T H E TIMES O F P, S AND SKS, AND T H E VELOCITIES
O F P AND S
Harold Jeflreys, F.R.S.
(Received 1939 March 9)
The foregoing study of Japanese deep-focus earthquakes * has provided
a set of corrections to the tables of Sd, Sr and SKS, whose uncertainties
are checked or estimated by the comparison of different earthquakes, and
some data concerning the loop of the P curve. The distribution of weight
with regard to distance is, however, very uneven for Sr, and the data for
SKS extend only to 100' with any appreciable weight, on account of the
positions of the bulk of the stations. It is therefore necessary to include
the information provided by the normal t and southern 1 earthquakes
already used. In the first case we consider Sr beyond 30' and SKS, since
there are some special difficulties of comparison for Sr between 20' and 30'.
The normal northern earthquakes that are useful at this stage are 1925
June 28 (Montana), 1928 October 15 (Baluchistan), 1929 November 18
(Newfoundland), 1930 October 24 (Marianne Islands) and 1931 May 20
(Azores). 1928March 3 1 (Anatolia) and 1930July 23 (Italy) were also used
The deep shock of
previously, but have not many observations beyond 30'.
1929February I (Hindu Kush) is also not used, because it is already known
to be consistent with the other data, and adds little new information. For
the five retained I kept the epicentres already found, since ellipticity effects
had already been taken into account in determining them. Comparison
of residuals, however, was with the trial tables used in discussing the deep
Japanese shocks. For P the table in question was the one in the 1937
paper mentioned above.
The 2 phenomenon is more troublesome in normal shocks than it has
so far appeared to be in deep ones. In the latter we saw that the variation
of the mean SKS residual from the times calculated with the focal depth
found from P alone was consistent with a slight and permissible increase
of the uncertainty of the latter. Sr had an additional variation, which
could be represented by a constant for each earthquake but varied with
standard error 16.3 in different earthquakes. This is therefore all that
remains of the Z phenomenon in deep shocks not accounted for by variations
of focal depth. There was no trouble with multiplicity in any of the deep
shocks used. In normal shocks, however, multiplicity is common and has
accounted for a number of cases of positive Z such as have been attributed
to high focus.§ Negative 2 could be attributed to focal depth within the
* M.N.R.A.S., Geophys. Suppl., 4,424-460, 1939.
M.N.R.A.S., Geophys. Suppl., 4,225-250, 1937.
t
1 M.N.R.A.S., Geophys. Suppl., 4,281-308,1938.
5 Stoneley, B.A. Rept., 1937 ; M.N.R.A.S., Geophys. Suppl., 4,461-468,1939;
Tillotson, Gerlands Beitr., 52, 377-407, 1938.
Times of P, S and SKS, and Velocities of P and S
1939 June
499
upper layers, but the latter cannot in practice be found from P alone, and
we still have no alternative to treating Z as a complication to be estimated
from the S residuals themselves. The deep foci, however, provide a
standard of comparison for known depths, and enable Z to be eliminated.
As the results from them cover a greater range of distance than in any
single normal earthquake it appeared best to compare the normal earthquakes with them directly.
The uncertainty of such a comparison is in two parts: first, the uncertainty of 2, derived from the comparison of each normal shock with
the results for a surface focus derived from the deep ones, is given by the
total weight of the differences of the means by ranges of distance ; second,
the intrinsic uncertainty of the mean residual in each range of distance.
It is clear that both are relevant; if there are no observations in some
range of distance in a particular earthquake it gives no information about
the correction there ; and if it could happen that all the observations were
in a range where the deep foci had none, a possible correction at that
distance could not be separated from 2. Both must therefore be taken
into account. In previous discussions I have neglected the uncertainty
of 2, combining the earthquakes in such an order that the comparisons
would have the greatest weight possible ; but in this case it needs to be taken
into account because in some of the earthquakes a large fraction of the
observations are in ranges where the deep ones have little weight, and the
uncertainty of Z will considerably increase that of the estimates for these
ranges.
I classified the residuals for the normal earthquakes by ranges of
distance corresponding as closely as possible with those for the deep earthquakes, but for 1930 October 24 the many observations beyond 95" were
classified more closely. A preliminary inspection showed that the weights
used for the deep shocks were suitable. The weights of the values of
Y + Z ranged from 17 to 27 for Sr and from 3 to 17 for SKS. The lower
weights are not worth keeping; 1930 October 24 gives the only useful
contribution for SKS. Y + 2 was subtracted from each mean by distance,
and allowance was made for its uncertainty ; then the results for the normal
earthquakes were combined. The results were :
-
Sr
A
A
--
-With Deep
Shocks
Weight
323
Mean
- 0.3
40
-0.2
22
-0.4
+0.2
29
+O.I
I0
20
-0.7
13
-0.7
-0.7
- 1-3
+0.8
0
50
0.0
60
-0.1
70
80
90
973
+o-2
-0.5
+ 1.1
4
2
- 1-3
4
I023
-0.1
1053
+0.6
9
5
I00
Mean Weight
-0.1
-0.1
+0.6
15
26
44
33
74
123
72
4
9
9
5
With Deep
Shocks
SKS
A
a
85
91)
I00
II 0
I20
7-
7
-
Mean Weight
-0.3
13
-2-4 50
Mean
-0.8
-2.3
Weight
-1.5
-1.5
13
-1.4
6
-1.8
7
+0*5
2
3
6
31
Dr. Harold Jeflreys,
500
4,7
Sr for the comparison of normal and deep shocks gives x2 = 8.0 on 7 degrees
of freedom ; SKS gives 1.6 on 3 degrees of freedom. Applying the corrections to the means by ranges for the separate earthquakes, and redetermining
Y 2, I obtained the following values of x2 :-
+
xp
Degrees of Freedom
1930 OCt. 24
1931May 20
1925 June 28
I929 NOV.18
1928 Oct. 15
10.7
8.0
9
5
4
5
2.5
2
Total
29.9
2.8
5.9
25
The consistency of the data is therefore quite satisfactory. It may be
recalled that the contributions to x2 for Sr in the deep-focus earthquakes,
after allowing for additive constants for the former, were 45.3 on 44 degrees
of freedom. The randomness of the outstanding variation is therefore
adequately checked. There is no evidence for a variation of the form of
the curves for different earthquakes or different regions.
The values of Y and Y +Z for the various shocks are as follows. The
origin of time in each case is to as given in the I.S.S.
Y
Y + Z (SKS)
Y+Z(S)
- 5.0 f 1.0
- 7.2 f 0.4
I930 OCt. 24
1931May 20
1925 June 28
1928Mar. 31
1928Oct. 15
1929 Nov. 18
+
+
+
8-6f 0.32
4.2 i 0.6
7.5 i 0.3
+14-2f 0.5
- 10.0* 0.53
- 10.6 f 0.49
- 12.4 f 0.61
+
8.9 i 0.48
+ 1.9& 0.88
+ 7.2 * 0.63
+ 10.0 f 0.51
We must recall that the times of S need a reduction of 16.7 to make them
comparable with the 1937 P table for a surface focus, and SKS one of
18.6. 18.7 and 18.6 respectively should therefore be added to 2 ; the
standard error of 2 is u ( Y + Z ) f o&(Y). The nearest approach to a
convincingly positive Z is for 1925 June 28, and it is still consistent with
the unexplained variation found in the deep earthquakes.
With the exception of 1928 March 3 1 these earthquakes had few S
observations within 30'.
On examination it was found that the residuals
were consistent with corrections decreasing by about 48 f 18 between 20'
and 30°, which is what the deep shocks suggest. Between 18" and 20'
there was some sign of a concentration of readings about 48 after the expected
Sd, which might refer to a cusp in the time-curve, but they were not enough
to give a useful determination. For 1928 March 3 1 the uncorrected
residuals between 20' and 25' gave the following summary :-4
+I
+2
+3
+4
+5
+6
I
I
0
I
0
I
3
+7
3
+8
+9
I
I
The weighted mean is + 68.1 08.8.
This is +48.2 f 18.2 more than the
mean from the observations beyond 30°, and confirms the suggestion that
Times of P, S and SKS, and Velocities of P and S
1939 June
501
The residuals from 16"-5
a positive correction is needed in this range.
to zoo are as follows :-8
-6
-2
-I
+2
+3
+6
I
I
I
I
I
3
I
+8
3
+18
I
The weighted mean is +48.1& 08.9, whereas the deep foci have suggested
slightly negative corrections. But the early readings, which contribute
little weight to the mean, may be the true Sd. The residuals -2, - I and
+ z occur between 19' and 20". There is some suggestion that the bulk
of the readings refer to an upper branch about 2 s after the trial time and
extending back to about 17". On the whole the normal earthquakes add
little to the determinations within 30°, but so far as they go they appear to
be consistent with the interpretations suggested by the deep ones.
The residuals of SKS in the southern earthquakes were against a
different trial table, and the difference must be allowed for. They were
also at 5" intervals, whereas the present ones are mostly at 10'. This was
treated by placing half the weight at IO~', for instance, at IOO', and half at
I 10' before estimating the mean difference between the northern and
southern earthquakes. This was allowed for, but in combining the data
the southern values were left at the original distances. The adapted
southern values (save for the additive constant) and the combined ones are
as follows :A
Southern Mean
8
8
Total
8
8
I00
-2.0 f 0.5
105
- 1.9 f 0.5
- 1.72t 0.7
-3.1 i 1.2
-2.1 f 0.8
-0.5 2t 0.45
-0.5 f 0.5
- 2-4i 0.29
-0.8 f 0.47
- 1.5 f 0.29
- 1.6 f 0.5
- 1-6 f 0.5
-2.8
1.2
- 1.3 f 0.66
12.5
130
+0.2 2t 1.0
+0.5 f 1.0
I35
-2.2
140
- 1.02t 2.6
90
- 1.2f 0.7
-0.8 f 0.5
9If
95
-1.1
II0
115
I20
...
fO.47
-0.9 2t
1.5
* 1.6
*
-0.6 f 1.5
- 1.9 f 1.6
-0.7 f 2-6
The constant difference was - 08.3 08.3, and the standard error should be
combined with those of values derived wholly from the southern earthquakes,
but it is small enough not to matter.
Apart from an additive constant, 18.0 having to be added to the entire
P table," the times of P now need modification near zoo. The first arrivals
need no correction, but before we can determine the distribution of velocity
we must reconstruct the loop. This cannot be done from the observations
* The assumption that the changed estimates of the thicknesses of the upper
layers can be treated by simply adding a constant to the P times supposes that the
previous estimate of 10km. below the surface for the average depth of the earthquakes used in forming the P table is correct. There is no apparent reason to alter
it, but a check would be desirable if one can be obtained.
4, 7
Dr. Harold Jeflreys,
502
alone, since we have none on the uppermost side of the loop. But we can
set limits to the range permitted. In the limit, when there is a true discontinuity, the uppermost side of the loop becomes the time-curve for
waves reflected at it, and in such a case the curve must have 3-point contact
with the tangent to the Pd curve at its end. By analogy with PcP and
with some cases already worked out numerically for various types of crustal
conditions it seems likely that this branch will be nearly straight,* and its
curvature will be upwards.-f Again, in the extreme case, Pd, as calculated,
should run on to about 25" and Pr back to about 1 4 ~ . In the actual case
these limits will be brought nearer to 20". But we have observations
corresponding to Pd up to 22" and Pr back to 17", and the amplitudes
suggest that little extension of this range is permissible. The numerical
fitting of a suitable curve with two cusps appeared to be too complicated a
problem, and I proceeded graphically. The function
to=4m
08.0
+ II(A - 17")
was subtracted from the observed times, leaving the following :Pd
A.
0
0
18
+IG
17.5 +3*4& 0.7
19.5 +3.0 f 0.5
20
+3.7 k 0.3
+ 5 . 1 &0-4
21
+2*2
22
+I-4
17 -0.3
22
Quadratics were first fitted to these data, but it was found that an upper
branch connecting the values for Pr at 17"and Pd at 22" would either have
to be convex upwards or intersect one of the lower ones. With a permissible
readjustment of the extreme values, however, this additional condition
could be satisfied, and the results, read from a graph, are as follows :Pd
0
Upper Branch
8
17 -0.3
18 + I - Z
19 +2.5
20 +3*7
21 +4-6
22
+5'3
0
21
20
8
+4.8
+44
19 + + I
18 +3.9
17 +3.8
A.
