THE TIMES OF P UP T O 30"
Harold Jeffreys
(Received 1951December 19)
Summary
Times of P in Mediterranean, Japanese and Californian earthquakes are
analysed with the objects of testing the generality of the increase of the
velocity at short distances found in Europe and the evidence for Gutenberg's
layer of low velocity. The Mediterranean earthquakes do confirm the
increase at short distances, but the Japanese ones agree with the tables in use,
and there is clear evidence of a regional difference. The evidence from North
American earthquakes is conflicting. It does not definitely indicate any change
from the present travel times, but the uncertainties are larger than in the
other regions, chiefly because there are rarely enough stations in approximately
the same azimuth and at greatly different distances to give much information.
The difference of the times at '2 and 25' is about the same in Europe and
Japan, but in comparison with them the times in Europe at 8" are about 38
earlier. The time-curve in Europe is nearly linear up to 14'. There is
evidence in both regions for a strong curvature of the time-curve between
15' and 20°, but it does not decide whether dtldA is continuous or discontinuous in this interval.
The curvature of the time-curve is in any case surprisingly small, because
laboratory studies of the variation of elastic moduli with pressure indicate an
increase of velocity with depth several times that given by any set of travel
times. Increase of temperature with depth would presumably reduce the
rate of increase but does not appear sufficient to account for the discrepancy.
I. Several recent investigations have suggested systematic departures of
the times of P up to 2 5 O or so from my 1939 table* of the order of a few seconds.
Since the apparent uncertainties of the times found in the joint study of Bullen
and myselft were mostly well under a second, such errors are surprising. If
genuine and general they would imply either a systematic error in the observations
or a great underestimate of uncertainty.
In Western Europe the times of P from the Burton-on-Trent explosion,
confirmed from near earthquake studies and the Heligoland explosion, indicated
an error varying by about 4s in the first 8 O . f This was traced to systematic error
in the Tauern earthquake. The need to change dt/dA at short distances in
Western Europe from 14s-3/1" to about 138.8/1" seems well established. On
the other hand, it is not known how general this difference is. The " European "
earthquakes studied by Bullen and myself were mostly in the Mediterranean
region, extending into Asia Minor, and few of the epicentres lay in the region
covered by the special studies ; even for these, the near observations used were
few, only one lying within 3" and four within 6". They were slightly supplemented from North American earthquakes, and the solution was found by fitting
a cubic formula up to 19". In 1936s I introduced a few more "European"
* H. Jeffreys,M.N., Geophys. Suppl., 4 498-533, 1939.
t H.Jeffreys and K. E. Bullen, Bur. Centr. S&sm., Traw. Sci., TI, 1935.
1H.Jeffreys, M.N.,
Geophys. Suppl., 5, 105-119,1946.
8 H.Jeffreys, Bur. Centr. S&m., Traw. Sd.,I+ 1936.
The times of P up to 30"
349
earthquakes, better observed at short distances ; and some Japanese shocks
were used. Some time afterwards+ I used also some European near earthquake
studies and a set of small Japanese earthquakes; the agreement between the
separate solutions was satisfactory. The combined solution, after correction
for the ellipticity and allowance for the upper layers, is the basis of the 1939 P
table. Corrections differing by 4"at different distances, if general, would imply
systematic errors in all sets of data. It does appear, however, that there actually
was a systematic error in the readings for the Tauern earthquake, and those for
the Schwadorf earthquake had been re-interpreted to agree with it. A similar
error might be present in the small Japanese earthquakes. It would be unlikely
in the " European '' ones used from the I. S. S. because these were well observed
at much greater distances; but (I) the velocity might be higher in N.W. Europe
than in the Mediterranean region, ( 2 ) the uncertainty might have been underestimated.
Gutenberg? has recently asserted that dtldA for P is nearly constant at about
13*.8/1" up to 1.5". It suddenly decreases at about 17" and varies smoothly at
greater distances. He maintains that the zoo discontinuity does not exist, but
if his interpretation is correct he has not removed it but only shifted it to 17'.
However, his value for dT/dA up to 15" is less than our tables give, whereas in
a study of North American earthquakes I had foundf evidence for a difference
in the opposite direction, but with a rather large uncertainty. Gutenberg's
discussion of the times is open to criticism, and the question of generality remains
open.
In the present study I compare European, Japanese and North American
earthquakes with the object of finding the travel-times in the regions separately
and testing for regional differences ; the methods of estimating uncertainty are
also improved.
The method of solution used in forming the tables was essentially one of
relaxation. A trial table of travel times was available. The constants for each
earthquake were chosen to fit the table as well as possible, and residuals were then
classified by distance. The means were smoothed to give corrections to the
travel times, and the process was repeated. Now it is known that approximations
by relaxation methods are liable to appear to have converged when they are still
appreciably different from the correct answer, owing to correlation between
remaining errors. For this reason Richardson and Southwell have recommended
varying the procedures ; Southwell's process, known as " block adjustment ",
has been the more widely used. However, it appears that the smoothing that we
used redistributes the corrections according to " point adjustment " and 'that the
corrections at each stage consequent on changes of the trial table correspond to a
block adjustment ; the convergence was therefore probably genuine.
The uncertainties were estimated directly from the scatter of the observations
in the various ranges of distance, without allowance for the number of parameters
eliminated (three for each earthquake and one for each range of distance). This
would underestimate the uncertainty, but not by much in view of the large
number of observations used ; and the latter point was checked by the application
of x2 to the corrections.made in smoothing. In most cases, therefore, the
* H. Jeffreys, M.N., Geophys. Suppl., 3,401-422, 1936.
t B. Gutenberg, Bull. Seism. SOC.Amer.,38, 121-148,1948.
1H.Jeffreys, Bull. Seism. SOC.Amer., 30, 225-234, 1940.
350
Harold Jeflreys
uncertainty is probably not appreciably underestimated. However, this may
not always be true, as can be seen from a simplified case. Suppose that
t =PA,
and that the stations are on the x axis with the trial epicentre at x = 0,and the
trial origin time at t =o. Then if the true epicentre is at x and the true origin
time a, and the correction to p is b, the equations of condition are as follows,
suffix r indicating a station on the positive x axis, s one on the negative x axis:
a + bA, -px
= t:,
a + bA,+px
=t i .
