DETERMINATION OF EARTHQUAKE ORIGINS.
Approximate determination of an Origin—Earthquake-hunting
in Japan—Determinations by direction of motion—Direction
indicated by destruction of buildings—Direction determined by
rotation—Cause of rotation—The use of time observations—Errors
in such observations—Origin determined by the method of
straight lines—The method of circles, the method of hyperbolas,
the method of co-ordinates—Haughton’s method—Difference in time
between sound, earth, and water waves—Method of Seebach.
One of the most practical problems which can be suggested to the
seismologist is the determination of the district or districts in any
given country from which earthquake disturbances originate. With a
map of a country before us, shaded with tints of different intensity
to indicate the relative frequency of seismic disturbance in various
districts, we at once see the localities where we might dwell with
the least disturbance, and those we should seek if we wish to make
observational seismology a study. Before erecting observatories for
the systematic investigation of earthquakes in a country, it would
be necessary for us, in some way or other, to examine the proposed
country to find out the most suitable district. The special problem
of determining _approximately_ the origin or origins of a set of
earthquakes would be given to us. Having made this preliminary
investigation, the next point is, by means of observatories so
arranged that they could always work in conjunction with each other,
to determine the origins more accurately. By knowing the origin from
which a set of shocks spring we know the general direction in which
we may expect the most violent disturbances, and we can arrange our
seismometers accordingly.
_Approximate determination of origins._—In 1880 I obtained a tolerably
fair idea of the distribution of seismic energy throughout Japan, by
compiling the facts obtained from some hundreds of communications
received from various parts of the country respecting the number of
earthquakes that had been felt.
The communications were replies to letters sent to various residents
in the country and to a large number of public officers. By taking
these records, in conjunction with the records made by instruments, it
was ascertained that in Japan alone there were certainly 1,200 shocks
felt during the year, that is to say, three or four shocks per day.
The greater number of these shocks were felt along the eastern coast,
commencing at Tokio, in the south, and going northwards to the end of
the main island. These shocks were seldom felt on the west coast. It
appeared as if the central range of mountains formed a barrier to their
progress. Similarly, ranges of mountains to the south-west of Tokio
prevented the shocks from travelling southwards. Proceeding in this way
the conclusion was arrived at that the west coast, the southern part
of Japan, and the islands of Shikoku and Kiushiu, had their own local
earthquakes.
_Earthquake-hunting._—These preliminary enquiries having shown that
the northern part of Japan was a better district for seismological
observations than the southern half, the next step was to subject the
northern half to a closer analysis. This analysis was commenced by
sending to all the important towns, from thirty to one hundred miles
distant from Tokio, bundles of postcards. These were entrusted to the
local government offices with a request that each week one of these
cards would be returned to Tokio stating the number of shocks felt.
In this way it was quickly discovered that the majority of shakings
emanated from the north and east, and seldom, if ever, passed a heavy
range of mountains to the south. The barricade of postcards was then
extended farther northwards, with the result of surrounding the origin
of certain shocks amongst the mountains, whilst others were traced to
the sea shore. By systematically pursuing earthquakes it was seen that
many shocks had their origin beneath the sea—they shook all the places
on the north-east coast, but it was seldom that they crossed through
the mountains, forming the backbone of the island, to disturb the
places on the west coast.
The actual results obtained in three months by this method of working
are shown in the accompanying map, which embraces the northern half of
the main island of Nipon and part of Yezo. The shaded portion of the
map indicates the mountainous districts, which are traversed by ranges
varying in height from about 2,000 and 7,000 feet. The dotted lines
show the boundaries of the more important groups of earthquakes which
were recorded.
I. is the western boundary of earthquakes, which at places to the
eastward are usually felt somewhat severely. Some of these have been
felt the most severely at or near Hakodadi, whilst farther south their
effects have been weak. Occasionally the greatest effect has been near
to Kameishi. Sometimes these earthquakes terminate along the western
boundaries of III. or IV., not being able to pass the high range of
mountains which separate the plain of Musashi from Kofu.
[Illustration: FIG. 29.—Northern Japan. Mountainous districts shaded
with oblique lines.]
II. is the boundary of a shock confined to the plain which surrounds
Kofu. These earthquakes are evidently quite local. Many of the
disturbances have evidently originated beneath the ocean, having come
in upon the land in the direction of the arrows A or B.
III. This line indicates the boundary of a group of shocks which are
often experienced in Tokio. These may come in the directions D, E,
or F. It is probable that some of them originate to the eastward of
Yokohama, on or near to the opposite peninsula.
IV. V. and VI. The earthquakes bounded by these lines probably
originate in the directions C or D.
VII. The earthquakes bounded by this line probably come from the
direction E.