0
8
18 +3.6
I9 +3'3
21
+2*8
+2*2
22
+1.4
20
The intersection of Pd and Pr is at 19O.4, for which both give +38-0, SO
that the upper branch there is 18.2 later. The differences of the times on
the three branches are small, and a new onset would be so soon after the
first that it would usually be regarded as a part of the same movement.
The intervals beyond 20" are, however, reasonably consistent with those
found by Lehmann.
To adapt to a surface focus, with the modified thicknesses of the upper
layers, all P times must be increased by 1 8 . 0 , since the times at short distances
will now be 68.8 + 14.28A instead of 58.8 + 14-28A. T o be comparable with
* M.N.R.A.S., Geophys. Suppl., 4, 224; Lehmann, M.N.R.A.S., Geophys.
SUPPL 4,250-2719 1937.
t M.N.R.A.S., Geophys. Suppl., 3, 201-202, 1934.
1939 June
Times of P, S and SKS, and Velocities of P and S
503
the new P times, all Sr times must be decreased by 08.7 instead of 18.7, and
SKS by 08.6 instead of 18.6.
Applying the corrections to Sr to the trial times, we have the following :A
t
m
a
*
*
a
8 18.9 0.6
9 49.6 0.7
1 1 40.6 k 0.6
13 43.8 * 0 . 5
16 9.0 f 0.4
18 22.0 & 0.4
20 26.0 0.3
t
80
m
22
90
23
974
24
25
0
I00
I024
1054
*
26
a
a
16.9f 0.3
54.5 0.3
58.6
21-6
41.6
7.1
*
We already know that the times are expressible within a few seconds by a
quadratic over the entire range. The last four values do not determine
a significant curvature, and were replaced by summary values 25m 208.7 & 08.6
at 100" and 26m 28-9& 08.8 at 105' ; the linear solution gave xa= 1.8 on
2 degrees of freedom in this range. The rest were treated by first interpolating a value at 30" and differencing at 10"intervals. Where the fourth
difference suggested an increase or decrease of any value, it was first altered
by its standard error. The results were then smoothed by subtracting
of the revised fourth difference * ; since the general level of the table has a
standard error of 08.7 a more elaborate method did not appear worth while.
Interpolation was then fairly easy. The strong curvature about 25",
however, is confirmed.
For Sd it was found that the formula
l! =
10.4 -k 25.418 - 0.003 I4(A - I)3
fitted all the values within about their standard errors. It agrees with
Sr at zo0.8-that is, 1O.4 further on than for P. Extrapolating a little, and
assuming that the loop extends about as far from the point of crossing as
that for P, we should expect it to run from about 18" to 23". On this
hypothesis we can make a rough reconstruction of the loop. The complete
S table is then as follows :Upper Branch
Sd
o
m
a
o o 10.4
2
I
1.2
52.0
4
5 2 17.3
6
42'5
8 3 32.6
I 0 4 22-2
I2
5
15.1
I4
59.2
15 6 22.9
o
m
B
19 8 0.8
20
20.7
21
40.5
22 g 0.6
23
21.4
Sr
o
18
19
20
21
22
23
24
25
26
27
m
8
7 41.2
8 0.6
19.6
38.3
56.6
9 14'4
31.8
48.9
10 5.6
22.1
* The chief disadvantage of.this method is that it ordinarily makes smaller
corrections than are really needed-X2 for the original values against the corrected
ones is systematically too small. But the preliminary alteration by the standard
errors corrects this and also makes allowance for the weights.
Dr. Harold Jeffreys,
504
4, 7
Sd
o
m
Sr
m
s
a
16 6 46.4
I7 7 9.5
I8
32.3
I9
54.9
20 8 17.1
21
38.9
2i
35
12 28.2
g
40
I 3 44.5
22
23
0.3
21.4
29
38.3
54'3
30
1 1 10.1
25.7
41.2
31
32
45 I 4 57.9
50 16 8-6
5 5 17 16.8
60 18 22.6
65 I9 25.5
70 20 25-6
75 21 22.6
80 22 16.5
85 23 7.3
54.5
90
95 24 38.2
IOO
25 20.4
105 26 2.1
107
18-7
The Sr table is taken to 107' because some observations in the earthquake
of 1930 October 24 were at nearly this distance.
We appear now to have reached the limit of accuracy that is attainable
for P, S and SKS until special attention has suggested explanations of
various minor departures from independence of the errors. These are as
follows. Occasionally a strong group of stations gives a mean P residual
larger than the apparent accuracy of the solution for the elements of the
earthquake and its own internal consistency would suggest. This is shown
most clearly by the evidence that the correlations between the mean residuals
of other phases in the deep earthquakes implied errors in the focal depths
found from P, which were systematically about 1.9 times larger than the
standard errors found on the hypothesis of the independence of the errors.
There are also sometimes peculiar mean residuals over special ranges of
distance. For instance, the deep shocks suggested a mean for P of
- 0 s . 5 f 08.3 corresponding to surface distances from 20' to 2 5 O , and there
are normal shocks with similar negative residuals in this range ; but the
Marianne Islands and Azores earthquakes gave means of about + 18, the
former depending mostly on the same stations as the deep shocks. None of
these discrepancies would be significant by itself, but the evidence about
errors in the focal depths indicates that they must be taken seriously, and
my impression is that they occur oftener than the hypothesis of randomness
would imply. The differences are much less than some that have been
asserted by various authors concerning differences between earthquakes in
different regions and different earthquakes in the same region, such as
have been denied by Gutenberg.* But differences of the order of 18 in P
are not out of the question. The Pacific times seem to differ systematically
*
Bull. Seism. SOC.Amer.,
27, 337-347, 1937.
1939 June
Times of P, S and SKS, and Velocities of P and S
505
from the continental ones by amounts varying about 1 6 . 5 with distance.
For the S waves the outstanding anomaly is the variation of the mean Sr
residual after focal depth has been allowed for as accurately as possible.
This corresponds to a standard error of about 16.3 common to all readings
of the same earthquake, and is not shown by Sd and SKS ; and it is not
associated with any systematic variation of the form of the Sr curve, apart
from its general level. The smallness of the constant terms in the solutions
for the S waves, in comparison with the P ones, in near earthquakes is also
now unexplained. Various suggestions have been made and examined in
ways that should provide adequate checks, but none have been found
satisfactory. The most hopeful one had appeared to be that S may appear
to leave the focus about 28 before P, on account of some failure of the usual
elastic equations at the focus ; but this does not explain the smallness of
the sS-S residuals in comparison with the pP-P ones in the deep shocks.
This list of anomalies does not include late readings of P where it is weak
or misidentifications of S, both of which have, I think, been avoided: for
P, by using only the best stations at great distances in well-observed earthquakes ; for s, by using only earthquakes where the regular variation of
the residuals against the trial times left no reasonable doubt that the same
thing had been read as S at all distances. The former point is illustrated
by Dr. A. W. Lee's recent study of the P residuals in the I.S.S. for 1930-31."
He gives the median and the upper and lower quartiles for different ranges
of distance. These indicate a nearly symmetrical law of error up to about
92O, but beyond that the intervals between the first quartile and the median,
and between the median and the third quartile, are very different, indicating
a long tail of late readings, and a skew law of error that would give a systematic error if the mean was taken as the estimate. But when only stations
of reliability 0.8 and more are used the law is again practically symmetrical,
with about the same standard error as at shorter distances. Inclusion of
the less reliable stations is probably the explanation of the fact that some
tables, notably that of Dahm, give much larger values of dt/dA at large
distances than mine does ; I took this precaution when I first applied the
ellipticity correction to P, because difficulties at these distances had already
been found. The existence of the large movement often taken as S between
15' and 20° also remains unexplained. Lee gives medians and quartiles
here that would imply a " probable error'' of one observation of about 108,
which would be entirely impossible if two competent observers agreed
about what to read. The only reasonable interpretation is that several
different movements are taken as S in this range by different observers using
different types of instrument. It appears that the former period of about 25s
of the ,horizontal instruments at Kew made them unsuitable for detecting a
new movement of short period, such as was read by Durham in the earthquake of 1930 July 23 and confirmed by me, when there is a previous
disturbance-just as they gave less complete information than the Wiechert
and Milne-Shaw instruments did in the Jersey and Hereford earthquakes.
Lee recognizes the fact that independent evidence is needed to test which
* Meteor. Ofice, Ge@hys. Memoir, 76, 5, 1938.
G 34
Dr. Harold Jeffrey,
506
49 7
of the various movements is to be identified with the theoretical S, which
is the only one relevant to estimates of the velocities. The Kew period
has recently been shortened. Cf.Lee, Meteorological Ofice, Geophysical
Mem., 78.
I think therefore that the time has come when the velocity distributions
for P and S should be computed and an estimate of the core attempted
from the times of PcP and ScS. I n view of the above difficulties, no
finality can be claimed for such a solution (and indeed I have been insisting
for about twenty years that the claim of finality for any scientific inference
is absurd *) ; but it now appears that further progress must be directed to
the solution of these difficulties, and will be possible only if a consistent
standard of comparison is available. A discrepancy suggested by a single
earthquake is usually capable of two or three interpretations, and a decision
between them cannot be made without comparison with other material ;
and such a comparison will be invalid unless the standard used is selfconsistent. It will also be invalid if the method of reduction is such as
not to consider the possibility that the standard may be right. It is usually
possible, by choosing the elements of an earthquake suitably, to make the
times appear to agree with a number of possible tables. Evidence that a
table is wrong can arise only if it is shown that they are not consistent with
that table. Any departure from it must be estimated by the most efficient
methods (that is, by inverse probability or maximum likelihood), given
with its standard error, and tested by the xa or some more detailed test of
significance, before it can be said to be anything but a random variation.
Standard errors, or weights permitting their determination, should be given
to two figures ; in a significance test very different conclusions might be
reached according as an estimate of 0.4 0.2 means 0.35 f 0.25 or
0.45 1.0-15. The use of “ probable errors” should be abandoned, because
they merely involve unnecessary arithmetic, which has to be undone before
any modern significance test can be applied ; and when a standard error
has to be estimated from the scatter of a few observations the assumption
that the error with an even chance of being exceeded is 0.6745 times the
estimated standard error is seriously wrong-the true value has an even
chance of lying between the first two observations, and the probable error
is then equal to the standard error.
It seems to be necessary to insist again that a standard error is not a
final claim of any specified degree of accuracy. It is always subject to the
condition that no unforeseen complications may be present ; but the onus
of proof is on the advocate of such complications, and they must be stated
in a form leading to verifiable inferences before they can be discussed.+ So
long as none is demonstrated by a significance test, the standard error as
given remains presumably valid, and it is a necessary part of any test. It
*
t
Cf. Nature, 141,672 and 716,1938.
Otherwise we are in Alice’s position in her conversation with the Cheshire
Cat : “Which way ought I to go to get from here ? ” “That depends a good deal
said
on where you want to get to,” said the Cat. “ I don’t much care where-,”
Alice. “Then it doesn’t matter which way you go,” said the Cat.
1939 June
Times of P, S and SKS, and Velocities of P and S
507
is not desirable to increase the standard error to try to guard against the
risk of such complications. If they are known and can be calculated, the
standard error will remain applicable to the corrected estimate. A correction
of + 2 applied to an estimate o f I will give + 2 f I ; it will not give o =k 4 5 .
The latter would often make it quite impossible to estimate a departure
actually present. All that can ever be claimed for a scientific inference is
that it is the best on the information available ; and this does not apply to
estimates of uncertainty based on personal impressions and not on actual
calculation from the observations, for these are often grossly too large or
too small. Even if we do our best it is inevitable that mistakes will sometimes
occur ; but they will be the sort of mistakes that lead to new advances when
they are corrected, and they will contain in themselves part of the means
of testing whether they are correct or not.
