Apparent Triggering of Large Earthquakes in Los Angeles and San Francisco Regions
Related to Time Derivatives of Ellipsoidal Demand, Kinematically Derived:
Qualitative Interpretations for 1890–2014.1
Douglas W. Zbikowski, Institute for Celestial Geodynamics
Draft 4.0 - 22 November 2015 rev.
Abstract
[1]
This article documents a beginning investigation into time-derivative aspects of vertical crustal motions that may supply the
energetic impetus responsible for triggering large (M ≥ 6) earthquakes in the Los Angeles and San Francisco regions. These subtle,
in-depth temporal aspects of motion are, herein, qualitatively correlated to quake occurrences, and presumably, quake triggering.
This matching in time appears to indicate that, frequently, strong triggering impetus derives from the jerk and jounce of ellipsoidal
demand. Ellipsoidal demand is the amount of georadial adjustment of the crust that is physically required for an increment of time
that includes movement of Earth’s rotational axis with respect to the globe, which is measured as polar drift. Also, significant
triggering impetus appears, on occasion, to derive from the jerk and jounce of lunisolar earth-tide displacement of Earth’s surface.
The constructive combination of time-derivative aspects of motion of both modes of vertical crustal motion—ellipsoidal demand and
earth tides—may drive crustal rupture mechanism(s) that are responsible for the initiation of large quakes in these two regions. This
investigation has only begun, many details remain undeveloped and questions unanswered. However, if these preliminary results
and conclusions hold up to extensive scrutiny and testing, the seemingly intractable problem of forecasting earthquakes in California
may be closer to solution.
Introduction
[2]
Polar drift is a multi-year, residual change in the geographic position of Earth’s rotational pole. (The path of the North Pole is
recorded.) A variable annual ‘wobble’ to polar location exists, so, to determine the drift path with (mostly) semi-monthly resolution,
a pole-position dataset of sequential mean poles for every 0.15 year (54.79 days) in 1890–1900 and every 0.05 year (18.26 days) in
1900–2011.05 was utilized. In computing the mean poles, pole components were filtered to remove both the Chandler and annual
wobble terms. Using this drift path, analytical evidence was recently presented (Zbikowski, 2012), showing that fluctuations in the
time derivatives of polar drift magnitude and variations in its direction have both been related, since 1890, to great earthquakes and
large volcanic eruptions, worldwide.
[3]
Movement of Earth’s rotational axis with respect to the globe implies geodynamic and tectonic consequences. The spin rate of the
non-rigid Earth maintains its ellipsoidal shape and, because of that, tilting axial movement with respect to its body requires
proportionate figure migration via equipotential adjustments. Associations in time and location of the kinematic aspects of polar
drift with global seismic and volcanic events seem to suggest that mechanisms exist that relate crustal processes, regionally
advanced by equipotential adjustments, to these catastrophic events. To investigate for tectonic influences from such regional
deformation, this analysis translates speed of polar drift into location-specific speed of georadial adjustment of the crust by applying
a novel algorithm. The physical requirement of an amount of georadial adjustment of the crust at a location for an increment of
time that includes polar drift, we term ellipsoidal demand. Time derivatives of ellipsoidal demand (e.g. speed, acceleration, jerk, and
jounce) for a particular region are then compared to the record of local seismic and volcanic events for the duration examined. The
equipotential forces involved are inferred as sufficient to gradually deform the crust and produce these kinematic aspects of motion.
Much additional empirical work would be required to confirm this proposition. No focused attempt is made, herein, to measure or
prove that such crustal kinematics are materially evident; however, the success exhibited by the accurate matching in time of
posited crustal kinematics to many large seismic events provides supporting evidence suggesting the validity of this proposition.
Further, accurate resolution of polar positions for more than a century and frequent confirmation by modern precision geodetic
monitoring establishes that the precise ellipsoidal shape of Earth with respect to the rotational axis appears to have been
continuously maintained in recent decades.