Clearly the first set constitutes a set of equations for a -px and b, the second for
a + p x and b. Both a-px and a + p x can be adjusted by corrections to the
epicentre and origin time, and the whole of the information with respect to b is
the same as if we estimated it from the two sets separately and combined the
results afterwards. The separate weights for b will be Z,(A,-&)2
and
and the total weight their sum.
&(A, Suppose however that we simply classified all residuals together and
estimated b by comparing times at different distances irrespective of direction.
The apparent weight would be Xr(Ar-E)2+Zs(As-E)2, where E is now the
mean of all the distances together. If E,=E, the result will be correct. But if
E,#& this will exceed the true weight, and the difference will be serious if one
set consists of a few observations (perhaps one) at a short distance and the other
of a large number at nearly equal distances.
The second procedure is a simple application of the one actually adopted
and indicates how uncertainties may have been under-estimated, since in fact
there were often only one or two observations in some quadrant, all of them
nearer than the bulk of the stations. In some cases groups of data were rejected
for this reason, but no thorough revision has been undertaken. The conclusion
is that the estimates are correct but the uncertainties may be too low, especially
in the comparison of times at small and large distances.
It is an adequate procedure to group by quadrants and form separate solutions
for the quadrants, as in my 1946 paper ; even if dt/dA is not constant, this does not
matter provided the epicentres are reasonably accurate (say to oO.2) and dtldA
does not vary too much. If all four quadrants are available the estimates of a
from the N. and S. quadrants and from the E. and W. quadrants should agree, and
this will provide a further equation ; but in these earthquakes at least one quadrant
was always poorly represented at short distances.
The chief systematic error is that weak movements tend to be read late;
this rule is not, however, universal, because observers sometimes overcompensate
and read a microseism instead. Since we are here directly concerned with
possibilities of systematic error it seemed best to use only stations known to be of
high reliability. In the study of P at large distances and of PKP I found it
desirable also to use only stations with vertical component instruments, but this
seemed unnecessary at distances up to 30'. Earthquakes with many readings
marked e were not used.
2. European and Near Eastern earthquakes.-The earthquakes used are those
used by Bullen and myself in 1935,and by myself in the later number of the
The times of P up to 30'
351
Travaux SCientiJi4lles, together with a few from the I.S.S. for 1937-9. The
stations used are :
Helsinki, Copenhagen, Lund, Scoresby Sound ; Kew ; Hamburg,
Stuttgart, Gottingen, Jena ; De Bilt, Uccle ; Zurich, Chur, Neuchatel ;
Paris, Strasbourg, Besanson ; Rocca di Papa, Taranto, Treviso,
Padova, Moncalieri ; Granada, Almeria, Tortosa, Toledo ; Sverdlovsk
(Ekaterinburg), Andijan, Samarkand, Pulkovo, Baku, Tashkent ;
Theodosia, Simferopol, Yalta ; Ksara, Helwhn, Algiers.
For the 1937-9 earthquakes I included also Basel, Clermont-Ferrand, Rome,
Trieste, Moscow, Tiflis, Athens and Istanbul. The earthquakes discussed
are given in Table I. The reference after " epicentre" indicates the source of the
epicentre used. Some of them proved to be unsuitable for the reasons indicated.
TABLE
I
1926 Aug.
1927 July
July
July
Sept.
1928 Mar.
Mar.
Apr.
Apr.
May
Oct.
1929 May
Aug.
Sept.
1930 Mar.
May
July
July
Oct.
NOV.
1931 Mar.
Mar.
May
1938 Feb.
Apr.
1939 Aug.
Aug.
Sept.
Sept.
30
I
11
zz
11
27
31
14
18
z
15
18
4
15
31
24
5
23
30
I
7
8
20
10
19
2
3
15
22
Epicentre
Source
36".6 N. 23O.05 E. J.B. 1935
36O.7 N. 22".8 E.
,,
31°'9N. 35O-3E.
34'7 N. 54"'O E.
,,
44"-4 N. 34'03 E.
,,
46O.51 N. 13O-25E. H.J. 1936
38O-1 N. z7"-7E.
J.B. 1935
,,
42'2 N. qO-7 E.
,,
42'2 N. 2 5 O - z E.
,,
39O.6 N. 29O.1 E.
28O.6 N. 67O.z E.
,,
40O-23N. 38O.15 E. H.J. 1936
36O.11 N. 3 1 ~ ~ E.
22
>,
39O.90 N. 38O.86 E.
*>
39O.73 N. 23' 48 E.
,,
46"-6N. 10"*6E.
I.S.S.
37O.6 N.
4O.6 W.
41O.1 N. I ~ O - ~ E .
9,
43O.6 N. 13''s E.
,,
~o".o
N. 19''s E.
41''oN. zzO-5 E.
,,
79
41O.oN. zzO.5 E.
37"*4N. 15O.9 W.
,
34O.8 N. 26"-2 E.
>9
,,
39O.5 N. 33O.7 E.
39O.8 N. 29O.6 E.
$3
39O.8 N. 29".6E.
,,
39O.8 N. 29"-6 E.
39''oN.
26"mgE.
I,
9,
Deep focus
Deep focus
Deep focus
Deep focus
9 ,
9,
Multiple ?
9 )
Of the shocks noted as possibly deep, 1926 August 30 and 1927 July 11 were SO
noted by Bullen and myself and not used in forming the P table. 1930November 1
gave residuals with a double maximum of frequency. These five shocks are not
used further in this paper. It is remarkable that we should have hit on four deep
shocks, besides that of 1926 June 26 ; we recognized this one as deep but did not
tabulate residuals, and I afterwards studied it more fully."
The distances were corrected to geocentric ones where necessary by means
of Bullen's table. The times were taken directly from the I.S.S. and the 1939
travel times for a surface focus were subtracted. In the first place I considered
*H. JefTreys, M.N.,Geophys. Supibl., 3, 310-343, 1935.
Harold Jeffrqs
352
stations up to distance 8" and fitted linear formulae a+bA to the residuds for
each quadrant where at least two Observations were available. The quadrants
are indicated by N., E., S., W. In some cases all the distances lay within I" or z0
of one another and the data would give an estimate of b with too low a weight to
be useful, but the data remained available for estimates of a and the standard
error when b is found from other data. The results for b were as in Table 11, in
seconds per degree. n is the number of observations, n - 2 will be the number of
degrees of freedom when the standard error is estimated.