It is, of course, legitimate to mention possible complications, especially
if means of testing them can also be suggested. What I am objecting to
is ( I ) the use of inefficient methods of estimation, which often lead to
'results that are definitely inconsistent with the observations that they are
alleged to be based on, (2) ignoring the known uncertainty indicated by the
scatter of the observations themselves, which is a minimum uncertainty
(3) attempts to allow quantitatively in advance for effects that we have so
far no means of estimating.
The present P, S , PKP and SKS tables all rest now on observational
material through their entire course ; it would be impossible to alter any
of them to give better agreement with one old or new set of observations
without making agreement worse with another. Though we have apparently
reached a stage where further improvement is impossible for all earthquakes,
since small departures from independence of the errors are beginning to
reveal themselves, further development is unlikely to lead to the substitution
of any single set of tables for the present ones. It is more likely to lead to
the introduction of minor corrections, not always with the same sign, that
can be applied to the present tables in specified circumstances. Accordingly
the velocity distributions indicated by them are likely to be as good general
rules as are attainable, and should lead to results that will permit a test of
particular departures when occasion arises. It is particularly desirable
that the radius of the core should be re-estimated from PcP and ScS, since
most estimates yet made depend on the accurate location of the edge of
the shadow zone of P. This in itself is difficult, and the depth reached
by the corresponding ray has a further uncertainty on account of the
difficulty of estimating the gradient of an empirical curve at its very end.
A more accurate method is greatly needed to improve our knowledge of
the times in the central core itself, and this should be provided if suitable
series of observations of Pep, ScS, PcPPKPand ScSPKPare obtainable. Some
other core phases such as ScSP and ScSSKS may be useful if special attention
is given to them ; I have tried to use SKKS, but it is rather unsatisfactory.
The procedure in computing the velocities is similar to that already
adopted. The allowances for the upper layers, now taken as having thicknesses 15 and 18 km.,are as follows :-
Dr. Harold Jeffreys,
P
dtldh
Upper (One Passage)
S (")
4 (sea.)
14.3
I4
0.14
0.13
1.9
1.9
I3
0.12
2.0
I2
0.10
0.09
0.08
0.07
0.06
2.1
I1
10
9
8
7
6
5
0
S
dtfdA
Total (Two Passages)
T'
2.3
2-4
2.5
0.05
2.5
2.6
2.6
2.7
0.03
0.00
Upper (One Passage)
6 (")
T
' (secs.)
~~
~
0.17
10.5
0.00
10.9
Total (Two Passages)
.
2.8
2.9
24
23
0.13
3-2
22
21
20
0.12
0.11
0.10
0.09
0.08
0.06
3.3
3 *4
0.60
0.55
3.6
3.7
3.9
4'0
0.50
0.05
4.1
0.25
0.04
0.03
4'2
43
0.02
4.4
4.5
0.21
0.16
0.12
18
16
I4
I2
I0
8
6
0
0.00
3'1
>
~~~
0.16
0.16
0-14
25
10.3
0.21
6 (")
0.90
0.82
0.73
0.66
25.4
(secs.)
6.8
7.0
7.7
8.2
8.7
9.2
9.5
9.8
10.1
2.2
0.04
~~
4, 7
0.42
0.36
0.30
0.00
T'
(sea.)
10.7
11.0
11.8
125
13-1
13.7
14.2
15.1
15.9
16.6
17.1
17.6
17.9
18.2
18.5
Comparison of these results shows that values of 2 reaching - 18.7 could
be explained, for continental earthquakes, by foci at the base of the upper
layer, and - 38.8 by foci at the base of the intermediate layer. The appropriate modification for oceanic earthquakes is still unknown.
Subtraction of T' for the double passage gives times to corresponding
distances in the lower layer, with a second-order error, and the velocities
can then be calculated by the Herglotz-Wiechert method. It should be
noticed that the existence of a loop in the time-curve does not invalidate
the method, provided that the loop can be observed or reconstructed. The
following simple derivation of the solution was communicated to me
privately a few years ago by Dr. G. Rasch. Taking the integral equation
in the form
multiply by dp/z/(p2- p a ) and integrate from p = p to Rlc,.
Then
1939 June
Times of P, S and SKS, and Velocities of P and S
509
in which the factor after the first sign of integration in the latter integral
is independent of r ) and can therefore be placed after the second sign.
Changing the order of integration, the limits for p become p to r), and those
for r ) from p to Rlc,. But if r) > p
and therefore
(2) is
equivalent to
the value of r to be taken in the last expression corresponding to r/c =p.
Thus we have the Herglotz solution
Even if there is a loop and A therefore is not a monotonic function of p ,
this will still hold provided that r) always decreases with decreasing r. It
will fail if for some range of depth r ) increases with decreasing I , since then
for p in this range there will be values of r ) for smaller depths that do not
satisfy r ) > p. This is the case where there is a shadow zone. Further,
the Wiechert transformation
p = p cash q
(6)
has no branch-point within the range of integration (though it has one at
the end) and the result
is valid provided that dA is reckoned negative on the upper branch. It
merely becomes necessary to integrate round the loop. For distances on
the Pr branch where it is earlier than Pd, for instance, the integral will go
right round it. It is of some importance to observe that if the loop was
removed and the Pd and Pr times were reduced in each case to the first
arrivals, we should have a possible time-curve, with a discontinuity of slope
but no gap. The difference between log (R/r)for such a curve and for the
curve with a loop is just the integral round the loop. What I have called
the " continuous solution " is a close approximation to such a reduced timecurve, and differences from the results for it will indicate the degree of
importance of the loop. The contribution from the loop is always negative,
so that its presence makes the rays penetrate less deeply. At large distances
its effect is practically negligible. The reason for this is easily seen, because
if q was of the form a + bp its integral would be the difference of aA + bt at
the intersection and would therefore vanish. The only reason why the
Dr. Harold Jeflreys,
49 7
contribution is not zero is that q is not a linear function of p ; but the
larger A is, the more nearly linear the relation becomes within a specified
range of p. But just beyond the intersection its contribution, in the
limiting case of a true discontinuity in the velocity distribution with depth,
TABLE
I
A
P
0
14-28
14.13
13-68
12.93
12.74
12.5
5
I0
I5
16
I7
I8
I9
20
21
22
17cusp
20 Pr.
21
22
23
24
25
30
35
40
45
50
55
60
65
70
75
80
85
90
95
I00
105
12.3
12-14
11-94
11.7
11.6
10.95
10.4
10.2
10.0
9.8
9.65
9-50
8.88
8.53
8.28
7.99
7.64
7.24
6.84
6-44
6.08
5.72
5.33
4'99
4.69
4.55
4.49
4.42
Calculation of Velocities of P
log RJr
*JRP
rJR
0.07003
0
I .oooo
0.00317
0.9968
0.07054
0.07217
0.01277
0.9873
0.02970
0.9707
0.07507
0.07582
0.9659
0.03407
0.07690
0-96I 3
0.03955
0.0778
0.04429
0.9567
0.04782
0.9533
0.07853
0.05292
0.07944
0.9485
0.08053
0.05960
0.9422
0.08098
0.06247
0.9394
0.08514
0.07013
0.9323
0.08876
0.07986
0,923I
0.09008
0.08470
0.9188
0.09142
0.08970
0.9142
0.9093
0.09279
0.09504
0.09382
0.9054
0.09939
0.9012
0.09486
0 -10407
0.8806
0.12713
0.09917
0.8658
1.0150
0.14412
I .0289
0.8519
0.16026
0.8340
I a0438
0.18153
I .0620
0.8114
0.20905
1.0838
0.2425I
0.7847
1.1062
0.27885
0.7567
1.1293
0.3I843
0.7273
1.1498
0.6991
0.35798
1.1708
0.6697
0.40096
1.1932
0.6360
0.45253
1.2132
0.6054
0.50194
I a296
0.5767
0.55037
0.5616
1.2343
0.57702
1.2336
0.59086
0.5539
1.2301
0.60936
0.5437
w-ii
7.75
7.80
7.98
8.30
8.39
8.51
8.61
8.69
8.79
8-91
8.96
9.42
9.82
9.96
10.11
10.26
10.38
10.49
10.97
11.23
11-38
11-55
11-75
11.99
12-24
12-49
12.72
12.95
13-20
13.42
13.60
13.65
13.65
13.61
but with the same first arrivals, represents the whole difference between the
continuous and discontinuous solutions. Pr rays emerging at the same
distance in the latter will have penetrated less deeply.
The solution for the velocities of P is given in Tables I and 11. R is
the revised radius of the top of the lower layer, 6338 km., and the length
of a degree at this level is 110.62 km. The values of c are given in km./sec.
The first differences of c against r/R suggest a sudden change of dc/dr
(not of c ) at about r=o.gqR, corresponding to the cusp at 22'.
On this
interpretation it is, of course, the strong refraction just below that level that
Times of P, S and SKS, and Velocities of P and S
1939 June
51 I
produces the upper branch of the loop in the time-curve. No insistence
can be laid on the velocity distribution corresponding to values of p in the
loop, but it may be noticed that for Pr at 20' the whole contribution from
and at 105' is - O - O O O ~ . Having regard to
the loop to log R/r is - 0-0018,
TABLE
I1
Velocities of P
Solution for Loop
From
smoothed c
Solution for Continuous Case
-
I
A
r/R
P
rlRp
1.00
0
14.28 0.07003
0.99
8.88 13.81 0.07169
0.98 12.44 13'35 0.07341
0.97 15'15 12.90 0.07519
0.96 17.27 12.4 0.0774
0.95 19.72 12.00 0.07917
0.94 21.72 11.6 0.0810
0.93 18.00 10.8 0.0861
0.92 20.73 10.3 0.0893
0.91 22.85
9.8 0.0929
0.90 25-27 9.46 0.09514
0.89 27.61
9'14 0.09737
8.86 0.09932
0.88 30.18
0.86 37.15 8.42 0.10214
8.09 0.10383
0.84 43.45
0.82 48.21
7.77 0.105 53
0.80 52.20
7-47 0~10710
0.78 55.86
7'17 0.10879
0.76 59,42 6.89 0.11030
0.74 62.84 6.61 0.1 I195
0.72 66.29 6-35 0.11339
0.70 69.84 6.09 0.1 I494
0.68 73.30
544 0.11644
5.61 0.1176
0.66 76.45
5.38 0.1190
0.64 79'41
0.62 82.56
5'15 0.1204
0.60 85.85
4'93 0.1217
4'72 0.1229
0.58 89.35
0.56 95.92 4.54 0.1234
0.55 101.9
446 0.1233
'
0.54
0.53
0.52
(smoocthed) rJRp
7.75
7.747 7003
7.936 7174
7.93
8.12
8.131 7350
8-32 8.332 7532
8.56
8.539 7719
8.76
8.752 7912
8.96
8.971 8110
8588
9'50
9.52
9.88
9'91
8959
10.28 10.26
9275
10.52
10.55
9537
9736
10.77 10.77
9935
10.99 10.99
11.30 11.29 10206
11'50
10396
1 1 '49
11.67 11.67 I0550
11.85 10712
11.85
12-03 12.03 10875
11029
12.20
12.20
12-38 12.37 11191
12.54 12.54 11354
12.71 12.71 1 I490
12.88 12.87 11634
13.01 13.02 11770
13.16 13.16 1 1 897
c
13-32
13.46
13.60
13.65
13.64
P
14.28
13-80
13-33
12.88
12.44
12.01
11.59
10.83
10.27
9.81
9.44
9.141
8.858
8.427
8.080
7.773
7.468
7.172
6.891
6.612
6.341
6.092
5.845
5.607
5.380
12041 5.149
13.32
13.46 12168 4'931
13-60 12294 4718
4.541
13.64 12331
4461
13-64 12331
4'379
12331
12331
4.298
12331
4.217
A
P
14-28
8.91 13.81
12.82
13'37
15.23 12.91
17.22 12.45
18.00
11.95
I 8.89 I 1.41
20'05 10.88
21.21 10.38
22.61 9-91
9'49
24.67
27.28
9.12
8.88
30'34
8.4
36.82
8.09
43'13
48.00 7.78
7.47
52.13
7.18
55'91
6.89
59.46
62.94 6.62
66.42 6-35
69.88 6.10
5.85
73.21
5.63
76.37
5-38
79.45
5.16
82.62
85.93 4.95
89.57 473
95.98 4.54
102.35
447
0
rlRp
C
0.07003
0.07169
0.07330
0.07514
0.07711
0.07950
0.08238
0.08548
0.08863
0.09183
0.09484
0.09759
0.09910
0.1019
0.1038
0.1054
0.1071
0.1086
0.1103
0.1118
0.1134
0.1148
0.1162
0.1174
0.1190
7.76
7-94
8.12
8.32
8.54
8.80
9.12
9.47
9.81
10.17
10.50
10.81
10.97
11.28
11.49
11.67
11.86
I 2.03
12-21
12.38
12-56
I 2.7 I
12-87
13-00
0.1202
13-18
13.31
0.1212
13.42
0.1226
0.1233
13.58
13.65
13.62
0.1230
the uncertainties of the times on the loop, these may be taken as having
standard errors of about a quarter of their whole amounts. The contribution
from the uncertainty of the loop is therefore negligible in comparison with
that arising from the value of dt/dA at the end of the range of integration,
which gives a standard error of about 0.009 in log R/r at 105', and about
0.005 at 22'.