[4]
Destructive earthquakes along the San Andreas Fault system have supplied the stimulus for Californians to develop, arguably, the
most extensive and sophisticated seismic monitoring systems in the United States. The beginnings of this remarkable network and
its supporting legions of technical experts started effectively about the time of the Great San Francisco Earthquake of 1906. Thus, to
introduce a novel seismic analysis by examining the Los Angeles (LA) and San Francisco (SF) regions seems appropriate, because
earthquakes are common there and seismic records are relatively accurate and complete. Additionally, urban seismic risk is high, in
part due to the proximity of numerous active faults and the many millions of people that reside in each metro area.
Note:
Zbikowski, D. W. (2012). Global View of Great Earthquakes and Large Volcanic Eruptions Matched to Polar Drift and its Time Derivatives. Institute
for Celestial Geodynamics, latest draft available at: http://www.celestialgeodynamics.org/content/global-view
Calculating Time Derivatives of Ellipsoidal Demand
[5]
To calculate the speed of ellipsoidal demand for any particular global location, a kinematic algorithm was developed that was
introduced, in part, in section [26] of the article, Global View, which is referenced above. The complete algorithm is described here:
The speed of polar drift (SPD) times the cosine of the shortest longitudinal difference between the location of interest and the
meridian great circle that is aligned with the direction of polar drift (COS(∆long)), is multiplied by a georadial factor (radfactor)
comprising the georadius at the location’s latitude divided by the polar georadius (RL/R90). This equation yields the surface speed at
which the Earth ellipsoid migrates at that location. This product is multiplied by the ellipsoid's local slope along the meridian and, to
approximate georadial change in the direction of migration, is multiplied by |COS(∆long)|. The final result is the georadial speed of
ellipsoidal demand at the location, herein expressed in mm/year. A final check is advised to ensure consistency in the direction of
polar motion with the sense of local adjustment (rise or fall of the crust due to bulge migration), to achieve the correct direction of
local georadial movement.
[6]
Successive time derivatives of ellipsoidal demand are calculated simply by dividing incremental changes in speed, acceleration, or
jerk by appropriately matched time intervals, because interval values changed within the polar drift data. As the following graphs
describe in their legends, to calculate jerk (first time derivative of acceleration), a smoothed acceleration array (LOESS, span 0.08)
was used and to calculate jounce (first time derivative of jerk), a smoothed jerk array (LOESS, span 0.02) was used. The smoothing
reduced erratic behavior substantially and yet appeared to yield reasonable resolution of detail in the aspects of motion. The
procedure of differentiating smoothed time-derivative arrays has not been optimized, presently, and requires additional work in the
future.
Los Angeles Region: Graphs of Time Derivatives of Ellipsoidal Demand
[7]
The following graphs of kinematic aspects of ellipsoidal demand for the LA region are briefly described and qualitatively interpreted.
While graphs can be cognitively appropriate to help recognize relationships, the ultimate value of these plots may be to suggest and
justify attempts to determine quantitative contributions from individual terms of the aspects of motion to the effective cumulative
triggering influence for quakes of particular energy—ideally under matched circumstances of stress/strain condition, fault geometry,
nucleation maturity, and numerous other attributes. Thus, graphically deduced indications are to investigate the development of a
polynomial with terms of time derivatives of ellipsoidal demand, which equates to total seismic energy.