VIII. This line gives us the boundary of earthquakes which may come
from the direction B.
The above boundaries sometimes do not extend so far to the westward as
they are shown. At other times, groups like V. and VI. extend farther
to the south-west. These earthquake boundaries, which so clearly show
the effects of high mountains in preventing the extension of motion,
have been drawn up, not from single earthquakes, but from a large
series of earthquakes which have been plotted upon blank maps, and are
now bound together to form an atlas. To give an idea of the material
upon which I have been working, I may state that between March 1 and
March 10, 1882, I received records of no less than thirty-four distinct
shocks felt in districts between Hakodate and Tokio, and for each of
these it is quite possible to draw a map. In addition to the boundaries
of disturbances given in the accompanying map, other boundaries might
be drawn for shocks which were more local in their character. The
groups which contained the greatest number of shocks are III., IV.,
V., VI., and VII. By work of this description it was found that a very
important group of earthquakes might be studied by a line of stations
commencing at Saporo in the north, passing through Hakodate, down the
east coast of the main island, to Tokio or Yokohama in the south. A
further aid to the study of this group, together with the study of
an important local group, might be effected with the help of a few
additional stations properly distributed on the plain of Musashi, which
surrounds Tokio. With this example before us it will be recognised that
the choice of sites for a connected set of seismological observatories
will often be more or less a special problem. If earthquake stations
were to be placed in different directions around Tokio without
preliminary investigation, it is quite possible that some of them might
be so situated that they would seldom if ever work in conjunction with
the remaining observatories, and therefore be of but little value.
And this remark must equally apply to districts in other portions
of the globe. The method is crude, and, so far as actual earthquake
origins are concerned, it only yields results which are approximate.
The crudeness and the want of absoluteness in the results is, however,
more than counterbalanced by the certainty with which we are enabled
to express ourselves with regard to such results as are obtained. Even
when working with the best instruments we have at our command, unless
we are employing some elaborate system, this method of working gives a
most valuable check upon our instrumental records, and enables us to
interpret them with greater confidence.
_Determination of earthquake origins from the direction of motion._—If
we assume that an earthquake is propagated from a centre as a series
of waves, in which normal vibrations are conspicuous, and obtain at
two localities, not in the same straight line with the origin, and
sufficiently far separated from each other, the direction of movement
of these normal motions, by drawing lines parallel to these directions
through our two stations, the lines would intersect at a point above
the required origin. If instead of two points we had three, or, better
still, a large number, the results we should obtain ought to be
still more certain. Unfortunately, it seems that earthquakes seldom
originate from a given point, and, further, normal motions are not
always (sufficiently) prominent. Sometimes, as has already been shown
in the chapters on earthquake motion, they may be non-existent. It is
probable, however, that difficulties of this sort are more usually
associated with non-destructive earthquakes. Mallet regards the
destructive effects of an earthquake as almost solely due to normal
motions. If this be true, for destructive earthquakes, the problem is
shorn of many of its difficulties. In cases where normal vibrations
are not prominent, where we have only transverse vibrations, motions
due to the interference of normal or transverse motions, or directions
of motions due to the topographical or geological nature through which
the disturbance has passed, the determination of the origin of an
earthquake by observations on the direction in which the ground has
been moving appears to be a problem which is practically without a
solution. We will, therefore, only consider the determination of the
origin of those earthquakes which have predominating directions in
their movements, which directions we will consider as normal ones.
The question which is, then, before us, is the determination of the
direction of these normal movements. First of all we may take the
evidence of our senses. In exceptional cases these have given results
which closely approximate to the truth, but in the majority of cases
such results are not to be relied upon, as the inhabitants of a town
will, for the same shock, give directions corresponding to all points
of the compass. Much, no doubt, depends upon the situation of the
observer, and much, perhaps, upon his temperament. If he is sitting
in a room alone, and is accustomed to making observations on an
earthquake, on feeling the earthquake, if he concentrates his attention
on the direction in which he is being moved, his observations may be
of value. If, however, he is not so situated, and his attention is
not thus concentrated, his opinions, unless the motion has been very
decided in its character, are usually of but little worth.
_Direction determined from destruction of buildings._—When an observer
first sees a town that has been partially shattered by an earthquake,
all appears to be confusion, and it is difficult to imagine that in
such apparent chaos we are able to discover laws. If, however, we take
a general view of this destruction and compare together similarly
built buildings, it is possible to discover that similar and similarly
situated structures have suffered in a similar manner. By carefully
analysing the destruction we are enabled to infer the direction in
which the destroying forces have acted. It was chiefly by observing the
cracks in buildings, and the direction in which bodies were overthrown
or projected, that Mallet determined the origin of the Neapolitan
earthquake. From the observations given in Chapter VII. it would appear
that, with destructive earthquakes, walls which are transverse to the
direction of motion are most likely to be overturned, whilst, with
small earthquakes, these walls are the least liable to be fractured.
