THE DEPTH OF AN EARTHQUAKE CENTRUM.
The depth of an earthquake centrum—Greatest possible depth of an
earthquake—Form of the focal cavity.
_Depth of centrum._—The first calculations of the depth at which an
earthquake originated were those made by Mallet for the Neapolitan
earthquake of 1857. These were made on the assumption that the earth
wave radiated in straight lines from the origin, and, therefore, at
points at different distances from the _epicentrum_ it had different
angles of emergence. These angles of emergence were chiefly calculated
from the inclination of fissures produced in certain buildings, which
were assumed to be at right angles to the direction of the normal
motion. If we have determined the _epicentrum_ of an earthquake and
the muzoseismal circle, and make either the assumption that the angle
of emergence in this circle has been 45° or 54° 44′ 9″ (see page
54), it is evidently an easy matter by geometrical construction to
determine the depth of the _centrum_. Höfer followed this method when
investigating the earthquake of Belluno.
Other methods of calculation which have been employed are based on
time observations, as, for instance, the method of Seebach, the method
of co-ordinates, the method of hyperboloids or spheres (see pages
200–212).
By means of a number of lines parallel to twenty-six angles of
emergence, drawn in towards the seismic vertical, Mallet found that
twenty-three of these intersected at a depth of 7⅛ geographical miles.
The maximum depth was 8⅛ geographical miles, and the minimum depth 2¾
geographical miles.
The mean depth was taken at a depth of 5¾ geographical miles where,
within a range of 12,000 feet, eighteen of the wave paths intersected
the seismic vertical.
The point where these wave paths start thickest is at a depth not
greater than three geographical miles, and this is considered to be the
vertical depth of the focal cavity itself.
For the Yokohama earthquake of 1880, from the indications of
seismometers, and by other means, certain angles of emergence were
obtained, leading to the conclusion that the depth of origin of that
earthquake might be between 1½ and 5 miles.
Possibly, perhaps, the earthquake may have originated from a fissure
the vertical dimensions of which was comprised between these depths.
A source of error in a calculation of this description is that the
vertical motions may have been a component of transverse motions or
perhaps due to the slope of surface waves.
The following table of the depths at which certain earthquakes have
originated has been compiled from the writings of several observers.
+---------------------------------+-----------------------------+
| | In feet |
+----------------+----------------+---------+---------+---------+
| | | Minimum | Mean | Maximum |
| Rhineland | 1846 (Schmidt) | | 127,309 | |
| Sillien | 1858 (Schmidt) | | 86,173 | |
| Middle Germany | 1872 (Seebach) | 47,225 | 58,912 | 70,841 |
| Herzogenrath | 1873 (Lasaulx) | 16,553 | 36,516 | 56,477 |
| Neapolitan | 1857 (Mallet) | 16,705 | 34,930 | 49,359 |
| Yokohama | 1880 (Milne) | 7,920 | 17,260 | 26,400 |
+----------------+----------------+---------+---------+---------+
A table similar to this has been compiled by Lasaulx.[86]
With the exception of the determination for the two last disturbances
these calculations have been made with the assistance of the method
of Seebach, which depends, amongst other things, on the assumptions
of exact time determinations, direct transmission of waves from the
centrum, and a constant velocity of propagation.
Admitting that our observations of time are practically accurate, it
appears that the other assumptions may often lead to errors of such
magnitude that our results may be of but little value.
From what has been said respecting the velocity with which earth
disturbances are propagated, it seems that these velocities may vary
between large limits, being greatest nearest to the origin.
If we refer to Seebach’s method, we shall see that a condition of this
kind would tend to make the differences in time between various places,
as we recede from the _epicentrum_, greater than that required for the
construction of the hyperbola. The curve which is obtained would, in
consequence, have branches steeper than that of the hyperbola, and the
resultant depth, obtained by the intersection of the asymptotes of this
curve with the seismic vertical, indicates an origin which may be much
too great.
Another point worthy of attention, which is common to the method
of Mallet as well as to that of Seebach, is the question whether
the shock radiates directly from the origin, or is propagated from
the origin more or less vertically to the surface, and then spreads
horizontally. We know that earthquakes, both natural and artificial,
may be propagated as undulations on the surface of the ground, and
that the vertical motion of the latter, as testified by the records
of well-constructed instruments, has no practical connection with the
depth from which the disturbance originated.