TABLE
I1
b
1930 May 24 E. -1.66
IT. --0. 84
July 23
Oct. 30
1928 Mar. 27
I930 July
1931 Mar.
Mar.
1938 Apr.
N. -1.09
N. -0.84
S. f 1 . 8 8
N. + I . O Z
s.
-1.08
W. -0.08
E. - 1 . 2 5
W. f o . 3 4
8 W. - 0 . 2 0
19 W. - 1 . 0
5
7
weight
4'7
3 '2
8 '5
23.8
5. 8
I2
W(O-C)2
3
4
5
5 *6
6
3
I
4'2
2
4.8
21.8
-7
0'2
I -8
'7
10.6
I -8
5 '0
5.1
2
6.5
9 '5
8 '0
3
4
2 '4
5 '4
1'3
2
1.5
37'4
The weighted mean is b = - 0.44; the standard deviation found from the
residuals is 2S.o on 19 degrees of freedom. There is one outstanding residual,
that for 1930 October 30 S.,which contributes 31 to C ~ ( 0 - c
and
) ~therefore
nearly 8 to x2. This equation rests on three observations whose original residuals
were - 6s, - 5S,oS, and it is likely that the last is a mistaken identification. If we
omit this equation the weighted mean is
b = - 0.57 ;
weight 102.1.
This gives the values ~ ( 0 - cin)the
~ table, and
x 2 = CW(O-C)2/U2 =9'4.
As there are 11 entries, from which one unknown, b, has been estimated, this
value of x2 is on 10 degrees of freedom and is quite satisfactory. We can say
then that
b = ( - 0.57 & o.zo)S/IO.
The need for a substantial negative value of b is strongly indicated.
The use of x2 is explained fully in many places.* Here we need to notice
that the theory of its probability distribution rests on the normal law of error
and the assumption of randomness of the outstanding variation. If Y is the
number of degrees of freedom (d.f.) x2 will usually lie in the interval u + 2 / ( z u )
and will seldom lie outside u & z ~ / ( z u ) . Nearly all practical departures from
the theoretical conditions tend to increase x2, and a normal value of x2 is
provisional confirmation of the hypothesis under test.
In general, seismological observations depart from the normal law of error
in the sense that there are somewhat too many errors larger than 2 0 ; detailed
treatments of this point have been given, but in the present set of observations
* E.g., H. Jeffreys, Theory of Probability, 1939 and 1948.
The times of P up to 30'
353
the discrepancy is not large and it is good enough simply to reject residuals over
f 58. If there are only a few observations in a series it is impossible to say
directly which are the abnormal ones, if any ; but if such a series makes a specially
large contribution to x2 we can say that some observation in it is discrepant and
reject the whole series. If we find that the contributions to x2 of high weight
are normal but some of those of low weight are excessive, we can infer that the
latter are due to departure from the normal law of error and can be rejected for
the same reason as a single large residual can be rejected in a long series-namely,
that in an accurate treatment, taking account of departure from the normal law,
it would have low weight.
The Burton explosion and the revision of near earthquake studies gave
~=(-O~~~O.II)B/I~.
The agreement is perfect; the data together give
b = ( - 0.58 f o.og)8/1".
As a test of generality the solutions were grouped according to longitude of
epicentre at 10" intervals; they occur from 4".6W. to 33".7E., and negative
estimates of b occur in all ranges. There is also no association with quadrant of
azimuth. The results are therefore consistent with uniformity of b along the
Mediterranean region from Spain to Asiatic Turkey. They do not prove such
uniformity, because it is only for earthquakes near the Alps and Italy that the
number of observations at short distances is enough to give determinations
significantly different from zero, but there is nothing in the data against it.
The main difticulty about constructing tables of travel times is that they
have no prescribed functional forms, and the most we can do is to try to
estimate average corrections over given intervals of distance, and afterwards
try to fill in details by considering smaller intervals. The method used in
previous papers has been to select two particularly well-observed earthquakes,
estimate any constant difference between their residuals by the method of least
squares, reduce the data to a common standard, and combine. Other earthquakes could then be built into the scheme one by one. The order has to be
chosen so as at each step to get the greatest possible weight for the comparison.
The method is less accurate than a full least squares solution, which would
estimate simultaneously the constants for all the earthquakes ; on the other hand,
it is much less troublesome in practice, and not much weight is lost if the comparison
of highest weight is taken at each step.
I n the present paper I apply the method in cross-section ; instead of comparing
times at all distances for two earthquakes, I usually compare two ranges of distance
for all earthquakes, beginning with ranges where many observations lie. In
some cases I compared three ranges simultaneously. It is necessary to take some
range as a standard and estimate corrections relative to it. This method is more
convenient here because there are always two or three dominating ranges of
distance but not always two closely comparable earthquakes to use as a starting
point.
I took the times up to 8" to be of the form a + bA, and with the adopted value
of b they gave an estimate of a. a of course depends both on the earthquake
and on the quadrant. For European earthquakes I compared a with the means
for 8" to 12O.5 and 12"-5 to 17O-5. I did not use any earthquake unless it had at
least two observations in one of these ranges ; if an earthquake had only one in
Harold Jeffrey
354
each interval, and all the observations were normal, the loss of weight would be
trivial, and one abnormal observation in three is quite likely. The results were
that, relative to 12"-5 to 17O-5, mean corrections are needed :
0°-8", x =
+ 38.6 & 08.4;
S0-12'-5, y = - 08.1 & 08.4.
The weight of the difference corresponds to a standard error of oS.5. As a
check I estimated y from the separate sets, and the estimates gave x2 = 12 on 2 0
degrees of freedom; then with the adopted value of y the equations for x gave
x2 =28 on 2 1 degrees of freedom. These are quite normal. The positive value
of x of course arises from the smaller value of dt/dA now adopted up to 8"; with
a higher velocity at short distances we need a later origin time to give the same
arrival time at 8". There is no significant change in the differences of travel
times between distances in the interval 8" to 17O-5.
The interval 21'5 to 26'3 was next compared with I" to 17O-5 and gave a
mean relative correction of + 0s-7& 0s-36. The values of a were re-estimated.