From a critical examination of the _general_ nature of the damage
done on the buildings of a town, earthquake observers have shown that
the direction of a shock may often be approximately determined. The
direction in which a body having a regular form like a prismatic
gravestone or a cylindrical column is overturned sometimes gives the
means by which we can determine the direction from which a movement
came.
_The rotation of bodies._—It has often been observed that almost all
large earthquakes have caused objects like tombstones, obelisks,
chimneys, &c., to rotate.
One of the most natural and at the same time most simple explanations
is to suppose that during the shock there had been a twisting, or
backward and forward screw-like motion in the ground. Amongst the
Italians and the Mexicans earthquakes producing an effect like this are
spoken of as ‘vorticosi.’ In the Calabrian earthquake, not only were
bodies like obelisks twisted on their bases, but straight rows of trees
seem to have been left in interrupted zigzags. These latter phenomena
have been explained upon the assumption of the interference of direct
waves and reflected waves, the consequence of which being that points
in close proximity might be caused to move in opposite directions.
Reflections such as these would be most likely to occur near to the
junction of strata of different elasticity, and it may be remarked that
it is often near such places that much twisting has been observed.
Another way in which it is possible for twisting to have taken place
would be by the interference of the normal and transverse waves which
probably always exist in an earthquake shock, or by the meeting of
the parts of the normal wave itself, one having travelled in a direct
line from the origin, whilst the other, travelling through variable
material, has had its direction changed.
Mallet, however, has shown that the rotation may have been in many
cases brought about without the supposition of any actual twisting
motion of the earth—a simple backward and forward motion being quite
sufficient. If one block of stone rests upon another, and the two are
shaken backwards and forwards in a straight line, and if the vertical
through the centre of gravity of the upper block does not coincide with
the point where there is the greatest friction between the blocks,
rotation must take place. If the vertical through the centre of gravity
falls on one side of the centre of friction, the rotation would be in
one direction, whilst, if on the other side, the rotation would be in
the opposite direction.
Although the above explanation is simple, and also in many cases
probably true, it hardly appears sufficient to account for all the
phenomena which have been observed.
Thus, for instance, if the stones in the Yokohama cemetery, at the time
of the earthquake of 1880, had been twisted in consequence of the cause
suggested by Mallet, we should most certainly have found that some
stones had turned in one direction whilst others had been twisted in
another. By a careful examination of the rotated stones, I found that
every stone—the stones being in parallel lines—had _revolved in the
same direction_, namely in a direction opposite to that of the hands of
a watch.
As it would seem highly improbable that the centre of greatest friction
in all these stones of different sizes and shapes should have been at
the same side of their centres of gravity, an effect like this could
only be explained by the conjoint action of two successive shocks, the
direction of one being transverse to the other.
Although fully recognising the sufficiency of two transverse shocks
to produce the effects which have been observed in Yokohama, I will
offer what appears to me to be the true explanation of this phenomenon:
it was first suggested by my colleague, Mr. Gray, and appears to be
simpler than any with which I am acquainted.
[Illustration: FIG. 30.]
If any columnar-like object, for example a prism which the basal
section is represented by A B C D (see fig. 30), receives a shock at
right angles to B C, there will be a tendency for the inertia of the
body to cause it to overturn on the edge B C. If the shock were at
right angles to D C, the tendency would be to overturn on the edge D C.
If the shock were in the direction of the diagonal C A, the tendency
would be to overturn on the point C. Let us, however, now suppose the
impulse to be in some direction like E G, where G is the centre of
gravity of the body. For simplicity we may imagine the overturning
effect to be an impulse given through G in an opposite direction—that
is, in the direction G E. This force will tend to tip or make the body
bear heavily on C, and at the same time to whirl round C as an axis,
the direction of turn being in the direction of the hands of a watch.
If, however, the direction of impulse had been E′ G, then, although the
turning would still have been round C, the direction would have been
_opposite_ to that of the hands of a watch.
To put these statements in another form, imagine G E′ to be resolved
into two components, one of them along G C and the other at right
angles, G F. Here the component of the direction G C tends to make the
body tip on C, whilst the other component along G F causes revolution.
Similarly G E may be resolved into its two components G C and G F′, the
latter being the one tending to cause revolution.