Figure 1
Description
[8]
Figure 1 shows calculated speed and acceleration history of the rise and fall of the crust for the LA region since 1890 and projected
to 2022 based on the median of an ARIMA model. These kinematic aspects of ellipsoidal demand are calculated by using geographic
coordinates of the estimated city center: 34.05° N, 118.24° W. Georadial adjustments of the crust are controlled by the physics of
our spinning planet and are required to maintain Earth’s equilibrium figure during drift of the rotational pole. Equipotential forces
that drive such adjustments are inferred to apply multi-year pulses to the crust, which would result in a duality of internal
components of stress (compressive or extensive) and material dilatation (negative or positive). The geologically abrupt stress/strain
fluctuations that are generated could advance quake nucleation processes and trigger quakes on faults that are already loaded to
near-critical levels. To investigate this possibility, the history of speed and acceleration of ellipsoidal demand for the LA region is
compared to its record of large quakes for 1890–2014.17. For this analysis, quakes of M ≥ 5.5 were selected from an area within ±
2.5° latitude and ± 2.5° longitude of city coordinates. The record of quakes in the LA region is plotted on the smoothed acceleration
trace (black) with categorically sized markers. Quake markers are positioned along the acceleration trace by time of occurrence. No
other shared attribute is implied by marker superposition. Markers for 6.0 ≤ M ≤ 6.9 quakes that are hidden by other markers are
noted. Markers for smaller quakes that are hidden are not noted.
Interpretation
[9]
It is apparent along the smoothed acceleration trace for the period 1890–1966.5 that during times of great ellipsoidal-demand
jounce (second time-derivative of acceleration)—such as near points that are either high or low turning points or points of
inflection—quakes of M ≥ 6.0 appear to collect and often cluster. Similarly, along the speed trace (maroon) during times of great
ellipsoidal-demand jerk (second time-derivative of speed)—such as near points that are either high or low turning points or points of
inflection—other M ≥ 6.0 quakes appear to collect and often cluster. For this 76.5-year period, these two kinematic aspects, jounce
and jerk, appear to be primary factors correlated with occurrences of M ≥ 6.0 quakes and, presumably, influence the triggering of
such quakes.
[10]
For the next 28 years after 1966.5, many M ≥ 6.0 quakes occur during intervals between jounce and jerk extrema—with events
somewhat evenly dispersed. (The adjective or noun terms extreme/extrema will be used, herein, to mean related to, or being
maxima or minima of multiple prominent excursions in range on a trace and will not be limited to one preeminent point of
maximum and one of minimum.) Apparently, the necessity of jounce and jerk for triggering had diminished, relative to other
factor(s). One possibility to explain the change in triggering response is that regional strain levels in the crust may have generally
increased, thereby, reducing stress-gaps-to-criticality on many faults. With stress-gaps reduced other mechanisms that typically
produce less impetus for triggering ought to be operative. One explanation for a general increase in stress/strain levels is that the
context of regional crustal strain has changed markedly.
[11]
Crustal tectonics of the LA region involve primarily right-lateral, strike-slip shearing across major faults along with strong thrusting
on the LA side of, or close to, the prominent curve nearby in the trace of the San Andreas Fault—supplied by northward movement
of the Pacific Plate relative to the North American Plate. In 1960, the greatest quake instrumentally measured (M9.5) with a fracture
length of about 1000 km occurred near Chile on a plate margin of the Pacific Rim. In 1964, the second greatest quake instrumentally
measured (M9.2) with a fracture length of nearly 800 km occurred in Prince William Sound, Alaska on a plate margin of the Pacific
Rim. The San Andreas Fault system constitutes a plate margin of the Pacific Rim that is connected to and lies between these two
huge releases of strain. The change in occurrence pattern of quakes after 1966.5 may be evidence of far-field stress transfers along
the Pacific Rim that helped to reduce local stress-gaps-to-criticality for many faults in the LA region, thereby lessening dependence
on the presumably larger contributions to triggering from time-derivative aspects of ellipsoidal demand. The story is clearly complex
and is speculated to involve other significant factors for triggering, such as time-derivative aspects of lunisolar-tidal displacements of
the surface. (The term ‘tidal’ and related terms in this article and its appendices will refer to earth-tide displacements of Earth’s
surface and not ocean tides or ocean-tidal loading, unless ocean tides are specifically designated.)