The residuals in the comparison gave x2 = 16O.8 on 21 degrees of freedom. This
order of approach was chosen so as to leave the comparison of the critical interval
17O-5 to 21"-5 until a stage when little further improvement in the estimates of a
would be possible. As this interval is so important I thought it best to use only
series in which the weight of a at this stage was at least 8, so that the standard
error of the standard of comparison is not more than 08.7. The values of a were
subtracted from the residuals and gave the distributions in Table 111, centred
on intervals of 1'.
TABLE
I11
I 8"
W.
--I
-I
W.
W.
W.
N.
W.
Sept. 15 N.
--I
-2
1927 Sept. I I
1928 Mar. 31
Apr. 14
May 2
1929 May 18
w.
o
+I
20"
21°
--I
-. 2
--I
+I
0+2
-3
-1,o
--I
+ 3 -5
-3
--I
+-I
+I
+3
-4
-2
-2
+4
-1931May 20 E.
1938 Apr. 19 W.
Sept. 22 W.
Means
Weights
-I9O
-2-1
0 +2
+3
+2 +2
+I
0
0
-2-2
+2
16
-5
-6
+O.I
-0.1
-1,o
1 '4
I0
7
16
-2
-4 -4
-3-2
-0.6
The only possibly significant departure is at zoo. But this depends entirely
on the earthquake of 1938 April 19 ; without this the mean would be - 0.1. The
anomaly does not appear to be due to focal depth, because the interval 21O.5 to
26O.5 gives a mean residual of -08.7, weight 4. I have classified the residuals at
all distances, by simply subtracting the estimate of a (found by a different order
of approximation), and get the means in Table IV.
The weights are simply the numbers of observations, uncorrected for the
uncertainty of the origin times, but the latter is a random error and would be
expected to have cancelled. The interesting point is that this solution also
gives a correction of - 18.0 at 20°, with no appreciable changes on either side.
If this change is genuine it favours the hypothesis that there is no abrupt
change of dt/dA at any distance; but a strong curvature between 17' and 23"
persists.
The times of P up to 30"
355
3. Japanese earthquakes.-In most of the Japanese earthquakes used by
Bullen and myself there were few observations within 20'. These earthquakes
occurred from 1923 to 1928. They were not used for P on account of the
difficulty in fixing epicentres, but some use was made of them for S and SKS.
At that time there was no station to the south-east, and the nearest to the north-west
was Irkutsk, which did not always report. There was a great increase about
1930 in the number of Japanese stations reporting to the I.S.S., and this increase
has continued. Further, though there were already in 1930 many Japanese
observations at short distances, comparatively few stations appeared to be equipped
to record distant earthquakes. In this respect there appears also to have been
an improvement; the outstanding case is 1937 April 16, when there were
observations from 103 Japanese stations from 61' to 82'. (In this case the I.S.S.
uses depth o.o3R, but the pP-P intervals indicate about 0-o6R.)
TABLE
IV
0
8
9
I0
I1
weight
0
0' 0
I2
- I '0
-1.8
15
8
-1.1
8
17
28
I7
24
25
26
27
28
29
30
mean
I2
-0.9
I3
14
15
-0.5
-0.3
16
+0.6
17
+O.I
I8
19
22
+o-2
-0.3
- I '0
+0*3
+0*8
23
+I.I
20
21
-0.1
mean
weight
+O.I
21
-0.5
I1
+04
14
+I.O
2
-1.5
+0.6
4
8
-0.5
2
-2.0
2
fo.3
3
20
32
34
30
32
39
41
21
31
32
33
34
35
36
37
0'0
+0'3
...
...
-1.0
I
3
0
0
I
19
I have not used the data from our 1935 paper in the present study. I have
used some of the small Japanese shocks" (which are comparable with the larger
European near earthquakes), and some more recent ones over a considerable
range of intensity. The selection of suitable shocks was more difficult than in
Europe on account of the frequency of intermediate focal depth and multiplicity.
The former was most directly detected by the negative drift of the P residuals
at large distances. This could however be cancelled by lateness of reading if
the movement was small; but an additional check was often possible from the
interval between P and SKS-this would be diminished both by focal depth
and by lateness of reading of P. It is remarkable how few of the numerous earthquakes that occur are of any use in preparing or even checking the travel times.
The earthquakes actually used were as follows. All data are from the I. S. S.
1930 Mar. 2 2 ; May 16; Aug. 21d 10"; Sept. 4 ; Sept. 17; Sept. z8d 9";
Oct. z ; Oct. 16d 21h32m; Oct. 16d zxh 36m ; Nov. 25 ; Dec. zod 14h zm.
1931 Jan. 6 ; Feb. 16; Mar. 3 ; Mar. 6 1 ; Mar. 611; Mar. 11;
Mar. 19; Apr. 9; Sept. 8; Dec. 26. 1937 Jan. 5; Jan. 7 ; Feb. 21.
1938 Jan II ; Nov. 5 ; Nov. 76 oh; Dec. 6. 1939 May ~d sh.
* H. Jeffreys, M.N., Geophys. Suppl., 3, 401-422,
1936 (especially pp. 407-410).
Harold Jefreys
356
For the earthquakes of 1930 and 1931,I used only the stations known to have
been of high reliability in that period, namely Akita, Aomori, Chiufeng, Gihu,
Hamada, Hikone, Hong Kong, Hukusima, Kobe, Kosyun, Koti, Kumagaya,
Kumamoto, Kusiro, Kyoto, Matuyama, Misima, Mizusawa, Morioka, Muroto,
Nagasaki, Naha, Nemuro, Oiwake, Phu-Lien, Sapporo, Sendai, Simidu,
Siomisaki, Sumoto, Taihoku, Tainan, Taityu, Tientsin, Toyooka, Urakawa,
Wakayama, Yokohama, Zi-ka-wei, Zinsen. For the later period the stations
appeared to be much more uniform in quality and the amount of material was so
great that anomalous readings were easily detected.
The earthquake of 1938 November 5 and its large aftershock on November 7
were treated separately; these were the first and a successor in a series of several
hundreds. The others, however, mostly showed irregularity of the residuals,
pointing to weakness or multiplicity. I compared the residuals in these two
directly. Stations showing widely discrepant results were dropped (and afterwards treated with suspicion in dealing with other shocks, though not absolutely
rejected).