From this we see that if a body has a rectangular section, so long
as it is acted upon by a shock which is parallel to its sides or to
its diagonals, there ought not to be any revolution. If we divide
our section A B C D up into eight divisions by lines through these
directions, we shall see that any shock the direction of which passes
through any of the octants which are shaded will cause a _positive_
revolution in the body—that is to say, a revolution corresponding in
its direction to that of the movements of the hands of a watch; whilst
if its direction passes through any of the remaining octants the
revolution will be _negative_, or opposite to that of the hands of a
watch. From the direction in which any given stone has turned, we can
therefore give two sets of limits between one of which the shock must
have come.
Further, it will be observed that the tendency of the turning is to
bring a stone, like the one we are discussing, broadside on to the
shock; therefore, if a stone with a rectangular cross section has
turned sufficiently, the direction of a shock will be parallel to one
of its faces, but if it has not turned sufficiently it will be more
nearly parallel to its faces in their new position than it was to its
faces when in their original position.
If a stone receives a shock nearly parallel with its diagonal, on
account of its instability it may turn either positively or negatively
according as the friction on its base or some irregularity of surface
bearing most influence. Similarly, if a stone receives a shock parallel
to one of its faces, the twisting may be either positive or negative,
but the probability is that it would only turn slightly; whereas in the
former case, where the shock was nearly parallel to a diagonal, the
turning would probably be great.
_Determination of direction from instruments._—When speaking about
earthquakes it was shown, as the result of many observations, that
the same earthquake in the space of a few seconds, although it may
sometimes have only one direction of motion, very often has many
directions of motion. In certain cases, therefore, our records, if we
assume the most permanent motions to be normal ones, give definite and
valuable results. In other cases it is necessary to carefully analyse
the records, comparing those taken at one station with those taken at
another.
One remarkable fact which has been pointed out in reference to
artificial earthquakes produced by exploding charges of gunpowder or
dynamite, and also with regard to certain earthquakes, is that the
greatest motion of the ground is _inwards_, towards the point from
which the disturbance originated. Should this prove the rule, it gives
a means of determining, not only the direction of earthquake, but the
side from which it came.
_Determination of earthquake origins by time observations._—The times
at which an earthquake was felt at a number of stations are among the
most important observations which can be made for the determination of
an earthquake origin. The methods of making time observations, and the
difficulties which have to be overcome, have already been described.
When determining the direction from which a shock has originated, or
determining the origin of the shock by means of time observations,
it has been usual to assume that the velocity of propagation of the
shock has been uniform from the origin. The errors involved in this
assumption appear to as follows:—
1. We know from observations on artificial earthquakes that the
velocity of propagation is greater between stations near to the origin
of the shock than it is between more remote stations; and also the
velocity of propagation varies with the initial force which produced
the disturbance. If our points of observation are sufficiently close
together as compared with their distance from the origin of the
disturbance, it is probable that errors of this description are small
and will not make material differences in the general results.
2. We have reasons for believing that the transit velocity of an
earthquake is dependent on the nature of the rocks through which it
is propagated. Errors which arise from causes of this description
will, however, be practically eliminated if our observation points are
situated on an area sufficiently large, so that the distribution of the
causes tending to alter the velocity of a shock balance each other. It
must be remarked, that causes of this description may also produce an
alteration in the direction of our shock.
Other errors which may sometimes enter into our results, when
determining the origin of shocks by means of observations on
velocities, are the assumptions that the disturbance has travelled
along the surface from the _epicentrum_ and not in a direct line from
the _centrum_. Again, it is assumed that the origin is a point, whereas
it may possibly be a cavity or a fissure. Lastly, if we desire extreme
accuracy, we must make due allowance for the sphericity of the earth
and the differences of elevation of the observing stations.
I. _The method of straight lines._—Given a number of pairs of points
A_{0}, A_{1}, B_{0}, B_{1}, C_{0}, C_{1}, &c., at each of which the
shock was felt simultaneously, to determine the origin.
Theoretically if we bisect the line which joins A_{0} and A_{1} by a
line at right angles to A_{0}, A_{1}, and similarly bisect the lines
B_{0}, B_{1}, C_{0}, C_{1}, all these bisecting lines _a__{0}, _a__{1},
_b__{0}, _b__{1}, _c__{0}, _c__{1}, &c., ought to intersect in a point,
which point will be the _epicentrum_ or the point above the origin.
This method will fail, first, if A_{0}, A_{1}, B_{0}, B_{0}, C_{0},
C_{1} form a continuous straight line, or if they form a series of
parallel lines.
Hopkins gives a method based on a principle similar to the one which
is here employed—namely, given that a shock arrives simultaneously
at _three_ points to determine, the centre. In this case, the
relative positions of the three points, where the time of arrival was
simultaneous, must be accurately known, and these three points must
not lie in a straight line, or the method will fail. For practical
application the problem must be restricted to the case of three points
which do not lie nearly in the same straight line.