[12]
In this period (1966.5–1994.5), a significant pair of large quakes occurred off-peak of ellipsoidal-demand jounce—the M7.3 Landers
quake, on June 28, 1992 at 11:57 UTC (4.197 a.m. solar time @ epicenter) and the M6.5 Big Bear quake, three hours afterward at
15.092 UTC (7.303 a.m. solar time @ epicenter). The Big Bear quake, with an epicenter about 34 km from the Landers, probably
triggered as a result of stress-coupling to the Landers event, because Big Bear occurred on a fault with an orientation and slip that is
conjugate to the faults which slipped in the Landers rupture (SCEDC, 1992). In any case, if the previously described, far-field-stresstransfer scenario is correct and stress-gaps-to-criticality were reduced for many faults along the San Andreas in California, the
question arises, “What mechanism(s) beyond ellipsoidal demand could have supplied the energetic impetus to trigger the Landers
quake?”
[13]
Earth tides (or body tides) are sub-meter, vertical displacements of Earth's surface, which are caused preeminently by interaction
with both lunar and solar gravity. Earth tides raised by the Moon and Sun combine most effectively at syzygy, when the Earth,
Moon, and Sun are configured in a straight line, which is the case for full moons and new moons. Quite precise alignments occur,
which produce even larger tidal displacements, during lunar and solar eclipses. Solar eclipses produce maximum earth tides on the
side of Earth facing the Moon and Sun. Whenever the three bodies are in alignment, two antipodal tidal bulges are raised on Earth
along the same alignment vector, and tidal depression is produced on a great circle that is equidistant from the two bulge peaks. As
Earth spins, geographic locations experience cycles of tidal bulge alternating with tidal depression. The semidiurnal (~12 hour
periodicity) amplitude envelope of earth tides can reach about 55 cm at the equator. Earth-tide displacement is generally out-ofphase with crustal displacement from ocean-tidal loading, and the latter can cause displacements of the land near coastlines that
can exceed displacements due to the earth tide. Significant displacement components from ocean-tidal loading can extend inland
for scores of kilometers.
[14]
During several days before June 28, 1992, many repeated, earth-tide bulge/depression cycles were quite pronounced and may have
acted together strongly to help trigger the Landers quake, because four tide-optimal aspects were operative: (1) summer solstice
occurred on June 21 (7 days before), when lunisolar earth tides are at their annual maximum for southern California, and (2) the
Landers quake occurred 48.23 hours before a total solar eclipse on June 30th, at 12:11 UT (Espenak and Meeus, 2009)—therefore, at
the time of the quake, the 3-body configuration featured the Moon preceding occultation of the Sun by 24.5° (or 1.63 hours, solartime equivalent), and (3) surface displacement at the epicenter followed a daily maximum depression (crustal compression) by only
45 minutes, which would have added constructively to both the general tectonic and ellipsoidal-demand compressions prevalent in
the region, and (4) monthly lunar perigee occurred on July 2, at 1h UT (Meeus, 1995) (3.5 days after), so, at the time of the quake
the Moon was slightly closer than its average distance. Although the tidal peaks of these orbital-aspect functions are a little off
optimum synchronization, the combination thus produced was quite near maxima of annual, monthly, and daily components of tidal
depression/compression at the epicenter. Also, the rotation of Earth produced repeated, large amplitude tidal pulses leading up to
the quake, which generated prominent, periodic dilatation of the crust—thereby, possibly advancing quake nucleation processes
and predisposing the Landers quake for criticality. Further, the greater the tidal potential produced from near-optimal lunisolar
alignments and proximities with Earth, the greater the values of time-derivative aspects (e.g. jerk, jounce) of crustal motion. Earthtide displacement of the surface and its time-derivative aspects of motion for both the Landers and Big Bear quakes are calculated
and graphically presented in Appendix A, which follows this article. At this point, the reader may benefit from examining Appendix A
to witness the close association between the timing of both quakes and extrema of jounce and jerk. Other examples of significant
quakes in the SF and LA regions are similarly illustrated. The temporal matching of such quakes with extrema of jounce and jerk
helps one to appreciate the materially consistent, triggering impetus that is apparently derived from jounce and jerk across both
modes of vertical crustal motion—ellipsoidal demand and earth tides.
