoo
155
150"
I no-
+O.I
+O.I
+0.3
+0 . 3
0.3
0.3
t0.3
0.3
+O.l
-0.1
= 70'
1o0
0.0
0.0
20°
+O.I
30°
40°
-0.1
- 0.2
-0.1
SO0
-0.3
-0.1
60"
-
- 0.2
70°
0.4
- 0.5
80"
- 0.5
- 0.2
- 0.2
90°
-0.3
0.0
I0O0
- 0.1
+ 0.2
+ 0.2
+ 0.2
+ 0.2
+ 0.2
+0 . 3
+ 0.4
to.5
+ 0.5
+ 0.6
+O.I
+O.I
+O.I
+O.I
0.0
+ 0.2
to.3
+O.+
+0.5
+ 0.6
+ 0.7
+O.I
+0 . 3
+ 0.4
+0.4
+ 0.5
+ 0.4
+0.8
+0.5
+0.5
0.4
+0.3
+O.l
+O.I
+0.8
+0.8
+
+
+
0.0
+
+0.2
+O.l
- 0.2
-0.1
- 0.4
- 0.3
- 0.6
- 0.4
- 0.7
=80°
10°
zoo
+ 0.2
30°
+O.I
40°
SO0
60"
70°
80"
0.0
- 0.2
-0.3
- 0.4
- 0.5
+ 0.2
+ 0.2
f0.I
0.0
-0.1
-0.1
+O.l
+O.I
+0.3
+0.3
0.4
+0.3
0.5
+0.5
+0.5
+0.5
to.7
+ 0.7
+o.r
+0.3
30°
+0.2
+ 0.3
40°
+O.I
+0.2
60"
70°
- 0.2
80"
90'
looo
-0.5
-0-5
- 0.4
-0.4
+0.8
+O.I
20'
0.0
+0.7
+O.I
- 0.2
50°
+0.6
+ 0.6
+ 0.6
-0.1
loo
+0.5
+0.5
+0 . 3
+ 0.4
+ 0.5
+ 0.9
+ 0.9
+ 0.9
- 0.2
- 0.4
90°
I ooo
+0 . 3
+0 . 3
+0 . 3
+ 0.4
+ 0.4
+O.I
0.0
0.0
-0.1
-0.1
-0.1
+
+ 0.4
+ 0.6
+ 0.6
+ 0.6
+
+ 0.6
+0.8
+ 0.9
t x . 0
+ 1.0
+0.3
+ 0.4
+0.5
0.5
+ 0.5
0.5
0.5
+ 0.5
+
+
+
+ 0.5
+O.I
+0 . 3
+ 0.4
+ 0.4
+0.5
+0.5
+0.5
+ 0.6
+ 0.6
+ 0.6
+O.I
10.1
+0.3
+0 . 3
+0 . 3
+ 0.2
+0 . 3
+O.I
-0.1
0.0
+0.3
+ 0.2
+ 0.I
-0.3
-0.1
- 0.4
- 0.2
- 0.5
- 0.2
-
+O.I
+O.I
+0.3
0.3
+0.3
+
+ 0.2
+O.I
0.6
+ 0.2
+O.I
0.0
0.0
- 0.2
0.0
- 0.4
-0.1
- 0.5
-0.1
- 0.5
- 0.1
- 0.4
[TABLE
V.
Mr. K . E. Bullen,
156
41
TABLE
V
Ellipticity Corrections to Travel-times of S Waves
$
A
O0
60"
I zoo
1500
- 0.2
- 0.6
- 0.6
- 0.5
- 0.2
'\
I
180"
a = 30'
lov
zoo
30°
-- 1'3
- 1.2
40°
-
- "3
- 1.3
60"
1.4
- 1.4
- 1.2
70°
- 1.2
- 1.0
80"
- 1.0
90°
looo
- 0.5
- 0.8
- 0.4
- 0.2
- 0.8
- 1.0
- 1.0
- 0.9
- 0.7
- 0.6
- 0.5
- 0.2
-0.1
- 0.1
-0.1
- 0.3
lo'
- 0.2
- 0.2
- 0.2
zoo
30°
40°
- 0.7
- 0.7
- 0.5
- 0.1
- 0.4
- 1.1
- 0.9
- 1.2
- 1.1
- 1.3
- 1.1
- 1.1
- 0.6
- 0.7
- 0.6
- 0.4
- 0.3
SO0
- 0.3
- 0.9
- 0.2
- 0.9
- 1.1
- 0.4
- 0.4
- 0.3
- 0.3
- 0.4
- 0.5
- 0.8
- 0.2
- 0.2
- 0.2
- 0.4
- 0.4
- 0-3
- 0.3
- 0.4
- 0.5
- 0.8
- 1.2
- 1.7
- 0.5
- 0.5
- 0.3
- 0.3
- 0.3
- 0.4
- 0.6
- 1.0
- 1.5
u =40°
SO0
60"
70°
80"
90°
-0.1
-0.1
- 0.2
-0.1
- 0.4
- 0.2
- 0.2
-0.1
-0.1
- 0.3
-0.1
- 0.9
- 0.5
- 0.2
+0.2
+ 0.2
-t 0.1
+0.3
+O.I
0.0
-0.1
10°
-0.1
-0.1
-0.1
zoo
- 0.4
- 0.4
- 0.2
- 0-3
30'
-0.7
40°
- 1.0
500
- 1.1
- 0.6
- 0.8
- 0-9
60"
- 1.0
-
709
80"
- 1.1
- 0-7