It is important only for values of r corresponding to points
actually on the loop. A closer comparison can be made by interpolating
the present values to convenient intervals of r/R and doing the same for
the solution for the continuous case already considered, in which no loop
was assumed. It is seen that for most depths the differences are within
Dr. Harold Jeffreys,
4, 7
the errors of calculation, the exceptions being from r/R =0-95 to 0.91,
corresponding to distances from 18" to 22O.6 in the continuous case and
19O.7 to 22O.8 in the case of a loop.
For S (Tables 111, IV) the rays in general penetrate a little less deeply than
the Pones emerging at the same distances, except for Sr up to about 30'.
This
TABLE
I11
Calculation of Velocities of S
A
P
20
25.41
24.50
23.37
22.47
21.7
21
21-35
22
21-05
23 cusp
I8 cusp
55
60
65
70
75
20.88
19.4
18.7
18-35
17-85
17-55
17-15
16.8
16.2
15.65
15.44
14.94
14.36
13-86
13-38
12-84
12.26
I I .68
I I .06
80
10.44
0
I0
15
I8
20
21
22
23
24
25
27
30
35
40
45
50
85
90
95
I00
105
107
9-78
9.04
8.52
8.39
8.29
8.25
log Rlr
0
0.01207
0.02717
0.03982
0.05081
0.05591
0.06058
0.06581
0.07596
0.08251
0.08684
0.09332
0.09738
0.10346
0~10910
0.1 I973
0.13242
0 . I 3865
0.15818
0.18186
0.20439
0.22819
0.25748
0.291I I
0.32726
0.36867
0.41333
0.46454
0.52644
0.57469
0.58972
0.60337
0.60978
rlR
.oooo
0.9880
0.9732
0.9610
0.9505
0.9456
0-9412
0.9363
0.9268
0.9208
0.9168
0.9109
0.9072
0.9017
0.8966
I
0.8872
0.8760
0.8705
0.8537
0.8337
0.8151
0.7960
0.7730
0.7474
0.7209
0.6917
0.6614
0.6284
0.5907
0.5629
0.5545
0.5470
0.5436
rlRp
0.03935
0.04033
0.04164
0.04277
0.04380
0.04429
0.04471
0.04484
0.04777
0.04924
0.04996
0.05103
0.05 I 69
0.05258
0.05 337
0.05477
0.05540
0.05638
0.05714
0.05806
0.05881
0.05949
0.06020
0.06096
0.06172
0.06254
0.06335
0.06425
-6534
0.06607
0.06609
0.06598
0.06589
C
4.35
4-46
4.61
4.73
4.85
4'90
4.95
4.96
5.28
5-45
5.53
5.64
5.72
5.82
5.90
6.06
6-13
6.24
6.32
6.42
6.51
6.58
6.66
6.74
6.83
6-92
7.01
7.1I
7.23
7.31
7.3I
7-30
7.29
would mean, if correct, that the great change in the velocity is spread over a
greater range of depth for S than for P. Interpolation for S in this range is
difficult, however, and the loop is more uncertain than for P ; the details of
the difference are therefore not to be depended on, though it would be hard to
remove it without doing violence to the observations. The values found
for rlR for the extreme rays, at 1o5O for P and 107' for S, agree to 0.0001,
but the close agreement is accidental, since even if the edges of the shadows
were accurately located the standard errors of the depths reached would
Times of P, S and SKS, and Velocities of P and S
1939 June
513
of course, the accuracy is much
be about 0.005. Between 30' and IOO',
higher, since the values of p rest on centred differences over long ranges.
TABLE
IV
Velocities of S
From smoothed c
rlR
A
1-00
0.00
0.99
0.98
0.97
0.96
0.95
0.94
0.93
0.92
0.91
0.90
0.89
0.88
0.87
0.86
0.84
9.06
12.77
15439
18.20
0.82
0.80
0.78
0.76
0.74
0.72
0.70
0.68
0.66
0.64
0.62
0.60
0.58
0.56
0.55
0.54
0.53
0.52
20.10
22.25
20.23
22.24
24.33
26-35
28.84
P
25.41
24.66
23.94
23-15
22.39
21-66
21.00
18.6
17.8
17.05
16.4
15.85
38.36
43.45
48.67
54.00
15.12
14'54
13'99
'3.48
58.55
13.00
62.58
66.43
70.16
73.61
76.97
12.54
12.09
11.66
80.22
10.41
10.01
9.61
9.22
8.84
8.47
8-33
83-30
86-09
88.73
91.63
96.18
103.00
11-23
10.82
rlRp
0.03935
0.04105
0.04094
0.04190
0.04288
0.04386
0.04476
0.0495
0.05 I I
0.0528
0.0543
0.05552
0.05688
0.05777
0.05861
0.05935
0~06000
0.06061
0.06121
0.06175
0.06233
0.06285
0.06340
0.06394
0.06452
0.06508
0.06561
0.06612
0.06603
0.06603
0.06603
0.06603
C
4353
4.441
4'529
4.635
4.743
4.852
4.95I
5*24?
5.48
5.65
5-84
6.01
6.142
6.292
6.391
6.483
6.565
6.637
6.705
6.77I
6.831
6.895
6.952
7.013
7'073
7.137
7.199
7.258
7.3'4
7.304
c
(smoothed)
4.353
4.444
4.539
4.638
4.741
4.850
4.962
5.227
5.463
5.670
5.850
6.002
6.125
6.221
6.295
6.395
6.483
6.564
6.637
6.706
6.770
6.833
64393
6.953
7.012
7.074
7.137
7'199
7.258
7'314
7.304
r
rlRp
P
3935
4017
4103
4'93
4286
4384
4486
4725
4939
5126
5289
5426
5537
5624
5691
5781
5861
5934
6000
6062
6120
6177
623I
6285
6339
6395
25.41
24.65
23.89
23-13
22-40
21.67
20.95
19.68
18.63
17.75
17-02
16-40
'5.89
15.47
15.11
14-53
13.99
13.48
I 3 -00
12.54
I 2.09
11.66
11-23
The results were interpolated to convenient intervals of r/R by the
Scientific Computing Service, second differences being kept. The interpolation gives a slight smoothing, but for depths corresponding to the
distances where 'I intervals of distance had had to be used to give dt/dA
the values of c were rather irregular, and smoothing was done by fitting
quadratics down to r/R =o.g4 and at greater depths by the Darwin-Comrie
method. The depth of the second-order discontinuity at r/R = 0.94 rests,
of course, ultimately on the positions adopted for the cusps at 22' and 2 3 O ,
but it was satisfactory that with a permissible smoothing the distributions
*
Multiplied by rob.
Dr. Harold Jeflreys,
514
4,7
of velocity for both P and S were consistent with it. It is desirable to make
the adopted times consistent with this smoothed velocity distribution in
order that the results actually given shall be self-consistent. This makes
a few small but permissible changes in the times.
Times for ascending rays from deep foci were worked out by the previous
method for depths down to o.12R. The greatest depth yet recorded in the
I.S.S. is o-ogR, but Stechschulte has informed me of one case of O ~ I I R ,
and it is desirable to be prepared for a greater depth. Times and distances
for rays leaving the foci horizontally, for depths from o.05R to o-o7R, were
worked out at shorter interyals to obtain revised times on the loop. It was
found that the further cusp for P corresponded, as closely as the calculation
would permit, to the ray that grazes depth o.o6R, and should emerge at
21O.7.
The nearer was found to be at 18O.5 and the ray would reach depth
0-066R. The shift of the nearer cusp by 1 O . 5 is surprising; it must
apparently be attributed mainly to the changes introduced in smoothing
the velocities. If so, it means that the cusp cannot be brought back to
17' with any distribution of velocity just below the discontinuity that will
permit interpolation without closer intervals. The modification may therefore as well be adopted. The resulting times (given in seconds in Table V)
then can be interpolated and used as surface times in the computations for
descending rays.
TABLE
V
Recalculated Times
S
P
rlR
P
0.95
0.94
0.938
0.936
0.934
0.932
0.93
0.92
0.91
0.90
0.89
0.88
12.01
11.59
11.42
I I .26
11-10
10.96
10.83
10.27
9-81
9.44
9.141
8.858
A
19.8
21.6
19'4
18.8
18.6
18.6
18.6
P
t
21.0
275.3
296.6
27 I
264.4
262.0
262.2
262.2
287.4
22.4
301.4
25.6
28.0
332.4
354.2
372.4
30.0
a
0
21.67
20.95
20.68
20.41
20.16
19-91
19.68
18.63
17-75
17.02
16.40
15.89
A
t
20.0
497-I
548.4
506.8
494.6
490.5
490'4
490.4
5 17.4
5434
585.4
618.8
657.4
22.4
20.4
19.8
19.6
19.6
19.6
21.0
22.4
24.8
26.8
29.2
Little special comment is needed on the calculation, which followed the
previous lines, except in one minor respect. The integration for the ascending rays gives A and t - p A to the top of the lower layer. Previously I added
T' to give the time to the same distance over the surface, except for the ray
that leaves the focus horizontally ; for this I added 6 to A and Q(7' +pa)
to t. But it is a completely equivalent and somewhat easier process to
add 8 to A and T' to t -PA ; then the latter, increased by the new PA, with
the same value of p, gives t for the new A. In this form A can safely be
rounded to o O . 1 before adding t -PA, since the only error introduced is a
second-order one in the height of the tangent, not in its slope.
Times of P, S and SKS, and Velocities of P and S
1939 June
515
For depth 0-02R it was found that pPd should precede pPr up to 23' ;
for greater depths the first p P is always pPr. It should appear first at the
following distances : 0.02R, 19' ; o.o@, 22' ; o.o6R, 25' ; o.o8R, 30' ;
O ~ I O R37";
,
0.12R, 45". The last three and that for 0.02R are about
3 times the distance reached by the horizontal ray ; the others are decidedly
shorter. It was usually found that near these distances the time did not
vary more than about 0 8 . 1 , which is within the error of calculation, when
TABLE
VI
pP-P
.oo
-01
-02
'03
-04
-05
-06
61
62
70
-07
-08
'09
.IO
-11
.IZ
119
124
129
134
138
141
I44
I47
I49
152
152
153
124
I9
20
21
22
24
50
51
52
25
53
23
28
29
30
10
34
35
37
40
45
10
50
55
10
10
10
10
10
10
10
10
60
65
70
75
80
85
90
95
I00
10
10
10
10
10
11
11
22
22
22
22
23
23
23
24
24
24
25
25
25
26
26
26
26
34
34
34
35
35
36
36
37
38
38
39
39
40
40
41
41
41
45
46
46
47
47
48
49
50
51
52
53
53
54
54
55
55
55
56
57
58
58
58
60
61
63
64
65
66
66
67
68
69
69
69
66
67
68
69
69
71
72
74
76
78
79
80
81
82
82
83
83
75
77
78
78
80
81
83
85
88
89
91
92
94
95
96
96
96
81
82
83
86
86
87
89
91
93
96
98
90
93
94
95
97
99
102
105
108
100
100
107
104 I I O
107 114
IIO
118
114 122
117 126
120 129
102
IOO
111
102
113
115
123
125
132
I35
117
118
127
129
I37
I39
120
120
120
131
141
131
142
142
I04
105
107
108
108
108
131
130
136
141
146
150
I53
156
I59
162
162
163
the assumed distance of the point of reflexion was varied by about 3". At
most other distances the corresponding variation would occur with a
displacement of 0'-3 or so. Accordingly I doubt the remark of Gutenberg
that the cusp where p P and PP join is fundamentally different in character
from that of PKP ; it seems to be quite similar and should be associated
with large amplitudes. The range just mentioned is possibly shorter than
for the combination of PcP and K to give PKP, but that is because the
respective values of dt/dA and dat/dA*are smaller in the latter case. In
Tables VI and VII the calculated intervals pP - P and sS - S are given in
seconds for varying depths h/R, measured in earth-radii.