Notes:
Espenak, F., J. Meeus (2009). Five Millennium Canon of Solar Eclipses: -1999 to +3000 (2000 BCE to 3000 CE)–Revised, National Aeronautics and
Space Administration, NASA/TP–2009–214174, p. A-161.
Meeus, J. (1995). Astronomical Tables of the Sun, Moon, and Planets, 2nd ed., Willmann-Bell, Inc., p. 395.
SCEDC: Southern California Earthquake Data Center (1992). Big Bear Earthquake. http://www.data.scec.org/significant/bigbear1992.html
Retrieved 08 Oct 2014.
[15]
After 2001, speed of ellipsoidal demand is positive (rising crust) and steeply increasing, so this movement should apply an increasing
regional component of positive dilatation, helping to reduce accumulated compressive strain in LA’s rather compressive crustal
regime. The great speed that is projected to be reached by 2022 (0.81 mm/yr.) is 3.7 times the next highest positive speed since
1890 (0.22 mm/yr. in 1942.2). The trend of sharply increasing positive speed should also apply increasing relief from the ongoing
generation of highly compressive strain via tectonic loading near the prominent curve in the trace of the San Andreas Fault, and any
triggering of M ≥ 6.0 quakes in this region may eventually, again, depend more on impetus from the kinematics of ellipsoidal
demand, than it would without an added component of positive dilatation. Of course, the idea of triggering patterns being altered
by an increasing component of positive dilatation from sharply increasing positive speed are speculation only, because, presently,
the relative and absolute contributions of aspects of ellipsoidal demand and tectonic loading to the complete stress condition and
triggering process(es) are not known.
[16]
The LA quakes plotted in Figure 1 are much smaller (5.5 ≤ M ≤ 7.3) than the great quakes (8.0 ≤ M ≤ 9.5) plotted in Figure 3 of the
article, Global View (Zbikowski, 2012). For example of scale, a quake of magnitude Χ has only 3.1% of the seismic energy of a quake
of magnitude (Χ + 1). Thus, the ratios of quake energy to aspects of local ellipsoidal demand have much greater energy resolution
than the ratios of quake energy to aspects of global polar drift. This fact is encouraging, because, as Figure 1 shows, quakes as small
as M6.0, which are still potentially damaging, appear to associate with extreme aspects of ellipsoidal demand and so have a
possibility of being forecast by methods that project aspect values.
Sense of Ellipsoidal-Demand Speed (SEDS)
[17]
The speed trace in Figure 1 shows the calculated rate of rise and fall of the regional crust as required by ellipsoidal demand. Where
the trace crosses the horizontal ‘0.00’ line, the sense or direction of speed changes, which, to the extent that equipotential
adjustments of the crust involve a short time lag in response to polar drift, produces a transition in the crustal stress/strain
component applied from compressive to tensive, or vice versa. Sense of Ellipsoidal-Demand Speed (SEDS) transitions occur three
times for the LA region (i.e. 1935.2, 1948.4, and 2000.9). When examining these transitions at the zero line, one might suspect that
the inflectional curvatures might be artifacts of the method used to calculate speed, which is possible. However, aside from the
appearance of such inflectional curiosities, all SEDS transitions for LA are closely matched to M ≥ 6.0 quakes, as detailed in Table 1.
Such comprehensive matching would seem unlikely, if a causal link from transitions to quakes, either direct or indirect, were not
involved. However, the small number of events prevents definitive conclusions.
LA: SEDS TRANSITIONS
↑ = Comp. to Tens. ↓ = T to C
1935.2 ↑
1948.4 ↓
2000.9 ↑
QUAKE
DATE
1934.43
1935.00
1947.27
1948.93
1999.79
1999.79
QUAKE
MAGNITUDE
M6.0
M7.0
M6.5
M6.0
M7.1
M6.7
Table 1
[18]
During the last 124 years, the LA region has experienced five quakes of M ≥ 7.0. Two of these quakes (40%) precede SEDS↑
(compressive to tensive) transitions by an average of about eight months. Also, four quakes of 6.0 ≤ M ≤ 6.9 precede SEDS
transitions by an average of about seven months. A sample size of six large quakes is statistically indeterminate. To suggest valid
statistical inference, another two centuries of global polar motion and earthquake data for the LA region may be required.