In cases like these, the direction of cracks in buildings, and other
phenomena usually accredited to a normal radiation, may in reality be
due to changes in inclination of the surface on which the disturbed
objects rested. When our points of observation are at a distance from
the _epicentrum_ of the disturbance which, as compared with the depth
of the same, is not great, calculations or observations based on the
assumption of a direct radiation of the disturbance may possibly lead
to results which are tolerably correct. The calculations of Mallet
for the Neapolitan earthquake appear to have been made under such
conditions.
For smaller earthquakes, and for places at a distance from the seismic
vertical of a destructive earthquake, the results which are deduced
from the observations on shattered buildings, and all observations
based upon the assumption of direct radiation, we must accept with
caution.
Another error which may enter into calculations of this description
is one which has been discussed by Mallet at some length. This is the
effect which the form and the position of the focal cavity may have
upon the transmission of waves.
Should the impulse originate from a point or spherical cavity, then
we might, in a homogeneous medium perhaps, regard the isoseismals as
concentric circles, and expect to find that equal effects had been
produced at equal distances from the _epicentrum_. Should, however,
this cavity be a fissure, it is evident that even in a homogeneous
medium the inclination of the plane of such a cavity will have
considerable effect upon the form of the waves which would radiate from
its two walls.
For example, let it be assumed that the first impulse of an earthquake
is due to the sudden formation of a fissure, rent open from its centre,
and that the waves leave the walls at all points normal to its surface.
Then, as Mallet points out, it is evident that the disturbance will
spread out in ellipsoidal waves, the greatest axis of which will be
perpendicular to the plane of the fissure.
By taking a number of cases of fissures lying in various directions
and drawing the ellipsoidal waves which would result from an elastic
pressure, like that of steam suddenly admitted into such cavities, the
differences in effect which would be simultaneously produced by these
waves on reaching the surface can be readily understood. The following
example of an investigation on this subject will serve as an example to
illustrate the general nature of the many other cases which might be
taken.
[Illustration: FIG. 33.]
Let a disturbance simultaneously originate from all points of the
fissure _f_ _f_. This will spread outwards in ellipsoidal shells to
the surface of the earth _e_ _e_. The major axis of these ellipsoidal
shells will be the direction of greatest effect. In the direction _c_
_d_ the waves will plunge into the earth, and places to the right side
of the fissure will, to use an expression due to Stokes, when speaking
of analogous phenomena connected with sound, be in _earthquake shadow_.
The same expression has been employed, somewhat differently, when
speaking of the effects produced on buildings.
For places, like _s_ and _p_, situated at equal distances from the
seismic vertical, it is evident that the intensity of the shock will be
different, and also its time of arrival. It will also be observed that
the isoseismals will take the form of ovals or distorted ellipses, the
larger or fuller end of which being to the left of the fissure.
Other cases, like those just given, which are discussed by Mallet in
his account of the Neapolitan earthquake, are where the fissure forms
the division between materials of different elasticities. In the hard
and more elastic material the waves will be more crowded, the velocity
of a wave particle will be greater, and the transit will be quicker
than in the less elastic medium.
The result is that the distance of equal effect from the seismic
vertical will be greatest in the direction of the more compressible
material.
Unless these considerations are kept carefully before the mind when
investigating the depth and, we may add, the position and form of the
centrum of an earthquake, serious errors may arise.
_Greatest depth of an earthquake origin._—A curious but instructive
calculation which Mallet made was a determination of the greatest
possible depth at which an earthquake may occur. This calculation is
based upon the idea that the impulsive effect of an earthquake has an
intimate relationship with the height of neighbouring volcanoes, the
column of lava supported on a volcanic cone being a measure of the
internal pressure tending to rupture the adjacent crust of the earth.
Mitchell, in 1700, virtually propounded this idea, when he suggested that
the velocity of propagation of an earthquake was related to the height
of such a column.[87]
Mallet showed that there was probably considerable truth in such a
supposition by appealing to the results of actual observation. The
pressure gauge of the Neapolitan district would be Vesuvius, the height
of which has in round numbers varied between 3,500 to 4,000 feet. One
of the most destructive earthquakes in this district—namely, the one of
1857—projected bodies with an initial velocity of about fifteen feet
per second. The Riobamba earthquake, which projected bodies with an
initial velocity of eighty feet per second, appears to have been the
most violent earthquake, so far as its impulsive effort is concerned,
of which we have any record. It occurred amongst the Andes, where there
are volcanoes from 16,000 to 21,000 feet in height.