It has been mentioned already that the method of matching gradients
gives a second-order error in the computed times found by addition or
4, 7
Dr. Harold Jeflreys,
516
TABLE
VII
fl
I9
ss -s
-00
23
24
I3d
I3d
13d
14d
Igr
I5
25
I5
26
27
16
16
16
16
16
16
16
16
17
17
17
17
17
17
18
20
21
22
28
29
30
35
40
45
50
55
60
65
70
75
80
85
90
95
I00
18
18
18
18
*OI
32d
36d
36r
37
37
37
38
38
38
38
38
39
40
41
41
42
42
43
43
44
44
45
45
45
.02
SId
55r
57
57
58
59
59
59
60
60
61
63
64
65
66
66
67
68
69
70
71
72
72
'03
70d
747
76
78
78
79
80
80
81
83
84
86
87
89
90
91
93
94
95
96
97
98
'04
91
94
96
97
98
99
IOO
IOI
103
105
107
109
111
113
114
116
118
120
121
122
123
-05
.06
.07
.08
*Og
*I0
.I1
*I2
162
168
172
178
180
185
190
196
203
206
213
222
210
221
201
215
228
206
220
211
226
231
236
241
246
250
253
254
234
240
246
I09
112
114
116
117
118
120
122
125
128
130
132
134
137
139
141
143
145
146
147
126
130
132
I34
135
138
141
145
148
150
153
156
158
161
164
166
169
170
171
148
150
154
158
162
166
169
172
176
179
182
186
190
194
198
191
196
215
182
185
202
220
205
187
191
193
194
208
223
227
231
234
235
212
214
215
252
258
263
268
271
272
231
239
246
254
261
268
274
280
286
289
290
subtraction if the distances are found only to the nearest multiple of oO.5,
as was done in this and previous cases. This error was found in some
cases to reach 08.2 for P and 08.4 for S, and was taken into account.
PcP and ScS.-The extremes of the P and S tables have suggested about
o-543R for the radius of the core. Times of PcP and ScS for surface foci
were worked out for assumed core radii of o.52R to o.56R. For values of
r less than 0.55R the velocities were taken constant, and equal to 13.62 km./sec.
for P and 7.304 km./sec. for S. This gave the extrapolations in Tables I1
and IV. An error in the hypothesis of uniform velocity will make only a
second-order error in the depth found for the core. Contrary to what
might have been expected, the times were found to be still fairly sensitive
to the adopted depth of the core to a distance of about 7 0 ~ .
Original observations of PcP and ScS in the earthquakes of 1928 March 29
and 1931 February 20 are given by Stechschulte * and Scrase -f respectively.
I have given a rough discussion of these previously, 1 a complete one not
being possible at that stage. But it is necessary now to find the positions
and times of occurrence as accurately as possible. The solution in my last
paper needs two changes. The change in the P times due to the altered
* Bull. SeiSm. SOC.Aw., 22,81-137, 1932.
t Phil. Trans., A, 231,207-234, 1933.
1 Bur. Centr. S b . Int., Fasc. 14, 1936.
Times of P, S and SKS, and Velocities of P and S
1939 June
517
TABLE
VIII
A
0
0
5
I0
15
20
25
30
35
40
45
50
55
60
65
70
75
80
85
Calculated Times of PcP and ScS in seconds for Different Radii
of the Core and a Surface Focus
scs
PCP
-
-56
a54
e5.5
503.1 512.5
504.4 513.6
508.1 517-1
514.2 523.1
522.7 531.3
53-39 541.4
545.9 553.4
560.3 567.2
576.2 582.6
593.5 599.3
611.9 617.2
631.4 636.0
651.8 655.7
673.0 676.2
697.8
-53
-
I
-56
-52
521-7 530.9 540.1
522.8 532.0 541.2
526.2 535.3 544.4
531.9 540.7 549.6
539.6 548.1 556.7
549.4 557.4 565.6
561.0 568.6 576.3
574.4 581.5 588.7
589.2 595.8 602.5
605.3 611.3 617.5
622.5 627.9 633.6
640.8 645.6 650.7
659.9 664.1 668.6
679.6 683.3 687.3
700.1 703.2 706.6
721.0 723.6 726.4
744.4
765.4
*54
-5.5
Allowances for Focal Depth, for Core Radius 0-5@
0
5
I0
I5
20
25
30
35
40
45
50
55
60
65
70
c
(all negative)
,
1
_
0.04
0.05
0.06
0.04
0.05
0.06
36.6
36.6
36.5
36-4
36.3
36.1
35-9
35.8
35.6
35.4
43.9
43.9
43.8
43.7
43.5
43-3
43'1
42.9
42.7
42.5
42'3
42.1
51.1
51.1
65-1
65-1
64-9
64.7
64.5
64.3
64.0
63.7
63.3
63.0
62.8
62.6
62.4
62.2
62.0
78.3
78.3
78.1
77.8
77.6
77.2
76.7
76.4
76.0
75.6
75.3
75.0
74.7
74.5
74.2
91.2
91-2
91.0
90.7
90'4
89.9
89.4
88.9
88.4
35'3
35.2
35'1
35.0
34.9
42.0
41.9
41.8
'52
scs
PCP
A
a53
932.2 949.5 966.9 984.3
934.5 951.7 969.0 986.3
941.2 958.2 975.2 992.4
952.2 968.7 985.3 1001.9
967.1 983.0 999.0 1o15.o
985.8 1000.9 1016.2 1031.5
1008.2 1022.4 1036.8 1051.3
1033.7 1047.0 1060.4 1074.1
1050.0 1062.1 1074.4 1086.8 1099.5
1082.1 1092.9 1104.1 1115.6 1127.4
1116.1 1125.8 1136.0 1146.5 1157.3
1152.8 1160.7 1169.7 1179.2 1188.9
1190.9 1197.2 1204.8 1213.4 1222.1
1229.8 1235.1 1241.3 1248.6 1256.8
1269.5 1274.1 1279.0 1285.4
1313.9 1318.0 1323.4
1357.9 1362.0
1398.2 1401.3
914.8
917.2
924.3
935.8
951.3
970.8
994.1
1020.6
50.9
50.8
50.6
50.4
50.0
49.8
49.6
49.4
49'1
48.9
48.7
48.6
48.5
88.0
87.6
87.2
86.9
86.7
86.3
thicknesses of the upper layers would be adjusted by decreasing the focal
depths by o.ooo6R and the times of occurrence by 08.54, without change of
epicentre. Supplementaryinformationfrom other phases, however, indicated
that the depths found from P needed increases of +O.OOI@ f 0.00038R
and +O.OOIIR o-ooo33R respectively, and these will apply also to the
determinations from the present P tables. This can be allowed for by
Dr. Harold Jefreys,
49 7
TABLE
IX
1928 March 29
A
Zinsen
Zi-ka-wei
Ootomari
Manila
Phu-Lien
Irkutsk
Batavia
Sverdlovsk
Honolulu
10.92
14.08
PCP
- 1
Chiu-feng
Zi-ka-wei
Hong-Kong
Manila
Amboina
Medan
Hyderabad
Batavia
Abisko
Baku
Colombo
Upsala
Lemberg
Copenhagen
Potsdam
Hamburg
Berkeley
Ksara
Vienna
Glittingen
Belgrade
Durham
De Bilt
Zagreb
Stuttgart
Uccle
Strasbourg
Kew
Oxford
Pasadena
Zurich
- 15
- 4
-1
+2
-1
-2
-I2
-I0
+I
0
-I0
- I1
-4
-5
scs
0-c
- 9
+4
- 27
- 6
+5
- I1
-3
- I1
- I5
-4
-8
- 4
+2
20
PCP
15.05
17.36
28.33
32.40
48.55
+14+22
52.11
+ 4
+ 6
54.62
56.77
57-92
60.43
61.08
64.17
68.64
69.11
71.43
71 a67
72.04
73-25
73.25
73-29
73.94
74.48
74.57
75'25
75.80
75-92
76.51
76.94
76.97
77.01
77.18
- 14
-I0
1931February
A
0-c
+I 4
15-35
23.06
29.99
31.87
47.98
57-26
57-72
scs
- 16
+I5
+I4
+ 9
-
6
8
5
4
5
0
0
+ 6
+ I1
0
-3
-1
0
-1
+4
+4
+ 2
+5
+ 3
+6
- 4
- 5
-1
-2
0
- 3
- 3
+ 3
0
+6
+ 21
- 2
- 1
+I
-1
+I1
- 4
- 7
- 4
+ I
+4
+Z
-4
-1
+ 4
dropping the normal equation for the depth and replacing h in the equations
for Y , x and y by the increases suggested. The results are :
1928 March 29. Y = +3.2 f 0.4; x = - 0.09 f 0.08 ; y = - 0.06 f 0.07 ;
Depth =o.o608R f o.ooo@.
1931 February 20. Y = +0.7 f 0.3 ; x =o.oo f 0.03 ; y = +0.07 f 0.04 ;
Depth =o.o4ggR f o~ooo3R.
Times of P, S and SKS, and Velocities of P and S
1939 June
519
The displacements of the epicentres from the positions used already are
about their standard errors, and it is not necessary to revise the distances.
The times of occurrence may be taken as 5 h 6 m 68-8and 5 h 33m 258.8.
Table IX gives the residuals of PcP and ScS after allowing for the
revised times of occurrence and focal depths, with a core radius of oms@,
and for the ellipticity effect,* which exceeded 18 for ScS at some stations.
It is evident on inspection that PcP is much less satisfactory than ScS.
The latter shows a systematic trend, such as would be produced if the trial
radius of the core was too small, but subject to this the residuals run steadily,
with only a few exceptions. The PcP values are fewer and much more
ragged. Both Scrase and Stechschulte call attention to the greater difficulty
of reading PcP. ScS is a large and sharp movement, and it appears impossible that it could be read later than these observers have done ; but
the smaller PcP may easily have been read late. Scrase points out in
explanation that a P wave incident on the core would be mainly transmitted
and finally emerge as PKP. It may be added that an SH wave incident
on a liquid core would be totally reflected at any angle of incidence, and
that inspection of the records reproduced indicates that ScS is mainly of
SH type. It appears therefore that ScS may be trusted, but both the
theoretical considerations and the appearance of the residuals themselves
indicate that no useful information is to be obtained from PcP. The
following critical table gives the reductions in the times of ScS corresponding
to a change of the adopted radius of the core from 0.g.R to o.55R :A
Ii.0
22.5
I7
16
28.7 I 5
14
33.9 I3
39.0 I2
43.6 I I
48.5 .I
53.8
56.8
60.4
63.9
67-6
5
72.5
;
Taking the radius of the core as o . ~ ~ + o . o Ithe
x , least squares solutions
were : 1928 March 29, x = + 0.87 ; 1931 February 20 (omitting Hong-Kong),
x = + 0.72. With these values the distribution of residuals was as follows :-9
-5
-4
-3
-2
-I
1
1
4
1
1
7
o
5
I
2
3
4
5
1
3
2
4
1
6
2
If the residual - 9 is omitted, the mean is +0.4, the standard error
3-02, and the number of observations 32. On this basis the expectation of
*
Bullen, M.N.R.A.S., Geophys. Suppl., 4, 332-336, 1938.