II. _The method of circles._—Given the times _t__{0}, _t__{1}, _t__{2},
&c., at which a shock arrived at a number of places A_{0}, A_{1},
A_{2}, &c., to determine the position from which the shock originated.
Suppose A_{0} to be the place which the shock reached first, and that
it reached A_{1}, A_{2}, A_{3}, &c., successively afterwards.
Let _t__{1} - _t__{0} = _a_
_t__{2} - _t__{0} = _b_
_t__{3} - _t__{0} = _c_, &c.
With A_{1}, A_{2}, A_{3}, &c. as centres, describe circles with radii
proportional to the known qualities _a_, _b_, _c_, &c., and also a
circle which passes through A_{0} and touches these circles. The centre
of the last circle will be the _epicentrum_. The radii proportional to
_a_, _b_, _c_, &c. may be represented by the quantities _ax_, _bx_,
_cx_, &c., where _x_ is the velocity of propagation of the shock.
It will be observed that the times at which the shock arrived at three
places might alone be sufficient. If, instead of taking the times of
arrival of a shock, the arrival of a sea wave be taken, the result
would be a closer approximate to the absolute truth.
It will be observed that this method is not a direct one, but is
one of trial. If, however, an imaginary case be taken, and three
given points of observation, A_{0}, A_{1}, A_{2}, be plotted on a
piece of paper, it will be found that it is not a difficult matter to
determine two numbers proportional to _a_ and _b_ which will allow
you to draw two circles so that they may be touched by a third circle
drawn through A_{0}. This problem has practically been applied in the
case of the arrival of a sea wave at a number of places on the South
American coast, at the time of the earthquake of May 9, 1877. This is
illustrated as follows. The places which were chosen were Huanillos,
Tocopilla, Cobija, Iquique, Mejillones.
In the following table the first column gives the times at which the
sea wave arrived at each of these places in Iquique time; in the second
column the difference between these times and the time at which it
reached Huanillos is given; in the third column the distances through
which a sea wave, propagated at the rate of 350 feet per second, could
travel during the intervals noted in the second column is given.
+------------+----------+----------+------------+
| | Arrival |Time after| Distance at|
| | of sea |arrival at|350 feet per|
| | wave | Huanillos| second |
+------------+----------+----------+------------+
| | h. m. | minutes | miles |
| Huanillos | 8 30 | 0 | 0 |
| Tocopilla | 8 32 | 2 | 8 |
| Cobija | 8 38 | 8 | 32 |
| Iquique | 8 40 | 10 | 40 |
| Mejillones | 8 46 | 16 | 64 |
+------------+----------+----------+------------+
The distances marked in the third column are used as radii of the
circles drawn round the places to which they respectively refer.
The centre of the circle drawn to touch the circles of the first
column, and at the same time to pass through Huanillos, is marked C.
The position from which the shock originated appears therefore to have
occurred very near to a place lying in Long. 7° 15′ W. and Lat. 21° 22′
S.
[Illustration: FIG. 31.]
The actual operations which were gone through in making the
accompanying map were as follows. First, the places with which we had
to deal were represented on a map in orthographical projection, the
centre of projection being near to the centre of the map. This was
done so that the measurements which were made upon the map might be
more correct than those we should obtain from an ordinary chart where
this portion of the world was not the centre of projection. Next, a
number was taken as equal to the velocity with which the sea wave had
travelled. The first velocity taken was about 400 feet per second—this
being about the velocity with which, theoretically, it must have
travelled in an ocean having a depth equal to that indicated upon the
charts—also it seemed to have travelled at this rate from the various
times of arrival as recorded at places along the coast. Circles were
then drawn round Tocopilla, Cobija, Iquique, and Mejillones with radii
equal to 2, 8, 10, and 15, each multiplied by (60 × 400). It was then
seen _by trial_ that it was impossible to draw a single circle which
should touch four circles and also pass through Huanillos. These four
circles were, in fact, too large. Four new but smaller circles, which
are shown in the map, were next drawn, their radii being respectively
equal to the numbers 2, 8, 10, and 16, each multiplied by (60 × 350),
and it was found that a circle, with a centre C, could be drawn which
would practically touch the four circles, and at the same time would
pass through Huanillos.