Comparing these two earthquakes together, we see that the Riobamba
shock had a destructive power 5·33 times that of the Neapolitan shock,
and we also see that the Riobamba volcanoes were about 5·33 times
higher than Vesuvius. The accordance in these quantities is certainly
interesting, and tends to substantiate the idea that volcanoes are
barometrical-like pressure gauges of a district.
Carrying the argument still further. Mallet says that if the depth of
origin of earthquakes were the same, then the _area of disturbance_
would, for like formations and configuration of surface, be a measure
of the earthquake effort, and also some function of the velocity of the
wave. From this we may generally infer ‘that earthquakes, like that of
Lisbon, which have a _very great area_ of sensible disturbance, have
also a very deep seismal focus, and also the greatest depth of seismal
focus within our planet is probably not greater than that ascertained
for this Neapolitan earthquake, multiplied by the ratio that the
velocity of the Riobamba wave bears to that of its wave, or, what is
the same thing, by the ratio of the altitudes of the volcanoes of the
Andes to that of Vesuvius.’
Now, as the depth of the Neapolitan shock may be taken at 34,930
feet, the greatest probable depth of origin of any earthquake impulse
occurring in our planet is limited to 5·333 × 4,930 feet, or 30·64
geographical miles.
Ingenious as this argument is, we can hardly admit it without certain
qualifications.
First, we are called upon to admit the identity of the originating
cause of the volcano and the earthquake—as to what may be the
originating cause of earthquakes we have yet to refer, but certainly in
the case of particular earthquakes, as, for instance, those which occur
in countries like Scotland, Scandinavia, and portions of Siberia, the
direct connection between these phenomena are not at first sight very
apparent.
Secondly, even if we admit the identity of the origin of these
phenomena, it is not difficult to imagine that the fluid pressure
brought to bear upon certain portions of the crust of the earth may
possibly in many instances be infinitely greater than that indicated
by the height of the column of liquid lava in the throat of a volcano,
the true height of which we are unable to obtain. Further, in certain
instances such a column only appears to be a measure of the pressure
upon the crust of the earth in the immediate vicinity of the cone.
Thus, in the Sandwich Islands, we have lava standing in the throat
of the volcano of Mauna Loa 10,000 feet higher than it stands in the
crater Kilauea, only twenty miles distant. That these columns should be
measures of the same pressure, originating in a general subterranean
liquid layer with which they are connected, is a supposition difficult
to satisfactorily substantiate.
Another measure of the impulsive efforts which subterranean forces
may exert upon the crust above them is evidently the height to which
volcanoes eject materials. Cotopaxi is said to have hurled a 200-ton
block of stone nine miles. Sir W. Hamilton tells us that in 1779
Vesuvius shot up a column of ashes 10,000 feet in height; and Judd
tells us that this same mountain in 1872 threw up vapours and rock
fragments to the enormous height of 20,000 feet. This would indicate an
initial velocity of 1,131 feet per second.
Notwithstanding Mallet’s calculation that thirty miles is the limiting
depth for the origin of an earthquake, the origin of the Owen’s
Valley earthquake of March 1872 was estimated as being at least fifty
miles.[88]
_Form of the focal cavity._—Among the various problems which are put
before those who study the physics of the interior of our earth it
would at first sight appear that there was none more difficult than the
attempt to determine the form of the cavity, if it be a cavity, from
which an earthquake originates. Almost all investigators of seismology
have recognised that the birthplace of an earthquake is not a point,
and have made suggestions about its general nature. The ordinary
supposition is that the earthquake originates from a fissure, and if
the focus of a disturbance could be laid bare to us it would have the
appearance of a fault such as we so often see exposed on the faces of
cliffs.
A strong argument, tending to demonstrate that some of the shakings
which are felt in Japan are due to the production of such fissures,
is the fact that the vibrations which are recorded are transverse to
a line joining the point of observation and the district from which,
by time observations, we know the shock to have originated. The most
probable explanation of this phenomena appears to be that one mass of
rock has been sliding across another mass, giving rise to shearing
strains, and producing waves of distortion.
The first seismologist who attacked the problem of finding out the
dimensions and position of such a fissure was Mallet, when working on
the Neapolitan earthquake of 1857. The reasons that the origin should,
in the first place, have been a fissure, rather than any other form of
cavity, was that such a supposition seemed to be _a priori_ the most
probable, and, further, that it afforded a better explanation of the
various phenomena which were observed, than that obtained from any
other assumption.