520
Dr. Harold Jeflreys,
-4,7
the number of residuals over twice the standard e r r o r t h a t is, over +6.4
or below - 5-6-is I .6 ; but outside these limits there is only one observation,
that at - 9. It appears that there is no possibility of estimating a departure
from the normal law from the data, and we may as well retain the observation
at +9. Then the standard error of one observation is 38-4,and the
solutions are
X = + 0.87 f 0.095 ;
X = 4- 0.72 f. 0.097.
These are consistent ; combining them we have
x
=
+0*797f 0.068,
and the radius of the core is
0.5480 f 0.0007R =3473 f 4.2 km.
The ScS residuals from this solution are given as 0 - C.
The standard error looks very small, but it corresponds to one of '18-2
in the time of arrival at the epicentre. The uncertainties of the solutions
for the elements of the earthquakes are negligible in comparison. A possible
complication needs attention, however. We have seen that there is a
variation between different earthquakes in the mean Sr residual, and that
this is not shared by SKS. If it is not shared by ScS, it will not affect the
above results, but if it is we should subtract the mean Sr residual from
all readings of each earthquake before making our estimates. The previous
values for the two earthquakes were -38.8 and 0 8 . 0 ; allowing for the
various changes in the solutions these are altered to - 18.1 and + 18.5.
These were allowed for, and the new solutions were x = + 0.791and + 0.915,
with the same standard errors ; the weighted mean is x = + 0.85 I -C 0.068,
and the radius of the core is 3476.5 f 4.2 km. The difference between
the solutions is practically the standard error, and there is little to be said
between them as regards internal consistency. This ambiguity therefore
has little effect on the results. Some ground for suspicion is also provided
by the large standard error of one observation, 38.4 against 28.5 for Sr
and 28.0 for SKS. This suggests an unexplained variation, which may
have a systematic part. The distribution of the residuals, however, is not
noticeably skew, and the standard errors found for S and SKS would have
been larger if we had not been in a position to allow for the 2 term. It
does not appear, therefore, that the standard error found is too low. The
radius of the core, however, is such an important parameter that other
checks by independent methods should be obtained.
The main tables give the times of the first arrivals of P and S. Where
the first arrival for one tabulated depth is Pd, and that for the next deeper
one is Pr, interpolation between them will sometimes give errors of a second.
This can be avoided by using the following supplementary table, which
gives Pd where later than Pr, Pr where later than Pd, and the upper branch.
Times of P, S and SKS, and Velocities of P and S
521
P Upper Branches
Surface
Pd
o
m
s
0.00
0.01
0.02
0.03
Pd
Pd
Pd
Pd
o m
4 37.6
21
49.6
21.5
55.4
Pu
21.5 4 55-7
21
49'9
20
38.4
19
26.8
18.5 21.0
Pr
19 4 26.3 19
20
om^
B
4 33'5
45.4
51.2
19 4 16.9
20
29.2
20.5
35-1
Pu
Pu
4 51'5
45.7
342
22.6
16.7
Pr
4 21.9
4 29.5
18.0
6.4
0.6
Pr
18 4 5.9
20
19
18
17.5
om^
o m
0.04
Pd
B
18 4 1.6 16 3 34'4
19 13.6 17 46.5
18 58.3
Pu
Pu
19 4 13.9
I8
2.4 18 3 58.6
17 3 50-8 I7 47.1
Pr
16 35.5
I7 3 50-3
Pr
I 8 4 1.1
1.73 46.0
om
8
14 3 8.1
15
20.1
16
I7
43'7
32.1
Pu
I7 3 43'9
16 32-4
15 20.7
S Upper Branches
Surface
0.00
0.01
0.02
Sd
Sd
Sd
Sd
o
m
s
8 38.8
22
9 0.0
22.4
8.4
21
su
9 0.0
21
8 39.2
20
18.6
19.6 10.4
Sr
20
8 18.2
21
37-4
22
o m
8
o m s
B
32.2
20
8 2.8
53.2
9
su
I9
18
8 3.0
7 42.3
21.8
I9
Sr
7 42.4
20
32.1
11.4
3.2
Sr
su
su
1.8
8 53-3
I9 7 35.1
20
56.3
7 56.4
35.5
14.9
Sr
18 7 14.6
20
I9
I8
0.04
Sd
o m
B
17 6 41.8
su
16 6 21.1
Sr
16 6 20.8
17 6 40.2
10.8
30.0
[TABLES
G35
TAELB
X
TIMES OF P
2.5
3.0
3.5
42.6
49.7 7I
56.8
I
4.5
I 11.0
5.0
5.5
1
I
71
I
18.1 7I
25.2
I
I
07.6
14.7
21.7
71
o
,
70
I
I
I
3ga8
46*7
5se6
00.4
07.2
I
42. I
q8*4 64
63
54.8 b5
01.3
07.8
zi
I
14'4 66
b9
b9
68
68
69
14.1 68
20.9 68
I
I 21.0
:57''I::bIs8
55.2 49
56
1
I
I
03.2
62
09.4 62
I
I
15.6 64
I
I
I 22.0
I
00.4
06.0
11.7
:i
$
17.5 6o
23.5 6o
1
I
I
55.4 43
59.7 47
04.4
09.4 5530
14.7
54
1 20.1
I
25.7 56
57
Depth h =
A
0.06
0.08
0.07
0.09
0.10
0.12
011
o m
s
0.0
51.1
m
s
58.0
I
58.3
I
51-5 I2
52.7 21
59.3
I
I
54.8 28 I 01.0
24
57.6 33 I 03-4 z9 I
0.5
1.0
1.5
2.0
2.5
m
I
3.0
3.5
I
I
4.0 I
4.5 I
00.9
04.6 37
08.9 43
13.4 45
18.2 48
51
I
I
I
I
I
m
04.8
05.7
07.2
09.2
I
e
m
10.8
I
I 11.0
I 11.8
Is I 13.1
20
I 14.8 I7
26
I
I
I
I
23
r n s
s
16-8
17.0
17.7 I:
18.8
I
22.8
I 23.0
I 23.6
I 246
15
m
e
a I 28.8
I
I
I~
I
I3
20.3 2o I 25.9 I8 I
29.0
29.5 5
30.3
I2
31.5
15
06.3
I 11.8
I 17.1
I 22.3
I 27.7 2I I 33.0 I9
09.7 34 I 14.8 30 I 19.8 27 I 24.7
I 29.8
I 34.9 23
29
13.4
I 18.1 33 I 22.8 30 I 27-5
I 32-3 28 I 37.2
25
17.5
I 21.7 36 I 26-2 34 I 30.6 31 I 35-1
I 39.7 27
31
21.9 44 I 25.7 40 I 29.7
I 33.7 36 I 38.3
I 42.4 3o
zi
1:
46
5.0 I 23.3
5.5 I 28.4
6.0 I 33-7 53
6.5 I 39.2 55
7.0 I 44.8 56
26.5
31.3 48
I 36.3
I 41.4
I 46.5
7-5 I 50.4 57
8.0 I 56.1
8.5 2 01.8 57
9.0 2 07.5 57
I
I
56
e
045
I
I
53
51.8
:z
:i
29.9
33.5
34.3 $ 37-6 t:
38.9
41.8
43.6 ti
46.1 43
48.4
50.6 45
42
I
I
I
I
I
I
I
I
I
I
I
I
I
I
49 I
z:
46
:;
37.3 36 I 41'4
I 45.4
40-9
I 44.8 34 I 48.6 32
44.8 39 I 48.2 34 I 51.9 33
48.9 " I 52.0 38 I 55.3 34
36
53.1 42 I 55.9
I 58-9
43
37
57-4
I 59.9 4I 2 02.6
01.7 43 2 04.0
2 06.5 39
06.2 45 2 08.2 42 2 10.5 40
10.7 45 2 12.5 43 2 14.5 40
15.3
2 16.8 43 2 18.6 41
:z
53.3
I 55.2
I
58.3
I 59.8
2
2 04.5
2
2 02.5 54 2 03.3
2 09.3 48 2
2 07.9 54 2 08.4
2 13.3 54 2 13.5
2 14.1 48 2
9.5 2 13.2 57
58
53
51
48
44
10.0 2 19.0
2 18.6
2 18.6Io2 2 18.9
2 19.9
2 21.2 88 2
107
11.0dz 30.6 II I16
2 28.8 IoI 2 28.7 98 2 29.1 92 2 30.0
2
a 2 29.3
12.0r 2 41.8111 2 39.9
2 38.9 Io2 2 38.6 99 2 38.6 9s 2 39-0 90 2
Io6
2 50.5 Io4 2 49.1IoI 2 48.5 99 2 48.1 95 2 48.1 " 2
13.0 2 52.9
14.0 3 03.6 Io7 3 00.9 Io2 2 59.2 Ioo 2 58.2 97 2 57.6
2 57.2
2
57.1 53
I
I
$
i:
97
104
i:
41
22-7
31.2
39.8
48.5
57-3
85
86
8
7
88
89
101
4 03.8
4 00.3
3 57.3
4 13.4 96 4 09.7 94 4 06.5 9a
22.0 4 22.9 95 4 19.0 93 4 15.7 92
23.0 4 32.0 91 4 28.1 91 4 24.7 90
4 33.7
24.0 4 41'1 91 4 37.1
90
20.0
21.0
25.0 4
3 548
3 52.8 9o
4 03.9 " 4 01.8
4 13.0 91 4 10.8
4 21.9 89 4 19-6 88
4 30.8
4 28.4 g7
3 51.3 g9
4 00.2 88
4 09.0 87
4 17.7
4 26'4
::
3 50.3 g7
3 59.0 87
4 07.7 86
4 16.3 86
4 24.9 g5
i: if
::
::
4 3ga6
4 37"
4 35.O 86 4 33.4
4 46*1 4 42.6
4 45.8 ii 4 43.6
4
:
i
4 51-4
4 48.4
84
88 4 57.I :
:4 54.4 4 44 41-9
O3.9
5O.3 g4
88
88
g5
26.0 4 59.1 90
85
55.0
88
27.0 5 08.0 89 5
5 00'2
85
S2"
28.0 5 16.8
5 12.6
5 09.0 86 5 05.7 86 5 02.9
5 00.6
4 58.7 84
5 07.1
29.0 5 25.5
5 21.4 86 5 17.6 86 5 14.3 g5 5 11.4 g5 5 09.0 84
84
83
:;
30.0 5 3 4 2
::
5 30.0
5 26.2
5 22.8
5 19-9
5 17.4
5 15.4
Depth h =
A
-
I
0
30
31
32
33
34
35
36
7 01.8 84 6 55.6 84 6 49.7 84 6 4-084 6 38.5 83
