Keywords 
Mathematical model; Earthquake; Prediction; Poisson’s
distribution 
Introduction 
Objective 
The objective of this research is to improve upon a previously
developed Innovative Mathematical Model for Earthquake Prediction.
The result was the development of multiple rigorous algorithms to reach
a similar prediction confirming the Spatial Connection Theory, which
states that all earthquakes within a fault zone are related to each other. 
Previous literature 
Seismologists have been endeavoring to predict earthquakes
for many decades. In the past, an M8 algorithm has been developed
to predict earthquakes and has an accuracy rate of predicting past
earthquakes close to 100% [1]. Other seismologists have predicted
earthquakes based on the presence of an earthquake cloud, vapor that
is released through crevices of rocks just prior to an earthquake [2].
Previously, researchers have used the gamma distribution to model the
pattern of seismic events and earthquakes [3]. Geoelectrical signals
have also been a key point of interest for seismologists working on
earthquake clustering in Japan. Before seismic activity, there are great
changes in low frequency range geomagnetic and geoelectric fields.
Simultaneous changes in different stations noticed, signal a possibility
of an earthquake. One such example is the Izu island earthquake in
Japan. The geoelectric and geomagnetic dipoles at the WakAir and
BoeAir stations experienced great frequency changes before the
earthquake. The changes occurred in the ultra low frequency range of
0.0001–0.03 Hz, but the greatest changes occurred in the 0.006–0.03
Hz frequency range. A concern was expressed over the interruption
in the dipoles by telephone cable noise, but this concern was verified
as not true because the changes occurred both in WakAir and Boe
Air dipole stations [4]. An earthquake is located by its epicenter  the
location on the earth’s surface directly above the point of origin of the
earthquake. Earthquake ground shaking diminishes with distance from
the epicenter. Thus, any given earthquake will produce the strongest
ground motions near the earthquake with the intensity of ground
motions diminishing with increasing distance from the epicenter [5].
Larger magnitude earthquakes affect larger geographic areas, with
much more widespread damage than smaller magnitude earthquakes.
However, for a given site, the magnitude of an earthquake is NOT a good measure of the severity of the earthquake at that site. Rather, the
intensity of ground shaking at the site depends on the magnitude of
the earthquake and on the distance from the site to the earthquake
[6]. However, Moment Magnitude is a good measure of the amount
of energy released during an earthquake, which is not dependent on
ground shaking levels or level of damage. The Mercalli scale is used to
measure moment magnitude. It reflects factors that are characteristic
to the rupture of the fault that produces the earthquake [7]. Studies by
a Spanish researcher show that earthquakes of larger magnitudes cause
other large earthquakes to happen back to back in a short amount of
time. This research contradicts the average person’s thinking, which
is: If an earthquake of extremely high magnitude occurs, another
major earthquake won’t be due for a long time. The theory of the
Spanish researcher parallels the magnet theory of Earth. The magnet
theory is that a metal becomes magnetic when clusters of atoms feels
another’s magnetic force and align their magnetic moments in the same
direction [8]. Geological analysis has also been conducted to identify
changes in the rock before, during, and after earthquakes. Dilatancy
is the inelastic volume increase in rock due to great pressure. Due to
inelasticity, this sends high bursts of energy throughout the crust’s
rock. Immediately after great levels of dilatancy, there are earthquakes
in the area where the energy was released from the dilatory rock. The
volume increase can be calculated by Vp/Vs ratio (Poisson’s ratio). The
Poisson’s ratio is Velocity of P wave: Velocity of S Wave. Interestingly,
very low levels of Poisson’s ratio are exhibited before an earthquake.
Examples: There was a decrease in dilatancy in Gram, USSR before a
6magnitude earthquake and in the Adirondack region of New York
State before an earthquake occurrence was noted [9]. Possible factors
to experiment on are land deformations, tectonic movements, seismic
activity, and differences in seismic wave velocities of different world
regions, geomagnetic and geoelectric phenomenons, and active faults. There are many intense laboratory studies in US, Japan, and the Soviet
Union that focus on similar, geographic factors when trying to establish
a method for earthquake prediction and are funded by the public and
private sector [10]. 