III. _The method of hyperbolas._—The method which I call that of
hyperbolas is only another form of the method of circles. It is,
however, useful in special cases, as, for instance, where we have the
times of arrival of earthquakes at only two stations. Between Tokio
and Yokohama, at which places I frequently obtain tolerably accurate
time records, the method has been applied on several occasions with
advantage. In the preceding example let us suppose that the only time
records which we had were for Huanillos and Mejillones, and that the
wave was felt at the latter place sixteen minutes or 960 seconds
after it was experienced at the former. Calling these places H and M
respectively, round M draw a circle equal to the 960 multiplied by
the velocity with which the wave was propagated. It is then evident
that the origin of this disturbance must be the centre of a circle
which passes through H and touches the circle drawn round M. Join H
M, cutting the circle round M in Y. Bisect Y H in V. It is evident
that V is one possible origin for the disturbance. Next, from M, in
the direction of H, draw any line M Z P; join Z H; bisect Z H at right
angles by the line O P N. Because PH = PZ, it is evident that P is
a second possible origin. Proceeding in this way a series of points
lying to the right and left of V on the curve R V T may be found, and
we may therefore say that the origin lies somewhere in the curve R
V T. By increasing or decreasing our velocity we vary the position
of the curve R V T, and, instead of a line on which our origin may
be, we obtain a band. As the assumed velocity increases, the circle
round M becomes larger, and has its limit when it passes through H,
where the two arms of the curve R V T will close together and form a
prolongation of the line M Y H as the assumed velocity diminishes.
The circle round M becomes smaller until it coincides with the point
M. At such a moment the curve R V T opens out to form a straight line
bisecting M H at right angles. The curve R V T is a hyperbola with a
vertex V and foci H and M. Inasmuch as PM - PH = a constant quantity.
If we have the time given at which the shock or wave arrived at a third
station as at Iquique, it is evident that a second hyperbola R′ V′ T′
might be drawn with Iquique and Huanillos as foci, and that the mutual
intersection of these two hyperbolas with a third hyperbola, having for
its foci Iquique and Mejillones, would give the origin of the wave.
The obtaining of a mutual intersection would depend on the assumed
velocity, and the accuracy of the result, like that of the method
of circles, would depend upon the trials we made. The method here
enunciated may be carried farther by describing hyperboloids instead
of hyperbolas, the mutual intersection of which surfaces would, in the
case of an earth wave, give the actual origin or _centrum_ rather than
the point above the origin or _epicentrum_.
IV. _The method of co-ordinates._—Given the times at which a shock
arrived at five or more places, the position of which we have marked
upon a map, or chart, to determine the position on the map of the
centre of the shock, its depth, and the velocity of propagation.
Commencing with the place which was last reached by the shock, call
these places _p_, _p__{1}, _p__{2}, _p__{3}, and _p__{4}, and let the
times taken to reach these places from the origin be respectively _t_,
_t__{1}, _t__{2}, _t__{3}, and _t__{4}.
Through _p_ draw rectangular co-ordinates, and with a scale measure the
co-ordinates of _p__{1}, _p__{2}, _p__{3}, and _p__{4}, and let these
respectively be _a__{1}, _b__{1}; _a__{2}, _b__{2}; _a__{3}, _b__{3};
_a__{4}, _b__{4}. Then if _x_, _y_, and _z_ be the co-ordinates of the
origin of the shock, _d_, _d__{1}, _d__{2}, _d__{3}, and _d__{4}, the
respective distances of _p_, _p__{1}, _p__{2}, _p__{3}, and _p__{4}
from this origin, and _v_ the velocity of the shock, we have
1. _x_^2 + _y_^2 + _z_^2 = _d_^2 = _v_^2 _t_^2
2. (_a__{1} - _x_)^2 + (_b__{1} - _y_)^2 + _z_^2 = _v_^2 _t__{1}^2
3. (_a__{2} - _x_)^2 + (_b__{2} - _y_)^2 + _z_^2 = _v_^2 _t__{2}^2
4. (_a__{3} - _x_)^2 + (_b__{3} - _y_)^2 + _z_^2 = _v_^2 _t__{3}^2
5. (_a__{4} - _x_)^2 + (_b__{4} - _y_)^2 + _z_^2 = _v_^2 _t__{4}^2
Because we know the actual times at which the waves arrived at the
places _p_, _p__{1}, _p__{2}, _p__{3}, _p__{4}, we know the values
_t_—_t__{1}, _t_—_t__{2}, _t_—_t__{3}, _t_—_t__{4}. Call these
respectively _m_, _p_, _q_, and _r_. Suppose _t_ known, then
_t__{1} = _t_ - _m_
_t__{2} = _t_ - _p_
_t__{3} = _t_ - _q_
_t__{4} = _t_ - _r_.