Alternatively, to effectively increase sample size for analysis, these data may be added to a global study, roughly equivalent, that
includes transitions and quakes from other regions that exhibit similar tectonic regimes. Of course, any analysis that compares
generalized tectonic relationships outside of the LA region to the LA region seems a rudimentary approach, due to the inherent
complexities of crustal structures, faulting, and tectonic regimes.
[19]
In considering the apparent matching of crustal jerk and jounce, generated near SEDS transitions, to M ≥ 6.0 quakes in the LA region,
one might surmise that the quakes are causing crustal jerk and jounce rather than triggering from it. However, this order seems
implausible; because it appears that the speed trace without the inflectional features would cross the zero line at nearly the same
times. Consequently, this observation begs the question: “By what means could multi-year progression of the speed trace and its
crossings at zero, anticipate and synchronize with M ≥ 6.0 quakes?” Because ellipsoidal-demand speed is calculated entirely by using
global polar drift and Earth’s ellipsoidal geometry at the location, significant control of the multi-year progression of polar drift
seems implausible by a few episodically spaced 6.0 ≤ M ≤ 7.3 quakes in California. Therefore, the dominant arrow of causality seems
to point from the kinematic aspects of ellipsoidal demand to the quakes. However, more work needs to be done to clarify how large
a quake Mw is required at another global location to sufficiently affect polar drift and subsequently produce local jerk, jounce, and
possibly pop (first time derivative of jounce), so that the resulting energetic impetus would trigger local quakes. In any case, close
temporal matching of SEDS transitions and the triggering of M ≥ 6.0 quakes provides evidence that suggests both the reality of
ellipsoidal-demand kinematics and the short time lag in response to polar drift. Of course, more global data and analysis are needed
to confirm that such matching of crustal speed transitions and increased seismicity is not just coincidental.
SEDS Loose-Coupling Transition - Hypothesis A
[20]
SEDS transitions in the crust may share behavior similar to when a train engine pulls a long line of railroad cars, then slowly brakes
and reverses. The point in time of transition of applied tension to compression is accompanied by numerous jolts in the linkage
system, because of the many discrete ‘slops’ accumulated in the long series of loose mechanical couplings. Likewise, a SEDS
transition in the crust may involve numerous jolts from loose ‘couplings’ (e.g. irregular gaps between the asperities that arrest
movement along a fault). However, the crustal system may be far more complex than a linear train with couplings in series, because
a planar fault allows two dimensions of relative movement, and a network of stress-coupled faults may produce a complex response
including a mix of jolts that are coupled in series or in parallel, or both. Further, the time-derivative aspects of stress/strain applied
from a SEDS transition in the crust probably vary across a region, relative to local stress-gaps-to-criticality. Such variation in
proportion may alter the combined effectiveness of these time-derivative aspects as triggers. Regardless of the complexity of the
crustal system, such jolts appear to supply sufficient energetic impetus to trigger M ≥ 6.0 quakes in the LA region.
Figure 2
Description
[21]
Similar to Description of Figure 1: Figure 2 shows calculated jerk (blue trace) that is intrinsic within the speed and acceleration of
ellipsoidal demand for the LA region since 1890, which were plotted in Figure 1. The projected jerk was derived from the projected
acceleration in Figure 1. Jerk was calculated by dividing incremental changes in the array of smoothed acceleration (LOESS, span
0.08) by numerically matched time intervals. This smoothing reduces erratic behavior and yet yields reasonable resolution of detail
in the jerk trace. A trace of smoothed jerk (LOESS, span 0.02) (black trace) is superimposed over the jerk. Quakes of M ≥ 5.5 during
1890–2014.17 were selected from an area within ± 2.5° latitude and ± 2.5° longitude of city coordinates. The record of quakes is
plotted on the smoothed jerk trace with categorically sized markers. Quake markers are positioned along the trace by time of
occurrence. No other shared attribute is implied by marker superposition. The notation of hidden markers follows the same
treatment as in Figure 1.