Forms a + b A were fitted to the observations (by quadrants) up to 8" and
gave the results in Table V.
TABLE
V
1930 Mar. 22
-4ug. 21
Sept. 4
I930 OCt. 2
Oct. 16 I
Oct. 16 I1
Nov.
25
(Dec. 20
1931 Jan. 6
Feb. 16
1931 Mar. 3
Mar. 6 1 .
Sept. 8
Dec. 26
b
w. +0.8
S. -1.7
W. -0.41
S. -1.63
w. +O'&
E.
W.
N.
W.
E.
S.
S.
-2.1
+0.24
-0.08
-0.46
+1-05
-0.19
-0.29
w- +0*93
W. -0.25
N. -0.17
E. foe32
weight
11
2
2 '0
*8
17'4
2.7
9 '7
3 '9
9 '7
10.3
43'1
3
6
4
4
4'0
I
21.8
36-5
29'3
8 -6
7 '5
41 '7
31'5
N.
+0*21
Jan.
Feb.
1938 Jan.
7
21
II
S. +0-60
W. +0.09
W. -0.57
N. +0*09
E. +0.96
w. +0*08
1939 May
1
S. +om19
111-5
13.8
30.0
95 -8
17.0
104.3
19.3
58.9
41 '7
5
14'0
I '5
0'1
8
0 '0
N. -0.29
w. -0'02
28.2
162.3
190.5
3 '9
8
0.5
5
10.5
8
0.7
I0
0'0
I2
7 '3
54'9
37
0'0
I1
2'0
26
28
0.8
I2
9.5
5 '0
18
I '0
22
10-6
0.7
23
28
0'0
29.9
492'3
1938 Nov. 5 & 7
I 'I
5.8
3' 5
4
6
14
7)
255'7
E. +0'57
w(O-C)2
2 '0
I1
48
1' 5
0.3
I -8
The times of P up
to 30"
357
The first series gave b = -0.11, with weight 27.7.
The residuals led to a standard error of 16-95for one observation. The sum
Z;W(O-C)~
was 846, leading to x 2 = 2 2 . 3 on 15 degrees of freedom. 29.4 of
the sum, however, caFe from 1930 December 20, the residuals of which became
irregular beyond 6", and it seemed best to drop this series. This gives the
revised values and W(O-C)~,
with b now = - 0.16 f 0.13. x2 is now 14.5 on
14 degrees of freedom and is normal.
The residuals for the later period gave u = 28-01on 203 degrees of freedom ;
those for 1938 November 5 and 7, taken together, gave u = 18-49 on 55 degrees
of freedom. The decrease is, of course, because most of the data are means of
two observations.
The second series gave b = + 0.18 k 0.091, xa=7*4 on 8 degrees of freedom.
The data for 1938 November 5 and 7 give b = -0.06 f 0.11, xa=0.8 on I
degree of freedom. Finally the combination of the three series gives
Early series
b= -0.16f 0.13
1938 Nov. 5 & 7 b = - 0 ~ 0 6 ~ 0 ~ 1 1
Rest of 1937-9
b= +o.18f 0.091
I t was thought that the early series, being mostly small earthquakes, might give
a positive systematic error. The difference is actually in the opposite direction,
and we can conclude that no such error was present.
TABLE
VI
Mean n
8" -0.6
9' -0.9
12
9
10'
+-0.7 19
11'
-0.5
15
12" +o-2
8
13" +om4
14" -1.2
9
10
Mean
Mean n
91
15' -0.5
16" -0.8
17" + O * I
18' -2.0
19" -0.51
?
20'
6
22"
-0.4
5
9
9
4
23'
-2.0
24O
-3.7
-1-3
-1.2
5
21"
25"
26" -4.0
27' -2
28' -2.0
7
3
3
3
I
I
3
Mean
11
29" - 1 . 0 3
2
30' -1.5
0
31"
2
32" -0.5
0'0 2
33"
... 0
34O
... 0
35"
36" -1.0 I
...
The weighted mean is +0-03f0.05. This gives x2=5-5 on 2 degrees of
freedom, which is rather on the large side but calls for no special comment.
The conclusion is that the data for Japan are quite consistent, and that the times
up to 8" are unlikely to need any change exceeding 08.5. There is a clear regional
difference between Europe and Japan.
In extending the table I began by comparing the ranges 0"-8" and S0-zoo,
rejecting ragged series. This led to the mean difference [email protected]+@-08,
x2 = 12-5 on 12 degrees of freedom. Then 0"-20" was compared with 2oo-3o0
and gave - 18.7 f 08.4; x2 = 24 on 12 degrees of freedom. This would be rather
large if it stood by itself, but even if the hypotheses leading to the x2 theory are
correct large departures would be expected occasionally, and this is not too large
for a selected value. Then 0'-30" was compared with 30°-40" and gave
- 28.0 f e.9, xa =6-9 on 5 degrees of freedom. The general conclusion is that
the tables need no alteration up to 20" in Japanese earthquakes but possibly need
a reduction of about 2 s between 20" and 40".
T o study the distribution of the corrections in more detail the residuals were
corrected for the changes of origin time and classified at intervals of I". The
summaries are given in Table VI.
G 26
358
Harold Jeffreys
Unfortunately the data do not decide anything definitely about the times at 18"
to 20". The distributions of residuals near these distances are as follows:
Residuals
17"
I 8"
19"
-8
20"
2
-7
I
0
I
21°
-6
I
2
I
I
0
0
0
-3
I
-2
o
I
4
5
0
I
2
2
0
3
3
1
0
1
I
0
0
I
O
I
O
I
I
0
I
0
0
0
0
1
0
1
0
2
0
2
I
I
2
-4
-5
-I
1
2
0
0
0
2
1
Early readings occur occasionally at all distances, but from 18" to zoo they are
much commoner than elsewhere, and there is no definite concentration. Further
information was found in the Marianne Islands earthquake of 1930 October 24,
which gives a long series of highly consistent observations from 18" to 25" in
the north quadrant. All stations were used except that a few large residuals
were rejected. The means were:
Mean
n
10'
19'
20'
-I
18"
-0.6
1
5
0-0
-0.5
1
2
6
21"
22"
+1*5 +1*5
6
2
23" 24O
-I
I
25'
+z
+I
I
I
28"
o
1
Comparison with the main series gave a difference of + zS.2, weight 9-4, x2 =3*3
on 5 degrees of freedom. Correcting for this and combining with the main
series we have :
10"
Mean
+0-5
n
20
18"
19'
20"
-2-8
-2.2
-2.7
6
5
I2
21"
-1.1
XI
22O
-0.5
9
23"
-2.3
4
24"
-2.8
4
25"
-1-3
4
28"
-2.0
I
my previous study of these I took the
epicentres as definite, except for the Eureka earthquake of 1932 June 6. For this
the instrumental evidence and the field evidence led to epicentres about 0"-z
apart, and I used only stations in the south-east quadrant. The earthquakes
used were those in Table VII.