Subtracting equation No. 1 from each of the equations 2, 3, 4, and 5,
we obtain,
_a__{1}^2 + _b__{1}^2 - 2_a__{1} _x_ - 2_b__{1} _y_
= _v_^2 (_t__{1}^2 - _t_^2) = _v_^2 (_m_^2 - 2_t_ _m_)
_a__{2}^2 + _b__{2}^2 - 2_a__{2} _x_ - 2_b__{2} _y_
= _v_^2 (_t__{2}^2 - _t_^2) = _v_^2 (_p_^2 - 2_t_ _p_)
_a__{3}^3 + _b__{3}^2 - 2_a__{3} _x_ - 2_b__{3} _y_
= _v_^2 (_t__{3}^2 - _t_^2) = _v_^2 (_q_^2 - 2_t_ _q_)
_a__{4}^2 + _b__{4}^2 - 2_a__{4} _x_ - 2_b__{4} _y_
= _v_^2 (_t__{4}^2 - _t_^2) = _v_^2 (_r_^2 - 2_t_ _r_)
Now let _v_^2 = _u_, and 2_v_^2 _t_ = _w_.
Then
1. 2_a__{1} _x_ + 2_b__{1} _y_ + _u_ _m_^2 - _n_ _m_
= _a__{1}^2 + _b__{1}^2
2. 2_a__{2} _x_ + 2_b__{2} _y_ + _u_ _p_^2 - _n_ _p_
= _a__{2}^2 + _b__{2}^2
3. 2_a__{3} _x_ + 2_b__{3} _y_ + _u_ _q_^2 - _n_ _q_
= _a__{3}^2 + _b__{3}^2
4. 2_a__{4} _x_ + 2_b__{4} _y_ + _u_ _r_^2 - _n_ _r_
= _a__{4}^2 + _b__{4}^2
We have here four simple equations containing the four unknown
quantities _x_, _y_, _u_, and _w_.
_x_ and _y_ determine the origin of the shock. The absolute velocity
_v_ equals √ _u_. From _v_ and _w_ we obtain _t_. Substituting
_x_, _y_, _v_, and _t_ in the first equation we obtain _z_.
We have here assumed that the points of observation have all about the
same elevation above sea level.
If it is thought necessary to take these elevations into account, a
sixth equation may be introduced.
If the propagation of the wave is considered as a horizontal one, as
would be done when calculating the position of the _epicentrum_ or
point above the origin, by means of the times of arrival of a sea wave,
the ordinate _z_ of the first five equations would be omitted. Working
in this way the resulting four equations, viz.
2_a__{1} _x_ + 2_b__{1} _y_ + _u__m_^2 - _w__m_^2 = _a__{1}^2 + _b__{1}^2
&c. &c. &c.
remained unchanged.
Applying this method to the same example as that used as illustration
for the two previous methods, we obtain for the co-ordinates of
Mejillones, Iquique, Cobija, Tocopilla, and Huanillos, measured in
geographical miles, and the times in Iquique time at which the wave
reached each, as given in the following table; _ox_ and _oy_ being,
drawn through Mejillones.
+------------+--------------------------------+---------------+
| | Co-ordinates |Time of arrival|
+------------+----------------+---------------+---------------+
| | OX | OY | h. m. |
| Mejillones | _a_ or 0 | _b_ or 0 | 8 46 p. m. |
| Iquique | _a_{1}_ or 150 | _b_{1}_ or 96 | 8 40 „ |
| Cobija | _a_{2}_ or 36 | _b_{2}_ or 14 | 8 38 „ |
| Tocopilla | _a_{3}_ or 66 | _b_{3}_ or 31 | 8 32 „ |
| Huanillos | _a_{4}_ or 102 | _b_{4}_ or 58 | 8 30 „ |
+------------+----------------+---------------+---------------+
From this data we find the co-ordinates _x_ and _y_ of this origin to
be 85·8 and 56·7; whilst the velocity of propagation = 45 feet per
second.
Measuring these ordinates upon the map, we obtain a centre lying very
near Long. 71° 5′ W. and Lat. 21° 22′ S., a position which is very near
to that which has already been obtained by other methods.
If instead of Huanillos we substitute the ordinates and time of arrival
of the sea wave for Pabalon de Pica, another point for the origin will
be obtained lying farther out at sea. To obtain the best result, the
method to be taken will evidently be, first to reject those places
at which it seems likely that some mistake has been made with the
time observations, and then with the remaining places to form as many
equations as possible, and from these to obtain a mean value. This
is a long and tedious process, and as the time observations of this
particular earthquake are probably one and all more or less inaccurate,
it is hardly worth while to follow the investigation farther.
In this example, as in the preceding ones, it will be observed that
it has been sea waves that have been dealt with, rather than earth
vibrations. It is evident, however, that these latter vibrations may be
dealt with in a similar manner.
In these determinations it will also have been observed that it
is assumed that the disturbance has radiated from a centre, and,
therefore, approached the various stations in different directions.
If we assume that we have three stations very near to each other as
compared with their distances from the origin, so that we can assume
that the wave fronts at the various stations were parallel, the
determination of the direction in which the wave advanced appears to be
much simplified. The determination of the direction in which a wave has
passed across three stations was first given by Professor Haughton.