37 7 13.0 84 7 08.2
7 10.1
7 040
6 58.0 83 6 52.3 83
82 6 46.8 82
38 7 21.4 84 7 16.6
7
12.3
7
06.3
83
82 7 00.5 82 6 55.0 82
7
249
83 7 18.4
39 7 29.8 84
83
83
83
::
::
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
-
8 14.0
8 07.4
8 01.0
8 18.9
8 15.3 79 8 08.9 79
8 26.8 79 8 22.0
8 29.8 78 8 23.2 79 8 16.7 78
8 347
8 37.7 79 8 31.0 78 8 245 78
8 42.6
8 45.4
8 38.7
8 32.2 77
8 50.3
76
8 46-4
8 39-8
8 58.0 76 8 53.1
9 05.6 76 9 00.7 76 8 5 4 0 76 8 47.4 76
9 08.2 75 9 01.5 75 8 54-9 75
9 13.2
9 20.7 75 9 15.7 75 9 09.0 75 9 02.3 74
9 09.6
9 28.0 73 9 23.1 74 9 16.3
;:
;;
74
;;
73
;:
;;
7 54.8
7 48.9
8 02.7 79 7 56.7
8 10.4 77 8 04.5
8 18.2 78 8 12.2 77
8 25.8
8 19.8 76
7 43.3
7 51.1
7 58.8
8 06.5
8 14.1
;:
8 33.4
8 40.9
8 48.3
8 55.7
9 03.0
8 21.6
8 29.0 74
8 36.4 74
73
8 43.7 ,2
8 50.9
71
;:
;:
75
74
74
73
72
75
8 27.3
8 34-7 74
8 42.1 74
8 49.4 73
8 56.6
5:
78
77
77
76
75
TIMES OF P-continued
-
Depth h =
A
0
30
3
'
31
33
34
35
36
37
38
39
40
41
41
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
-
8
8
8
8
8
8 11.1
16.2
23.6 74 8 18.5 74
31.0 74 8 25.8 73
8 33.0
38.3
45.5 7I 8 40.1 71
;:
;:
8
8
8
8
8
06.4
13.7
21.0
28.1
35.2
8
;:
7
8
8
71 8 23.6 71 8
7r 8 30.6 70 8
70
70
02.1
73 8 09.4
73 8 16.5
58.1
05.3
12.4
19.5
26.5
7 54.4
7 50.9 71
72 8 01.6 72 7
71 8 08.6 70 8
" 8 15.6 70 8
70 8 22.6
8
69
g
58.0
05.0 70
11.9 69
18.8
ii
TIMES OF P-continud
Depth h =
--
A
Surface
0.00
0.04
0.03
0.02
0.01
-~
o
70
71
72
73
74
m
B
m
#
m
s
m
s
m
a
m
a
m
s
1 1 15.4 6I 1 1 10.2
I 1 03.1 6o I 0 56-1 6o I 0 49.1
I 0 42.3 6o I 0 35.9
I 1 21.5 6 o I I 16.3 61 I 1 09.1 6 o I I 02.1
I 0 55-1 :
:
1
0
48-3 6 0 ~ 41-9
0
I 1 27-5
1 1 22-2
15.1
I 1 08.0 59 1 1 01.1
10 54-3
10 47.8
I 1 33.4 59 I 1 28-2
I 1 20.9 58 1 1 13.9 59 I 1 07.0 59 I 1 00.2
I 0 53.7
I 1 39.2 $ 1 1 340 58 1 1 26.8 59 1 1 19.7
1 1 12.8 58 1 1 06.0 57 10 59.5
~ZII
i;
57
58
57
45.0
I I 39-8
50-7
45.5
56-3
1 1 51.2
12 01.8 55 I I 5 6 9
12 07.3 55 12 02.2
54
80
12 12.7
12 07.6
12 00.2
I I 52.9
1 1 45.8
II
12 18.0 53 12 12.9 53 12 05.5 53 I I 58.2 53 I I 51.1 53 I I
I 2 23.2 52 I 2 18.1 52 I 2 10.7 52 I 2 03.4 52 I 1 56.2
I1
12 28.4 52 12 23.2
12 08.5 'I 12 01.4 52 I I
12 15-8
12 33.5
28.3
12 20.8 "12 13.5 5 o x z 06.4 ' O I I
50
50
50
50
82
83
84
i211
11
11
II
32.5
38.2
57 I I 43.8
55 11 49.3
55 I I 54.8
54
ii
75
76
77
78
79
81
II
5711
II
"11
56 I I
55 I I
55 I I
54
25.4
11 18.5
31.0 5 6 24.1
~
36.6 56 I I 29.6
42.1 55 1 1 35.1
47-5
I I 40.5
II
~5 6
55 1 1
55 11
54 11
53
~ZIZ
85
86
87
88
89
12 38-5
12 33.3
12 25.8
12 18.5
12 11.4
12 43.5
12 38.2 49 12 30.7 49 12 23.4 49 12 16.3
12 4 8 4 49 12 43.1 49 12 35.6 49 12 28.2 48 12 21.1
12 53.2 48 12 47.9 48 12 40.4 48 12 33.0 48 12 25.8
90
91
92
93
94
12 49.8
13 02.7 46 12 57.4
I3 02.1 47 I 2 54-5
I3 07.3
13 11.9 46 13 06.7 46 12 59-1
13 16.5 46 13 11.3 46 13 03.7
13 15.8 45 13 08.2
13 21.1
46
12
58.0 48 12 52.7 48 12 45.1
47
47
f
95 13 25.7
:
13
30-3
I3
I3
34.8
39.3 45 13
43.8
13
20.4
24.9
29.5
34.0
38.5
13 48.4
13
101 I3 52.9 45 13
I02 I3 57.4 4 5 1 3
103 I4 01.8
104 I4 06.2
43-1
47.6
I3
97 I3
98 13
gg 13
100
0.05
-
li
52.1
56.5
00.9
53
38.9
44.1 52
49.3 52
5 4 4 'I
59.4
50
59
59
58
57
05.2
10.8
~ ~56
16.3 55
21.7 54
27.0 53
53
32.3
37.5
I 1 42.7
11 47.7
I I 52.8
II
11
52
52
50
51
49
12 0 4 4 49 I I 57.7 48
49 1 2 09.3
12 02.5
48 12 1 4 1 48 12 07.3 48
47 12 18.8 47 12 12.1 48
:;
l;
3;
12
47
46
46
45
12 42.4
12 35.2
12 28.2
12 21.4
I 2 47.1 47 I 2 39.8 46 I 2 32.8 46 I 2 26.0 46
12 51.7 46 12 44.5 47 12 37.5 47 12 30.7 47
12 56.3 46 12 49.1 4 6 ~ 42.1
2
12 35-3 46
13 00.8 45 12 53.6
12 46.6 46 12 39.8 45
37.7 47 12 30.5
47
46
12.8
13 05.4
17.3
I3 09.9
21.9
I3 14.5
26.4 45 13 19.0
31.0 46 13 23.6
45
13 35.5
13 28-1
45 13 40.0 45 13 32.6
4 5 1 3 44.4 4413 37.0
44~3
48.8 *IS 41-4
z 1 3 53.2 44 13 45.8
13
45 I3
46 I3
45 13
45 13
46
11.7
II
17.3
~ ~5 6
22.8 "11
28.3 55 I I
33.6 53 1 1
6o
:i
23.5
46 I3
45 13
46 13
58.2
02.7
07.3
11.8
16.4
13
45 13
4413
44 13
13
20.9
25.4
29.8
34.2
38.6
12
45 I3
45
12
16.7 46
47
li
li
46
*
12
46
12 51.2
45 I 2
46 I3
45 13
46 13
45
45" 44.4 46
55.7
I 2 49.0
00.3
I 2 53-5 45
048
12 58.0
09.4 46 13 02.6 46
45
45
li
13 13.9
13 07.1
45 13 18.4 45 13 11.5 44
& I 3 22.8 *I3 15.9 44
*13
*13
27.2 4 4 ~ 320.3
31.6 eF
TIMES OF P-continued
Depth h =
A
008
0.07
0a45
m
a
m
s
0.09
a
m
70
71
72
73
74
10
75
76
77
78
79
I0
11
11
11
11
80
81
82
83
84
1 1 25.9
1 1 19.8
11
1 1 31.1 52 1 1 25.0 52 1 1
" 11
1 1 30.1
1 1 36.2
1 1 35.1
1 1 41.3
11
40.1 5
0
1 1 46.3 "11
14.1
11
19.3 52 1 1
24.3 "11
11
29.4
3 ~4 3 ~4 9
85
86
87
88
8g
1 1 51.2
11
1 1 45.0
1 1 56.0 48 1 1 49.8 48 1 1
12 00.8 48 1 1 546 48 1 1
12 05.5 47 1 1 59.3 47 1 1
12 10.2 47 12 04.0 47 1 1
39.2
91
92
93
94
12
12
12
12
12
95
g6
97
29.9
10 24.1
10 35.8 59 10 30.0
10 41.7 59 10 35.8
10 47-5 58 10 41.6
10 47.3
10 53.2
zz
a
18.7
10 13.6
59 10 24.6 59 10 19.4
58 10 30.4 58 10 25-2
58 10 36.1 57 10 30.9
57 10 41.7 56 10 36.5
m
10
56
56
10
s
m
08.8
14.6 58
58 10
58 10 20.3
57 10 26.0
56 10 31.6
56
m
s
10 04.2
10 10.0
10 15.7
57
57 10 21.3
56 10 26.8
55
a
g 59.7 57
58
10 05.4
57 10 11.1
56 10 16.6
55 10 22.2
55
57
55
56
54
58.8
I 0 52-9
I 0 47.3
1 0 42-1
I 0 37.1
I 0 32.3
I 0 27.6
54
04.4 56 10 58-4 10 52.8 55 10 47.6
10 42.6 5 5 10 37.7 54 10 33-0
09.9 55 1 1 03.9 55 10 58.2 54 10 53.0 54 10 47.9 53 10 43.1 54 10 38-3 53
15.3 54 1 1 09.3 54 1 1 03.6 54 10 58.3 53 10 53.3 54 10 48-3 52 10 43.5 52
20.6 53 1 1 14.6 53 1 1 08.9 53 1 1 03.6 53 10 58.5 52 10 53.6 53 10 48.7 52
52
52
53
49
49
46
46
52
49
08.8
11
13.9
11
19.0
11
24.0 "11
28.9
~ ~4911
49
33.8
11
44.0'
38-6 48 1 1
48.7 47 1 1 43.3 47 1 1
53.4 47 1 1 48.0 47 1 1
58-1 4 7 52.7
~ ~ 11
11
48 1 1
:z
46
52
51
51
03.7
10 58.7
10 53-8
08.8 " 1 1 03.8
I 0 58.8
13.9
1 1 08.8 "11
03.8
18.8 4 9 13.7
~ ~49 1 1 08.7 49
23.8 5 0 18.6
~ ~$ 1 1
13.6 49
48
48
11
28.6
33.3 47 1 1
38.0 47 1 1
42.7 47 1 1
47.3
11
2
23.4
1 1 18.4
28.1 47 1 1 23.1 47
32.8 47 1 1 27.8 47
37.5 47 1 1 32.4 46
42.1 $ 1 1
37.0 46
46
14.8
12 08.6
12 02.7
1 1 57.3
1 1 51.9
1 1 46.7
1 1 41.6
46
19.4 46 12 13.2 46 12 07.3 46 12 01.9 4 6 56.5
~ ~46 1 1 51.3 4 6 1 46.2
~
06.4 45 12 01.1 4 6 55.9
~ ~4 6 50.8
~ ~46
24.1 4 7 ~ 217.8 4 6 ~ 211.9 4 6 ~ 2
45
46 1 2 16.5
12 22.4
28.7
12 11.0 46 12 05.7 46 1 2 00.5
1 1 55.3
46
12 21.1 46 12 15.5 45 12 10-3 46 12 05.1 46 1 1 59.9
33.2 45 Iz 27.0
46
45
12 31.5
12 37-8
12 42.4 46 12 36.0
12 46.9 45 12 40.6
12 51.5 46 12 45.1
CJCJ 12 56.0 45 12 49.6
98
44
roo
0.12
45
45
12 25.6
45
45
45
14.8
12 09.6
12 044
45 12 30.1 45 12 245
19.3 45 12 1 4 1 45 12 08.9 45
46
46 12 34.7 46 12 29.0 45 12 23.8 45 12 18.6 45 12 13.5
45 1 2 39.2 45 1 2 33.5 45 12 28.3 45 12 23-1 45 12 18.0 45
45
45 12 43.7 45 12 38.0 45 12 32-8 45 12 27.6 45 12 22.5
45
1 2 20.0
45
12
45 12
46
45
45
45
00.4
12 32.1
12 27.0
1 2 37.3
12 48.2
12 541
12 42-6
44
IOI 13 048
36.5 #IZ 31.4
58.5 4 4 ~ z52.6 4 4 ~ 247.0
41-7