4. North American earthquakes.-In
TABLE
VII
1931 Apr.
Aug.
1932 June
July
July
Dec.
1933 Mar.
Mar.
19
23
6
7
12
21
II
26
20O.7 N. 109"*1W.
40O.o N . 126".2W.
40°*75N. 124"*5W.
29"-IN. 113"*5W.
26"*6N. I I O O - I W.
38".7N. I I ~ O - ~ W .
33".6N. 118"*0W.
42O.5 N . 129'.0 W.
Unfortunately there were usually too few observations between 1"-5 and 8" to
permit a direct determination of the velocity at these distances, and I proceeded
- I ) ~directly to the residuals as given in the
by fitting cubics a + bA + O-OOIC(A
I. S. S., these being first classified by quadrants. (Actually the stations were
usually strongly concentrated about one azimuth.) The ellipticity corrections
applied to the distances and to the 1935 tables then used for comparison would
be small up to zoo and nearly cancel. In my previous paper I used this formula
to 19"; here, on account of the doubt at 17",I use it only to 15". It is already
clear from the residuals in my previous paper that no serious change is to be
expected, but the uncertainties found might have been a little too low. The
standard error of an observation is 18.92 based on 55 degrees of freedom. The
results were
c = - 1-28_+ 0.86 ;
b = + 0.51 & 0.19.
The times of P up
to 30"
359
My previous result was c = - 0.62 f 0.51 ; rejecting this as not significantly
different from o gave b = + 0-200 f 0.066. The same treatment here would make
b = + 0.26 0.09.
Unfortunately b = + 0.51 on inspection looked inconsistent with the trends
of the residuals up to 8",where the contribution from the c term would be only
about 0s-4. I therefore made also a set of solutions a + bA from the observations
up to 8"; these gave b = -0.03 ~ O * Z I .Having regard to the apparent uncertainties, and remembering that the data for the second solution are included
in those for the first, I doubted whether the cubic is in fact a satisfactory representation up to 15". This was tested as follows. For each earthquake a and the
residuals were estimated from the cubic formula ; these were classified for the
ranges 0°-5", 5"-10° and 10"-15". The mean residuals for the three ranges
were + 0.1,weight 14, - 0.3,weight 23 and + 0.4,weight 17. (The difference
of the weighted sum from zero is due to rounding-off errors.) Direct comparisons
of the ranges gave for 5"-10° against 0°-5", - 0.5, weight 6.2 ; and for 10~--15~
against 5"-10", +0-7, weight 7.9. These give x 2 = 2 - 5 and 2-3 respectively,
on 3 degrees of freedom. There is no evidence against the cubic formula up to
1 5 O , but the uncertainty of c is great and is reflected in that of b.
Gutenberg's 1948 paper gives data for 34 earthquakes and plots travel times
(observed time-origin time) on a diagram. The method seems to me profoundly
unsatisfactory. It is said that errors in the distances ' I rarely exceed one degree,
and usually they do not exceed half a degree". I n consequence, if the origin
times are right, errors in the calculated arrival times of 78 are to be considered
as normal, and of 148 as occurring occasionally. Without some means of compensating for thesethere is no guarantee that similar errors will not be transmitted
to the "corrections" to the tables. There must have been some compensation
because in fact no observation departs by more than 78 from a smooth curve, and
the reason is presumably that errors in the epicentres have largely been cancelled
by errors in the origin times, so that the mean time of arrival at the observing
stations is approximately correct. But in that case the times at the stations
themselves have been used in estimating the origin times and the uncertainties
are not independent. At any rate, not much progress could be made in determining epicentres of Pacific coast earthquakes without use of P at these stations.
In any case graphical methods of calculation are obsolete. It is well known
by this time that results obtained by such methods often differ from the least
squares solution by more than the standard error of the latter ; and sometimes in
fitting a linear formula they differ by more than would be expected if only two
data selected at random were used. It is really remarkable that physicists will
spend months in making observations and grudge the day or so needed to express
the results in a form that will make valid tests of consistency possible.
If the stations used are approximately in the same azimuth (which cannot be
checked from the paper, since epicentres are not given) it is possible to use the
methods used so far in this paper. This means ignoring all earthquakes whose
times are given for only one station, since the uncertainty of the origin time is
unknown. What I have done, to economize arithmetic, is to fit linear formulae
to the times for other earthquakes and replace the data for each by a difference
between times at two summary values of the distance ; these values are chosen
so that the difference is independent of thc curvature of the time-curve. The
standard error of one observation was found to be 18.8 on 53 degrees of freedon
G 26*
Harold Je#reys
The equations found for earthquakes both of whose summary values lay
within 15" were as in Table VIII.
Assuming that the correction to the difference between the two times of
transmission takes the form
b(Az - A,) + O.OOIC{(A,- I ) ~ (A, - I)~},
I was led to the normal equations for b and c:
96.ob + 25.62~ = + 24-2,
25.62b+ 7-692c= + 7-42.
whence
b = + 0.006 f 0.59.
c = + 0'92 f 2-08,
T h e large uncertainties are due, of course, to the fact that in most of the earthquakes the observations cover only a few degrees of distance, and there is a
considerable chance that errors will accumulate when the ranges are connected.