_Haughton’s method._—Given, the time of an earthquake shock at three
places, to determine its horizontal velocity and coseismal line.
The solution of this is contained in the formula
_a_ (_t_{2}_ - _t_{1}_) sin β
tan φ = ——————————————————————————————————————————————————————.
_c_ (_t_{3}_ - _t_{2}_) + _a_ (_t_{2}_ - _t_{1}_ cos β
When A, B, and C are three stations at which a shock is observed at
the times _t_{1}_, _t_{2}_, and _t_{3}_; _a_, _b_, and _c_ are the
distances between A, B, and C, and φ is the angle made by the coseismal
lines _x_ A _x_, _y_ B _y_, and the line A B, which are assumed to be
parallel.
This I applied in the case of the Iquique earthquake, but owing to the
smallness of the angles between the three stations A, B, and C, the
result was unsatisfactory. The problem ought to be restricted, first,
to places which are a long distance away from a centre, and, secondly,
to places which are not nearly in a straight line. This problem may be
solved more readily by geometrical methods. Plot the three stations A,
B, and C on a map, join the two stations between which there was the
greatest difference in the time observation. Let these, for example,
be A and C. Divide the line A C at point D, so that A D : D C as the
interval between the shock felt at A and B is to the interval between
the shock felt at B and C. The line B D will be parallel to the
direction in which the wave advanced.
_The difference in time of the arrival of two disturbances._—In the
various calculations which have been made to determine an origin
based on the assumption of a known or of a constant velocity, we have
only dealt with a single wave, which may have been a disturbance in
the earth or in the water. A factor which has not yet been employed
in this investigation is the difference in time between the arrival
of two disturbances; one propagated, for instance, through the
earth, and the other, for example, through the ocean. The difference
in the times of the arrival of two waves of this description is a
quantity which is so often recorded that it is well not to pass it
by unnoticed. To the waves mentioned we might also add sound waves,
which so frequently accompany destructive earthquakes, and, in some
localities, as, for instance, in Kameishi, in North Japan, are also
commonly associated with earthquakes of but small intensity. It was by
observing the difference in time between the shaking and the sound in
different localities that Signor Abella was enabled to come to definite
conclusions regarding the origin of the disturbances which affected the
province of Neuva Viscoya in the Philippines, in 1881; the places where
the interval of time was short, or the places where the two phenomena
were almost simultaneous, being, in all probability, nearer to the
origin than when the intervals were comparatively large. I myself
applied the method with considerable success when seeking for the
origin of the Iquique earthquake of 1877. The assumptions made in that
particular instance were, first, that the velocity of the disturbance
through the earth was known, and, secondly, that the velocity with
which a sea wave was propagated was also known.
A method similar to the above was first suggested by Hopkins. It
depended on the differences of velocity with which normal and
transversal waves are propagated.[85]
_Seebach’s method._—To determine the true velocity of an earthquake,
the time of the first shock, and the depth of the centre.
[Illustration: FIG. 32.]
Let the straight line M, _m__{1}, _m__{2}, _m__{3} represent the
surface of the earth shaken by an earthquake. For small earthquakes, to
consider the surface of the earth as a plane will not lead to serious
errors.
If an earthquake originates at C, then to reach the surface at M it
traverses a distance _h_ in the time _t_. To reach the surface at M_{1}
it traverses a distance _h_ + _x__{1} in a time _t__{2}. If _v_ equals
the velocity of propagation,
_h_ _h_ + _x__{1}
then _t_ = ———, _t_{1}_ = —————————————,
_v_ _v_
_h_ + _x__{2}
_t_{2}_ = —————————————, &c.
_v_
Seebach now says that _if we have given the position of_ M _or
epicentrum of the shock_, and draw through it rectangular axes like M
_m_{3}_ and M T_{3}, and lay down on M _m_{3}_ in miles the distances
from M of the various stations which have been shaken, and in equal
divisions for minutes lay down on M T_{3} the differences of time at
which M, _m_{1}_, _m_{2}_, &c. were shaken, then M_{1} T_{1}, M_{2}
T_{2}, &c. are the co-ordinates of points on an hyperbola. The degree
of exactness with which this hyperbola is in any given case constructed
is a check upon the accuracy of the time observations and the position
of the _epicentrum_. The apex of the hyperbola is the _epicentrum_.
The intersection of the asymptote with the ordinate axis is the time
point of the first shock, which, because the scale for time and
for space were taken as equal, gives the absolute position of the
_centrum_. This intersection is shown by dotted lines. Knowing the
position of the _centrum_, we can directly read from our diagram how
far the disturbance has been propagated in a given time.