Interpretation
[22]
General behavior of the jerk traces and the distribution of quakes along the smoothed jerk trace appear to loosely fit the framework,
similar to Figure 1, of three periods of different influence patterns of triggering (i.e. 1890–1966.5, 1966.5–1994.5, and 2001–
2014.17). (See Interpretation of Figure 1 for details.) During the first period (1890-1966.5) of more regular behavior, three
prominent peaks in jerk (i.e. 1902, 1927, and 1955) average about seven years before three correspondingly prominent peaks in
acceleration (i.e. 1908, 1933, and 1964) in Figure 1. Associations in time of M ≥ 6.0 quakes with abrupt turning points of the jerk
traces are, apparently, not as close as were evident for the traces in Figure 1. This appearance is interpreted to mean that jerk is
only one component of the impetus behind quake triggering and that jounce extrema, shifted in time slightly from jerk extrema, also
have an influence as suggested in section [9] of the Interpretation of Figure 1. In any case, of the five 7.0 ≤ M ≤ 7.3 quakes, at least
three (i.e. 1935.0, 1992.5, and 1999.8) appear to be closely associated in time (∆t ≤ 1 year) with turning points of jerk extrema.
Figure 3
Description
[23]
Similar to Description of Figure 2: Figure 3 shows calculated jounce (solid trace) that is intrinsic within the jerk of ellipsoidal demand
for the LA region since 1890, which was plotted in Figure 2. The projected jounce (dashed) was derived from the projected jerk in
Figure 2. Jounce was calculated by dividing incremental changes in the array of smoothed jerk (LOESS, span 0.02) by numerically
matched time intervals. This smoothing reduces erratic behavior and yet yields reasonable resolution of detail in the jounce trace.
In evidence, turning points of this trace are quite abrupt (needle-like). Quakes of M ≥ 5.5 during 1890–2014.17 were selected from
an area within ± 2.5° latitude and ± 2.5° longitude of city coordinates. The record of quakes is plotted on the jounce trace with
categorically sized markers. Quake markers are positioned along the trace by time of occurrence. No other shared attribute is
implied by marker superposition. The notation of hidden markers follows the same treatment as for Figure 1. For the years
1958.853–1995.114, a time-series trace representing Earth-Mars close encounters is positioned below the jounce trace. Each data
point in the series is valued as the inverse-squared of the nearest distance at the time of that event. The Y2 axis indicates the value
and units of these data.
Interpretation
[24]
General behavior of the jounce trace and the distribution of quakes along it appear to fit the same framework of three periods of
different influence patterns of triggering, although end points of the periods appear shifted somewhat from those of Figure 1 (i.e. for
Figure 3: 1895–1958.0, 1958.1–1994.5, and 1994.6–2014.17). (See Interpretation of Figure 1 for comparison of periods.) Of the five
7.0 ≤ M ≤ 7.3 quakes, four (i.e. 1935.0, 1940.4, 1952.6, and 1999.8) appear to be closely associated in time (∆t ≤ 1 year) with abrupt
turning points of jounce extrema. Close temporal proximities for four events, but failed matching of the 1992.5 event, lends support
to the interpretation that jounce is only one component of the total impetus behind quake triggering, but that it is a relatively strong
influence. Because all five 7.0 ≤ M ≤ 7.3 quakes are close in time (∆t ≤ 1 year) to either jerk turning points of extrema or jounce
turning points of extrema, or both, a shared dependence on these two kinematic aspects appears indicated for the triggering of 7.0 ≤
M ≤ 7.3 quakes in this region. Many 6.0 ≤ M ≤ 6.9 quakes appear closely associated in time with jounce turning points; however, a
statistical analysis would be required to confirm the accuracy of this impression.