TABLE
VIII
Difference of
Of A
corrections to
(degrees)
time
5 '6,
6.3,
8.4,
6.1,
7.6,
6.6,
8.3,
11.2,
11.2,
8 '4
9 '7
11.0
9'3
9.6
9 '4
12.9
+I.I
13- 2
13.2
+
weight
I
'5
I '0
I '0
+0.6
-0.3
+2'4
-0.6
I .8
0.8
+0'5
1 '3
+I-6
+1.6
1'2
I '2
I '2
1 '3
Californian data have been used directly for velocities at short distances by
Gutenberg." I have discussed the former set previously, but I now prefer to
fit a+bA for each earthquake separately and combine the estimates of b. As
Gutenberg's distances are given in kilometres, I take A to be in hundreds of
kilometres. Azimuths are not given, but the possibility that errors in the epicentres lead to an underestimate of the uncertainty can be checked by computing
xZ for the separate estimates.
For the solutions of appreciable weight from the 1932 paper, with a standard
velocity of 7-g4km/s, corresponding to 138*gg/1", the weighted mean for b is
+os~o3/1ookm. The standard error of one observation is found to be @-47,
based on 22 degrees of freedom, and the uncertainty would be @*074/100km.
Comparison with the separate estimates, however, gives x2 = 15.4 on 8 degrees
of freedom, and it seems safest to multiply the uncertainty by 1.4. Reducing to
degrees and to the standard 148-28/1" we have
b = - 0.26 f 0.11.
The 1 9 4 paper includes earthquakes from 1935 to 1941. The separate
observations are not given, but a determination of the velocity, with its standard
error, is given for each earthquake. No allowance has been made for uncertainty
of the epicentres. The summary value for the velocity is 8.064fo-o11km/s.
The separate values give x2 = 6.7 on 12 degrees of freedom, indicating either
that errors of the epicentres are small or that the bulk of the observations are near
* B. Gutenberg, Gerhnds Beitr., 35, 6-50, 1932; Bull. Seism. SOC.Amer., 34, 13-32, 1944.
The times of P up to 30"
361
one &muth. The standard error of one observation appears to be under 0s-4.
The determination is equivalent to b = - 08-50 f~ O Z O / I " .
Reducing the equations so that unit weight corresponds to standard error
",we have the following six equations with independent uncertainties :
I~/I
I.S.S. data
Gutenberg 1948
Gutenberg 1932
Gutenberg 1944
138*9b+27*9c=+ 35.68
3o*ob+ 8*0c=+
7-56
8o.ob
=20.8
2 5oo.ob
- 1250.0
-_
X2
6.2
0-0
1-50~=-1*82
026c=+0*10
Xa
21.7
1.3
3-9
1.0
11'1
23 so
The least squares solution is b = - 0.480k 0.018, c = + 2-62 f 0.33. Prima
fucie the value of c would confirm the suggestion that dt/dA is practically constant
However, x2 =34*1on 4 degrees of freedom and there is clearly some
up to 15'.
inconsistency. The first two series are the only ones that give estimates directly
of c and agree reasonably between themselves; the greater part of the anomaly
arises from the comparison of these two with the values of b found from the near
earthquakes.
In the 1944 series systematic corrections for the heights of the stations have
been applied, and certain stations have been omitted on account of thickness of
sediments or for other reasons. Gutenberg suggests that instrumental improvement is responsible for the low standard error of one observation. In fact,
however, it is not much less than that for the 1932 series; and the 1944 series
includes earthquakes from 1935 to 1941,and in the 1948 series, which includes
earthquakes up to 1946,the standard error of one observation was four times
as great.
I have a suspicion that highly sensitive instruments are not an unmixed
blessing, because they may pick up a small forerunner due to scattering in advance
of the main movement. There seems to be no immediate reason for rejecting
the I.S.S. data, since most of the earthquakes were recorded at much larger
distances and late reading of weak movements is unlikely. But at present all
that can safely be said is that there is an obvious anomaly with no obvious
explanation.
I have attempted a comparison with other ranges of distance for the North
American earthquakes, but without success. The trouble ultimately is that the
distribution of stations is seldom such that a long range of distance is covered in
a single quadrant, and consequently when times at widely different distances
have to be compared it is impossible to separate errors in the tables from errors
in the epicentres with any accuracy.
I have made solutions for those of Gutenberg's 1948 earthquakes that include
observations beyond 15". Six, numbers 18, 19,20, 21, 33 and 34, gave some
information. They indicated the following differences, in comparison with the
1939 table :
I7'-15'
20'22'-
19'
19'
23'-21'
- d * 9 f 08.4
fd . 5
-d-4f d . 3
-08.4+ 18.4
-08. I
Together they suggest a slight decrease in the times relative to 15',
them is significant by itself.
but none of
362
Harold Jejfreys
5. Comparison of regions.-For European earthquakes the times, relative to
the range 12".5-17".5, have been seen to need an increase of 38.6-0.58A up
to 8". My 1939 times for a surface focus at short distances correspond to 65.6
14'288, so that the total would be 108.4 + 13'7oA. We compute this in Table IX
and compare directly with the table.
+
TABLE
IX
Table
35 '4
63 '9
A
2
4
6
1o.4+ 13'70A
37'8
65 '2
92.6
92.2
8
I 20.3
120'0
1 47 '4
I4
148.0
161.7
175.3
188.7
201.9
I5
215'0
16
17
228.0
I0
11
I2
I3
Difference
161.1
174.8
188.5
202'2
215.9
229.6
243 '3
240.7
The mean found for 8"-12"-5 was - 0.12 0.4,so that the data up to 1 2 " 5
and indeed to 14",are consistent with a linear formula. This is remarkable,
seeing that the J.B. times are approximately equivalent to including a term
-0*0025(A-1)~, which would amount to -36.3 at 12". The change at short
distances has practically cancelled the cube term. This result may be compared
with one in which I included a fourth power up to 10" and got*
T=
-I- 0 7 5
r;l)s r134*
r133*
+ 14~102A-0*111 -
- 0.926
-
The standard error of the last coefficient was 0.69, and I dropped it as not
significant. The best fit by a cubic formula was
T = - 0.01 14-278A- 2.149
-
TABLB
X
o
m
s
3 35'9
16
49'2
I7 4 1'9
18
14.1
19
25.0
20
35'9
15
o
m
s
21
22
4 46.6
23
24
25
5
26
57.2
7'7
17.8
27'5
36.8
o
m
a
27
28
29
5 46.0
55.0
6 3.8
30
12.5
Thus even in 1936 there was a possibility that the cubic term was in fact negligible
in comparison with the fourth power term. The present data indicate no change
in the range 1 2 O . 5 to 1 7 O - 5 , which was taken as standard, and a mean increase of
about 08.7 from 21'-5 to 26O.5. There is no definite change at greater distances.