102 13 09.2 @13 02.9 * 1 2 57.0
51.4 #12 46.1 # 1 2 40.9
35.8 e4
103 13 13.6 ++13 07.3 ++I3 01.4 44
13
TABLE
XI
TIMES OF S
i
0.06
0.0
I
0.5
I
1.0 I
1.5
2.0
I
I
2-5 I
3-0 I
3.5 2
4.0 2
4.5 2
7.5 3
8.0 3
8.5 3
9'0 3
95 3
10.0 4
11.0d4
12.0r4
13.0 5
14.0 5
15.0
16.0
17.0
18.0
5
6
6
6
19.0 7
20.0
7
21.0 7
22-0 7
23.0 8
24.0 8
8
8
9
9
29.0 9
25.0
26.0
27.0
28.0
30.0 10
0.09
31.2 8
32.0 22
34.2
37.9 37
42-8 49
59
48.7
m
I
I
I
I
I
55.5
I
2
03.0 75
81
11.1 86
19.7 89
2
2
2
e
m
43.6
44.3 20
46.3
49.4 31
53.6 42
52
58-8 6o
048
11.5 67
18.8 73
26.6 78
83
I
I
I
2
2
2
2
2
2
2
e
m
0.10
e
m
55.5
2 06.9
56.0 I6 2 07.4
57.6
00.3 27
38
04.1 46
08.7
14.0 53
60
20.0 66
26.6
33.6 70
76
2
2 08.9
2 11.2 23
2
2
2
2
14.4 32
2
2
2
18.4
23.2 48
28.6
3 4 6 64
41.0 68
2
2
2
2
2
40
i:
2
2
2
0.12
0.11
e
m
17.9
18.3 If
19.6 2I
21.7
24.6 29
36
28.2
42
32.4 48
37.2
42.6 54
48.4
2;
2
2
2
2
2
2
2
2
2
e
m
28.6
4
29.0 I2
30'2
32.1 I9
34.6 25
e
2
2
39.0
39.4 .I
40'4 I7
42.1
23
++.4 29
37.8 38 2
41.6
2
46.0 44 2
50.9 49 2
56.2
3
47.3
34
50.7
54.6
59.1
04.1 5 0
32
$
2
2
::
53
TIMES OF S-cmtinued
Depth h =
A
Surface
o
30
31
32
33
34
m
I
u
m
II
I1
I2
u
I
0.01
/ m
Im
a
I
0.M
0.03
Im
a
u
I
0.04
Im
u
I
0.05
Im
8
I I 02.2
I I 10.2
I1
I
0.00
26.0 IS8
41.6
57.2
12.7
I1
II
:::
I1
I2
17.9
33.6
49.2
04.7
12 20.2
35 12 28.2
36 12 43.7
12 35.6
37 I 2 59-0 I 2 51.0
38 13 14.3 I S 3 13 06.2
39 13 29.4
I 3 21.4
:::
40 I 3 445
14913 36.4
41 I 3 59.4 14*13 51.3
43 14 14-2147 1406.1
43 I4 28.9 145 I4 20.7
44 I 4 43.4 145 I 4 35.3
45
46
47
48
49
I 4 57.9
14 49.7
14 37.9 143 I4 26.5
143 1452.2
14 40.7
15 12.2 143 15 04.0
'
4
1
15 26.5 143 15 18.2 15 06.4 '41 1454-9
'
4
1
15 20.4 140 15 08.9
15 40.6
15 32.4
15 22.9
15 54.7 139 ' 5 46.4
15 344
::
;:
:;
17 08.5 134 16 56-3I33 16 44.5
16 57.8
17 30.2 134 17 21.9132 17 09.6
17 22.8 132 17 10.9
I7 43.4 13' I7 35'1
131
17 56.6 13' 17 48.2
17 35.9 1 3 1 I7 240
131
18
01.3
18 09.7 13'
17
48.9
17 37.0
129
129
55 17 16.8
56
57
58
59
:ii
60 18 22.6 128 18 14.2128 18 01.8128 17 49.8
61
62
63
64
::
17 38.3
17 27.1 126 17 16.2126
18 14.6126 18 02.5
17 51.0 127 17 39.7
17 28.8
18 27.0
18 35.4
18 27.2 126 18 15.1
18 03.5 IZ5 17 52.2 12' 17 41.3
18 48.1
18 39.7
18 52.2 IZ5 18 39.8
18 27.6 12' 18 16.012' 18 04.6 124 17 53.6
19 00.7
18 40.0123 18 28-3 18 16.9123
18 05.9 121
19 13.21123
2'
19 04.7 123 18 52.2 IZ4
123
I22
i::
::
I9 17.0 I22 I9 04.5 122 18 52.3 122 18 40.5 121
65 I9 25.5
66 19 37.8
19 29.2 121 19 16.7121 19 0 4 5 120 18 52.6 121
19 04.7
67 I9 49.9 120 19 41.3 120 I9 28.8 12019 16.5
68 20 01.9 19 53-3 119 19 40.8
19 28.4
19 16.6 'I9
2
0
05.2 118 19 52-7 19 40.2
19 28-3'I7
69 20 13.8::
:::
::;
70
20
25.6
:;
:;
20
17.0
20
04.4
117
19 51.9
117
19 40.0
18 29.1 1 2 18
~
18.0 121
18 41.2 I 2 0 18 30.1 119
I8 53.2 118I8 42.O 118
18 53.8
19 05.0
117
19 16.8
19 05.5 116
:::
19 28.4
19 17.1
TIMES OF S-continued
-
Depth h =
A
0
m
n
Im
a
Im
Im
a
Im
a
Im
8
lm
a
B
30 [ O 02.0
9 54.0
9 47.0
9 41-0
9 35.9 152 9 31.6
9 28.0
10 02.5
9 56-4IS4 9 51.1
9 46.7
9 43.0 1-50
149
31 to 17.5 Is' 10 09.5
32 to 33.0 1-55Io 249 1-54Io 17.8 153'10 11.6 1-5' Io 06.3 Is2 Io 01.7 '5' 9 57.9 148
10 12.7
10 21.4"I 10 16.7
33 to 48.4 IS4 10 40.3 IS4 10 33.2 IS4 10 26.8
147
34 [ I 03.7 IS3 10 55.6 IS3 10 48.4 IS2 10 41-9 10 36.3 149 10 31.5 I@ 10 27.4
1-52
1-52
18.9
1 1 10.8
11
[ I 3 4 0 Is' 1 1 25.8 IS0 1 1
[ I 49.0 IS0 1 1 40.8 :$I1
12 03.9 149 1 1 55.6
11
[ z 18.7 I@ 12 1 0 . 3 147 12
1-51
35
36
37
38
39
4a
41
1 2 24.9
12 17.1
12 10.1
12 33.4
12 48.0 146 12 39.4 1
12 244 143
'
4
12 31.5
13 02.4
12 53-7143 12 45-8143 12 38.6
12 52.7 141
13 16.7 Iq3 13 08.0 143 13 00.0
13 31.0 143 13 22.1
13 1 4 1 141 13 06.8 141
4
43
44
146
I47
141
141
149
1-50
147
146
03.5
10 56.9
10 51.2
10 46.2
10 42.0
18.4 149 1 1 11.8 149 1 1 05.9 147 1 1 00.8 146 10 56.5 14'
33.3 1491 1 26-5147 I 1 20.6 147 I1 15.4 146 I 1 10.9 1 4 4
143
1 1 25.2
48.0 147 11 41.1 146 1 1 35.1 14' 1 1 29.8
02.6 146 1 1 55.7 146 1 1 49.6 14' 1 1 44-2 1 1 39.4 142
[I
I45
140
143
I44
139
12 03.9
12 18.1
12 32.3
12 46.3
13 00.2 139 I 2
139
141
142
1 1 58.4
12 12.5 141
12 26.6 141
140 12 40.6 140
11
12
12
12
544 138 I 2
138
53.5
07.6 14'
21-5139
35.4 139
49.2 138
137
4;
48
45
13 28.1
13 20.7
13 14.1
13 08.2
13 02.9
13 45.1
13 59.1 140 13 50.2 140 13 42.0 139 13 346 139 13 27.9 138 13 21.9 137 I 3 16.5 136
13 35,s 136 I 3 30.0 135
14 13.1 140 14 041 139 13 55.9 139 13 48.4 138 13 41.6
1426.9 138 14 17.9138 1409.6 137 14 02.1 137 13 55.2 136 13 49.0 13' I 3 43-4134
14 02.5 13' I3 56-8134
14 40.6 137 1431.6137 14 23.2 136 14 15.7136 1408.8
5c
5'
5s
5:
54
1445.2
14 36.8
14 29.2
1422.2
14 15.8
14 10.0
1454.3
15 07-9136 14 58.7 13' 1450.3 13' 1442.6 134 1435.5 133 14 29.0 13' 1423.2 132
15 21.4 135 15 12.213' 15 03.7 134 14 55.9 133 1448.7 13' 14 42.2 13' 1436-2
15 348 '34 15 25.5 133 15 16.9 13' 15 09.1 '3' 15 01.9 13' 14 55.2 130 1449.2
15 48.1133 15 38.8 133 15 30.1 13' 15 22.2 131 15 149 1 3 0 15 08.2
15 02.0 I 28
45
4f
5!
5(
5:
5j
5!
6c
61
6
6:
64
6!
6(
6:
6(
6!
7'
-
13 36.2
t:z
I37
132
136
131
136
131
135
130
134
129
133
:zi
132
:zi
TIMES OF S-continued
I
A
Depth h =
Surface
o
m
70
71
72
73
74
20
20
20
m
a
25-6
37.3 'I7
48.8 'I5
:if
11.4
21 00.2
21
20
20
20
20
112 21
a
44.5 108 Z1 35'9
80
16.5
22
1102~
140I I 0 21
Io92~
25.0
21
82
22
83
84
22
22
21
55.3 I o 7 2 1 46-7 Io621
06.0 Io5 21 57.3 Io5 21
::
81 22 26.9 Io4
37.2
47.4
57.4
22
22
22
1o022
9922
s
I I2
75 21
76 21
77 21
78 21
79 22
22.6
33.6
m
17.0 11620 04.4
28.6
20 15.9
40.2
20 27.4
51.6 ::f20 38.7
02.8
20 49.8
07.8
18.2
28.5
38.6
48.6
::
Io4
21
22
22
Ioo22
99
22
m
0.04
0.03
0.02
a
m
s
19 51.9
19 40.0
'I5 20 03.4 'I5 19 51.5
'I5 20 1 4 8 I I 4 2 o 02.9
20 26.1 'I3
I I2 20 1 4 1
111 20 37.3 I I o 2 0 25.2
:::
m
0.05
a
m
II0
00.9 I I 0 20
11.9
20
22.8 Io9 2 1
33.5 I o 7 2 r
44.2 "721
48-3 I I o 2 0 36.2 I09 20 2 4 4
20
59.3 1 0 8 47.1
~ ~ I o 8 2 ~35.3 Io920
107
10.1 Io7 20 57.9 Io6 20 46.0
20
20.8 I o 6 2 ~08.5
20 56.6
20
21 19.1
21 07-1 ''5 20
31.4
547
05.1
15.3
25.4
35.4
41.9
105
21
I02
22
I01
10022
9822
::
29.5
39.8
02.4 IoI 21 49.9
12.5
22 00.0
22.4 9 9 2 2 09.9
Io4 21 52.2
98
104
104
10.5
21
21
a
19 28.4
19 17.1
'I5 I9 39-9 'I5 I9 28.5
::f I9 51.2 'I3
I9 39.8
I I2
I 1 1 20 02.4 IIo 19 51.0
20 13.4 I I o 2 0 02.0
::
:::
'I4
I10
109
12.9
23.7
34'4
45.0
55.5
17.5 I o 2 2 ~05.8 102
27.7 IoI 21 16.0 I 0 0
IoI 21 37.8 Ioo 21 26.0 I 0 0
21 47.8
21 36.0
97
9 9 2 ~57.6 9 8 2 ~
45.7
97
21
21
97
97
TIMES OF S-continued
Depth h =
A
I
75
76
77
78
79
I
I
20
20
20
20
20
01-7 Io8 I9
12.5 Io6 20
23.1
20
20
33.7
44.1
20
:
O.I1
51.2
19 41.3
19 32.2
19 23.6
19 15.1
01.9 Io7 19 52.0 Io7 19 42.9 Io7 19 34.2
19 25.6 Io5
I 06
12.6 Io7 20 02.6
19 53.4 Io5 19 44.6 Io4 19 36.1 Io5
23.1 Io5 20 13.0 Io4 20 03.8 Io4 19 5 4 9
19 46.4
33.4 Io3
20 23.4 : z z o 14.1 Io3 20 05.1
103
I01
101 19 56.6 100
::
:
0.12
19 07.1
19 17.6 Io5
19 27-9 Io3
19 38.2 103
IoI
19 48.3 Ioo