Methods 
In the past, six zones were analyzed, with California offering the
most promising results [11]. The Californian fault zone was split into
two for further validation of the Innovative Mathematical Model for
Earthquake Prediction. The San Andreas Fault was split into two so that
each zone covers a major population center. Also, these two zones act
as cross validation for the prediction model. The rectangular latitudelongitude
range was chosen to cover a band of landmass on both sides
of the San Andreas fault. The following two different zones, Northern
California and Southern California, were analyzed using the fault lines
and past earthquake data. Depending on the occurrences of earthquakes
and their magnitudes, the rectangular area for the data collection for each
zone was chosen as shown in Table 1. The North and South Californian
fault zones are strikeslip fault zones where the movement of plates is
horizontal, building a lot of stress to release through earthquakes. The
1906 San Francisco earthquake and the 1994 Northridge earthquake
were major tectonic events reshaping the geological formation of the
region. The majority of Californian earthquakes are less than 16 km
below the crust [12]. In Northern California and Southern California,
most earthquakes occur in the 5 to 7 magnitude range. Using the
National Earthquake Information Center (NEIC) database, the data
for the North California and South California listed above were
downloaded in KML format. The data was analyzed using the spatial
connection theory, based on logical assumption that every earthquake
within a fault zone is related to the previous earthquake. In this research
the reverse Poisson’s distribution and exponential distribution were
used, conforming to the spatial connection theory by showing that
earthquake occurrences are not independent of each other. Using the
North California fault zone data points, Spatial Connection lines were
drawn between the first and second earthquake, then between the
second and third earthquake, and so on and so forth. Using the South
California fault zone data points, Spatial Connection lines were drawn
between the first and second earthquake, then between the second and
third earthquake, and so on and so forth. There is a relationship existing between the earthquake occurrences with respect to distance, direction,
and time. After spatial connection between earthquake’s epicenters
were carried out a relationship equation between angle of turn and time
to predict a distance range for the next earthquake was carried out as
follows. In Figure 1, consider the two lines between first, second and
third location of earthquakes. If the angle between the lines is ‘theta’,
distance between first and second location is ‘x1’ and between second
and third location is ‘x2’, then Poisson Range Identifier (Pri) is Poisson
Range Identifier (Pri) = [(x1 * time lag 2)/ [ (COS (theta) * x2 * time
lag1). 
To arrive at a statistically adept group for finding the average (Pri)
for the zone two of the highest values are omitted and cumulative value
and mean for rest of the values are found. The Poisson distribution or
Poisson law of small numbers is a discrete probability distribution that
expresses the probability of a given number of events occurring in a
fixed interval of time and/or space if these events occur with a known
average rate and independently of the time since the last event. 
Applying Poisson’s Distribution to the Pri Data, 
Df = POISSION DIST [Pri; Pri(mean); Pri (cu)] 
Similarly other Distance factors were worked out for rest of the
earthquakes in the zone. To arrive at a statistically adept group for
finding the average (Df) for the zone two of the highest values are
omitted and cumulative value and mean for rest of the values are
found. Using the Pri value and distance factor the predictions for North
California and South California were made. 
The exponential distribution was carried out using the cosine of the
angle of change, distance, and time between the various earthquakes.
The exponential distribution results were utilized to calculate the
exponential factor for the zones. The exponential factor was utilized
along with the Range Identifier function to predict future earthquake
occurrences based on time as outlined in Table 2. 
The difference between the Poisson’s distribution and Exponential
Distribution is the presence of a reverse distribution. For the Poisson’s
distribution method, a reverse Poisson’s distribution concept is applied
to show the earthquake occurrences are not independent events. For
the Exponential Distribution method, a prediction is carried out using
the Range Identifier function and the exponential factor. 
Results 
The thickness of the lines refers to the sequential order of the
earthquake occurrences in the fault zone with the thinnest being the
earliest earthquake (Figures 2 and 3). 
Discussion 
Interpretation of the results 
The Spatial Connection Theory was confirmed when the California
Fault Zone was split into two zones for analysis. The exponential
distribution confirmed the results of the Poisson Range Identifier
Analysis and allowed for improvement of the model. There were some
interesting patterns that were observed in the spatial distribution of
earthquake occurrence in each of the two zones. In the North California zone, earthquakes primarily clustered off the coast and minimal
earthquakes occurred on the landmass. In South California, the
earthquakes primarily clustered in a linear pattern and no occurrences
were reported off the coast of Southern California. Geological analysis
must be done to identify more information to allow for the development
of optimal earthquake prediction methods. Further methods must
be analyzed and triangulated to arrive at an accurate prediction of
earthquakes (Table 2). 
Applications 
After catastrophic earthquakes, there have been major economic
consequences. Earthquakes have resulted in major devastation causing financial losses to all scientists, investors, engineers, doctors,
etc. working in the impact area. Insurance companies have filed for
bankruptcy because of the large demand of money from their clients’
losses. Ports were extremely dangerous because they were built on
loose soil and they brought income to a particular region. Due to
the destroyed highways, railroads, and bridges, the goods cannot be
transported easily from one business to another. Potential future losses
after earthquakes are rising rapidly demanding we find a mechanism to
predict earthquakes [13]. 
The researched method of Innovative Mathematical Model (IMM)
can be applied to predict future earthquakes within a fault zone.
With over 24 billion dollars in losses from earthquake damage for
the 1994 North Ridge California Earthquake, the potential practical
implementation of these principles could save several hundred millions
of dollars and precious lives. The emergency management organizations
like FEMA can allocate and position their resources in the right location
to assist people in evacuation and supply and save lives. 
Future Research 
The researcher would like to team up with University Seismology Research departments to conduct scientific studies on Innovative
Mathematical Model (IMM) and develop a mathematical algorithm
for the most reliable earthquake prediction. That would help disaster
management agencies like Federal Emergency Management Agency
(FEMA) to be ready for safe evacuation of the population and protection
of public property. 
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