The method on which Mallet worked to determine the form and position of
the assumed fissure, which method was subsequently more or less closely
followed by other investigators, was as follows:—
From an observation of the various phenomena produced upon the surface
of the disturbed area, a map of isoseismals was constructed. These were
seen, as has been the case with many earthquakes, not to distribute
themselves in circles round the _epicentrum_, but as distorted oval or
elliptical figures, the major axes of which roughly coincided with each
other. Further, the _epicentrum_, did not lie in the centre of these
ovals, but was near to the narrow end where they converged.
This at once showed, if the reasoning respecting the manner in which
waves are propagated from an inclined fissure be correct, that the
fissure was at right angles to the major axis of the curves, dipping
from their narrow end downwards, in the direction of their larger
widespread ends.
The next weapon which Mallet employed to attack this problem was the
sound which was heard at different points round about the focus. These
sounds appear to have been of the nature of sudden explosive reports
accompanied by rushing, rolling sounds. The form of the area in which
these sounds were heard was closely similar to that of the first two
isoseismals. Except in the central area of great disturbance, no sound
was heard to accompany the shock.
Those at the northern and southern extremity of the sound area all
described what they heard as a ‘low, grating, heavy, sighing rush, of
twenty to sixty seconds’ duration.’ Those in the middle and towards the
east and west boundaries of this area described a sound of the same
tone, but shorter and more abrupt, and accompanied with more rumbling.
The nature of the arguments which were followed in discussing the sound
observations will be found in the chapter relating to these phenomena.
A portion of the argument which it is difficult to follow relates to
the maximum rate at which it can be supposed possible for a fissure to
be rent in rocks, which rate depends on the density and elasticity of
these rocks and other constant factors.
Next it was observed that the paths of the waves drawn on the surface,
although generally intersecting in a point, did not do so absolutely,
but along a line passing through the main focus some 7½ miles in
length. This, coupled with the observations of sounds, led to the
supposition that the centre of disturbance, considered horizontally,
originated along a curved line passing through the chief focus and the
various intersections of the wave paths.
The last phenomena brought forward to assist in the solution of this
interesting problem were a study of the tremulous movements that
preceded and followed the shock, and their relation to the sound
phenomena.
If the earthquake originated by the formation of a fissure, after the
rending has gone on for a certain time the focal cavity is enlarged
to a certain extent, and the great shock takes place. This would be
followed by concluding tremulous waves. A succession of phenomena like
those accompanied the shock about which Mallet writes.
By observations such as these, coupled with what has been said about
the maximum and mean depths of the focal cavity, Mallet came to the
conclusion that the focal cavity was a fissure, the rending open of
which produced the earthquake. The vertical dimensions of this cavity
were not more than 5·3 miles, but were probably limited to three miles.
From the intersection of the wave paths upon the surface and the
observed emergences, this fissure followed horizontally a curve of
double flexure, about nine geographical miles in length. The area of
this fissure was twenty-seven geographical miles. The time of rending
it open in Apennine limestone would be about 7½ seconds, which
should be the same as the period during which tremors were felt. The
time actually recorded was six or eight seconds.
Briefly, this is, then, the line of reasoning which was followed by
Mallet in an investigation the results of which are as interesting
as they are startling. Since the line of investigation has been
opened, and the existence of new problems has been indicated, other
investigators, although not exactly following Mallet’s method in
all their details, have, when endeavouring to attain the same ends,
employed similar weapons.
Thus, for example, Seebach, when determining the depth and nature of
the origin of the earthquake of Middle Germany, reasoned somewhat as
follows:—
Had the origin been more or less of a spherical cavity, then the region
of most violent disturbance upon the surface would, according to a
theorem we have already mentioned, have been upon or near a circle of
about 8·8 miles in radius round the _epicentrum_. This region, however,
was found by observation to lie along a curved band about forty miles
in length, altogether on one side of the _epicentrum_.
To explain this anomaly Seebach followed Mallet, and assumed that the
origin was not a spherical cavity, but a fissure.
The depth and strike of this fissure was determined by the observation
that the area of greatest disturbance was along a curved line lying
radial to the _epicentrum_. Such a condition it was assumed indicated
that the fissure of origin must be inclined towards this area of
greatest disturbance. A line was then drawn from this area to the
_centrum_. A second line at right angles to this one gave the dip of
the fissure.
Höfer, when working on the earthquake of Belluno, came to the
conclusion that the disturbance originated from two faults meeting
each other at an angle of 60°. In this determination he was chiefly
influenced by the form of a certain homoseist which was of the form of
an elongated ellipse met on one side by a second ellipse, the principal
axes of the two ellipses giving the strike of the two faults.