In the gap we can take the corrections from 06 to - 16.0. It is clear that a curvature
begins about 15". Now it is just in this range that Lehmannt found that the
amplitudes of P were small, becoming large before 18". It may therefore be
supposed that appreciable curvature begins between 15" and 18".
On the hypothesis of a change of curvature at 15" and continuous curvature
up to 30" the times in Table X are inferred.
* H. Jeffreys, Bur. Centr. S&m.,Trav. Sci., 14, 50, 1936.
t I. Lehmann,Geod. Inst. Copenhagen, Medd., 5, 1934.
The times of P up to 30"
363
This is only one of many possible interpretations of the data. It should be
recalled that Lehmann found what appeared to be a second P a little beyond zoo
and attributed it to Pd, a continuation of the branch up to 19". On examination
I found the readings to be a few seconds too late for such an interpretation. If in
fact there is a discontinuity of slope, but it enters at some distance before IS", her
interpretation would be possible and the hypothesis of two intersecting branches
is tenable. In dealing only with first arrivals the cubic formula has a great
advantage, since it contains four adjustable constants, the same number as two
linear forms. If besides a sharp change of slope we have to estimate a curvature
on each side of it, as many parameters have to be found as for a quintic, which
could be made to fit almost anything.
At short distances in Japan dt/dA is as well determined as in Europe, but the
values differ by 085/1". If the times about 15" are taken as standard, the 1939
table is about right up to 17", but needs a reduction of about 28 beyond 22".
This reduction may begin about 18". I n European earthquakes a slight increase
was found beyond 22" on the same standard. The differences between the times
at, say, 2" and 25" are about equal.
It seems probable that in Californian earthquakes dt/dA at short distances
is nearly as small as in Europe, but the data show serious anomalies, and no
satisfactory explanation seems available.
It appears that up to 15" the cube term is practically absent in Europe, but the
Japanese times agree reasonably with the table ; the Californian ones may agree
with European ones, on some interpretations of the data.
The times in Europe about 20" depend on the treatment of a single earthquake.
The rest of the data would support a sharp bend about zoo as in previous
solutions ; but if this earthquake is retained the bend would have to be moved
to a shorter distance, possibly 15".
It is desirable that the comparison should be extended to greater distances,
but this would require revision of the epicentres and recalculation of the distances.
It should be possible, for instance, to fix Japanese epicentres sufficiently accurately
to test whether the times of transmission to equal distances in Europe and
America differ.
6. The variability of the cube term may be connected with an anomaly that
has been known for a long time, but, so far as I know, has not been mentioned
in print. One naturally interprets increase of velocity with depth as due to
increase of elastic moduli under pressure. According to my solution for the
velocities from the 1939 table the velocity of P varies by about 3 per cent for
64 km of depth, say I per cent in 20 km. For most rocks tested in the laboratory*
the compressibilities drop by about 15 per cent in the range*of pressure from
z x 1o9 to ro10dyne/cm2, which would correspond to depths from 6 to 30 km;
thus the expected variation of the velocity would be about 6 per cent in 20 km.
There is a discrepancy by a factor of 6. It was thought that the effect of pressure
might be largely cancelled by that of temperature. Data on the latter are rather
scanty, but data on the rigidity given by Birch, Schairer and Spicer, p. 83, suggest
a variation of about 1.5 x I O - ~ parts per degree for dunite. If the temperature
gradient in the ultrabasic layer is 7 deg./km, as seems reasonable, this would suggest
a variation of about z per cent in 20 km, which is of the right order of magnitude
* F.Birch, J. F. Schairer and H. C. Spicer, Handbook of Physical Constants., Geol. SOC.Amer.,
Spec. Papers No. 36, p. 62, 1942.
Harold Jeffreys, The times of P up to 30"
364
but still too small. With any estimate of the cube term in the times of P there
is a difficulty in explaining why it is so small. It seems to be necessary either
that (I) the effect of pressure diminishes rapidly with depth, ( 2 ) that of temperature increases rapidly with depth, or (3) there is increasing admixture of lowvelocity material. But whatever the reason, some unconsidered factor must
cancel five-sixths of the effect of pressure, even in Japan, and it would not be
surprising if in some regions it should exceed it and give a layer of low velocity.
So far as the data from times are concerned such a region is most likely to be
in Europe. Data on amplitudes could help, but they would require detailed
study of the variation of amplitude over a long range of distance. The amplitudes
in a single earthquake usually vary rather irregularly (possibly owing to irregularities in the MohorovitiC discontinuity) and we should need to be able to
compare any systematic variation with distance with peculiarities of individual
earthquakes and stations.
It is in Japan that the near observations of S are far more consistent in deep
shocks than in normal ones. Gutenberg has pointed out to me that this would
be in agreement with the existence of a layer of low velocity, since waves from
a shallow focus would spread out in such a layer, but those from a deep one would
penetrate it steeply.+ Somewhat against this is the fact that the times in Japan
give the best evidence for increase of velocity with depth.
After finding that normal shocks gave no decisive evidence for or against a
surface of discontinuity of velocity as an explanation of the curvature of the timecurve of first arrivals near 20°, I foundt that in Japanese deep-focus earthquakes
the times in a critical range of distance were in agreement with the hypothesis of
discontinuity and a little too early for the contrary hypothesis. If, however, the
times in normal Japanese earthquakes beyond 20" need a correction of about
-z*, this argument loses its weight.
Gutenberg maintains that P actually disappears about 15" and that there is
a gap before the new branch enters at 16". I have occasionally found residuals
of about + 6s at several consecutive stations, but at all distances there is a definite
concentration about a value close to the tabular time and I have not found these
positive residuals to be notably more frequent at one distance than another.
I 60 Huntingdon
Road,
Cambridge :
1951December 17.
* B. Gutenberg and C. F. Richter, Bull. Seism. SOC.Amer., u),531-538, 1939.
t H.Jeffreys, M.N., Geophys. Suppl., 4,452, 1939.
to P from deep shocks in the Andes.
The data refer