- 1.0
-0.5
- 0.5
+ 0.2
- 0.1
t 0.4
0.0
+0.2
+0.7
I 00°
0.0
-0.1
- 0.4
0.7
- 0.9
- 0.9
- 1.0
0-8
- 0.3
- 0.3
-0.1
0.0
0.0
0.0
- 0.1
- 0.3
0.0
0.0
--
- 0.5
+O.I
fO.1
- 1.0
- 0.6
- 0.6
- 0.6
- 0.5
- 0.5
-0.1
0.0
+ 0.3
+0.2
+ 0.3
+0.5
+0.7
+0.9
+O.I
-0.1
+O.I
- 0.2
- 0.2
-0.5
+ 0.2
0.0
-0.1
0.0
+ 0.1
+ 0.2
+ 0.4
t 0.5
+ 0.6
+ 0.6
+ 0.6
0.0
+ 0.2
+0.3
+ 0.4
0.6
+ 0.7
0.9
+
+
+ 1.0
+ 1.0
+ 1.0
0.0
1
0.0
0.0
+ 0.2
+ 0-3
+ 0.3
+ 0.4
+ 0.4
+ 0.3
- 0.5
- 0.9
-
1.4
0.0
+0.2
+0.2
+0.2
+0.2
f0.2
f0.1
f0.1
- 0.1
- 0.1
- 0.4
- 0.4
- 0.8
- 1.2
- 1.7
- 0.2
- 1.2
+O.I
+O.l
+0.3
+ 0.3
+ 0.4
+ 0.2
0.0
+O.I
- 0.8
+0.5
+ 0.6
+ 0.6
+0.7
+ 0.6
1
- 0.2
- 0.4
- 0.7
- 1'3
-1.7
+O.I
0.0
+ 0.4
-0.1
0.0
+O.I
0.0
0.0
0.0
0.0
- 0.1
- 0.9
- 0.9
- 0.7
- 1.2
IOOO
90°
-0.1
+ 0.4
+ 0.4
+ 0.4
+ 0.2
0.0
f0.1
+ 0.3
+ 0.4
+0.3
+0.2
-0.1
- 0.4
- 0.2
- 0.7
- 0.6
- 1.0
- 1.2
- 1.5
1937 May Ell+ticity Correction to Travel-times of P and S Waves 157
-
d
h
1=
O0
30°
+O.I
60"
180"
90°
70'
+O.I
10°
+O.I
20°
+O.I
+ 0.2
30"
-0.1
+O.I
403
- 0.3
-0.1
soo
- 0.6
- 0.3
60"
- 0.7
- 0.4
+0.5
70°
80"
- 0.9
+0.7
- 0.9
- 0.4
- 0.4
90"
looo
- 0.6
-0-1
-0.1
+ 0.2
+0.9
10"
+O.I
+O.I
20°
30°
40°
+ 0.3
fO.2
0.0
+ 0.4
+0.3
+ 0.2
+0.3
i-0.4
+ 0.4
+ 0.4
+0.8
+ 1.1
+ 0.4
+ 0.6
+ 0.7
+ 0.9
+ 1.0
+ 1.2
+ "4
+ 1.4
t-1.4
+O.I
+ 0.2
+0.5
+0.5
+0.6
+ 0.6
+0.6
+0.8
50"
- 0.3
0.0
60"
- 0.5
- 0.2
+0.7
7oo
80"
- 0.8
- 0.3
+o.x
- 0.9
- 0.3
+0.9
90°
- 0.7
-0.1
IOOO
- 0.4
+ 0.1
+O.I
+ 1.0
+ 1.1
+ 0.9
+ 1.1
+ 1.2
+O.I
+ 0.5
+0.7
+ 0.8
+ 0.9
+ 0.8
+0.9
+ 0-9
+ 0.7
+ 0.6
+ 0.2
+ 0.6
+0.7
+0.8
+0.9
0.9
0.9
+
+
+ 1.0
+ 0.9
i0.I
+0.5
+ 0.5
0.6
+
+0.5
+0.2
0.0
+O.I
+0.5
+ 0.5
+ 0.4
+ 0.2
-0.1
- 0.4
- 0.2
- 0.8
- 0.5
- 1.1
-
0.7
+ 0.2
+0.6
+ 0.6
+0.5
+ 0.4
-
1.3
+ 0.2
+ 0.5
+0.5
+0.3
+O.I
+O,Z
- 0.2
-0.1
- 0.5
- 0.2
- 0.4
- 0.8
+0.9
- 0.4
- 1.0
+ 0.2
+0.2
+ 0.2
+0.6
+0.6
+0.5
+ 1.4
+ 1.6
+ 1.6
+ 1.6
- 1.0
=90°
30"
40°
+ 0.2
+ 0.5
+ 0.4
+ 0.2
50'
-0.1
60"
70°
- 0.4
IOU
20°
80"
90°
looo
+ 0.2
+0.5
+O.S
+ 0.4
+ 0.2
0.0
- 0.7
- 0.8
-0.1
- 0.9
- 0.2
- 0.7
-0.1
- 0.2
+ 0.2
+0.6
+ 0.8
+0.8
+ 0.9
+ 0.9
+0.9
+ 1.0
+ 1.0
+ 1.1
+0.9
+ 1.0
+ 1.2
+ 1.3
+ 1.5
+ 1.7
+ 1.7
+ 1.7
+ 0.8
+0.8
+ 0.9
+ 0.8
+ 0.9
+ 1.0
+ 1.0
+ 1.1
+0.5
+0.4
+ 0.2
0.0
-0.1
+0.2
+ 0.5
+ 0.4
+ 0.2
-0.1
- 0.4
- 0.7
- 0.2
- 0.8
- 0.2
- 0.9
- 0.1
- 0.7