[25]
The second period mentioned (1958.1–1994.5), exhibits curious behavior of the jounce trace. Because trace excursions in this
period span very little range, an underlying component frequency of small amplitude is clearly revealed. Upon examining the entire
graph, it appears that this sawtooth-like component may exist broadly. Presumably, this component also resides in the polar-drift
data, because ellipsoidal demand is calculated from only that and geometric parameters which are constant for a global location.
The polar-drift data were filtered to remove the Chandler wobble (~433 days) and annual wobble (~365 days), but this frequency has
a calculated average cycle of ~778 days (2.13 year). Further work by spectral analysis is needed to determine the full extent and
precise attributes of this intriguing component, so that a candidate source(s) may be identified.
[26]
A viable conjecture proposing the cause of this fine sawtooth signal in the jounce trace is that Earth and the planet Mars experience
a close encounter every 779.94 Earth days on average, which will herein be termed the synodic period. A synodic period is
conventionally defined as planetary opposition-to-opposition, however, the author will use nearest encounters to define synodic
periods, because the governing dynamics of an isolated pair of planets is solely gravitational and, therefore, distance-based. Also,
during the last century, nearest encounters of Mars have not varied more than 8.5 days from Martian oppositions—over centuries,
average periods of the nearest-encounter and opposition series become identical. Therefore, separation distance will be tracked
and utilized as an important parameter in approximating gravitational potential and, ultimately, gravitational torque. During an
encounter—including its approach and departure of, effectively, about six months each—gravitational torque on Earth by Mars is
modulated by both the inverse-squared of the separation distance and an (approximated) orientation factor that involves the
declination of Mars with respect to Earth. Further supporting the Mars conjecture is the observation that, in Figure 3, extrema
(often minima) of the sawtooth signal appear to be synchronized with and lag Earth-Mars’ close encounters (e.g. 2003.66, 2005.85,
and 2007.98) by a few months—a lag time which may reflect a brief period of increased gravitational torque integrating during the
most influential portion of the encounter. Also, the time-series trace of inverse-squared, nearest separation distances for EarthMars’ close encounters (this trace is displayed under the jounce trace) shows an undulation that is sympathetic with the jounce
trace—a matching in both wavelength (15.78 years) and synchronization, for these 36.4 years. This matching suggests that the
inverse-squared factor is, at least, partially determining the jounce of LA’s ED and, likely, similarly determining the jounce of polar
drift (Figure B1 in Appendix B). Lending further support to the Mars conjecture, statistical evidence has been published that
correlates the timing of very large quakes (M ≥ 7.4), worldwide, to the Martian synodic cycle (opposition-to-opposition) (Tikhonov,
2008). To fully evaluate the validity of time-derivative aspects of motion passing through the governing dynamics—from celestialmechanical origins to polar drift, then to ellipsoidal demand, and ultimately to statistically enhancing the triggering of large quakes—
further work is required to describe Earth-Mars’ gravitational interaction and its resulting planetary torque dynamics. Appendix B
contains a short, preliminary study showing correlation between the jounce of polar drift, M ≥ 8.0 quakes worldwide, and time
derivatives of Earth-Mars’ gravitational potential. The reader may benefit most by perusing Appendix B after reading this article,
because observations in Appendix B are logically connected by a framework of hypotheses that are developed herein.
Note:
Tikhonov, I. N. (2008). The Synodic Periods of Planets in the Solar System and Synchronization of the Timing of Large Earthquakes. Journal of
Volcanology and Seismology, Pleiades Publishing, Ltd., 2010, Vol. 4, No. 3, pp. 213–221.
San Francisco Region: Graph of Speed and Acceleration of Ellipsoidal Demand
[27]
Crustal tectonics of the San Francisco region involve primarily right-lateral, strike-slip shearing across major faults and some
deformation between major faults. However, the SF region experiences less compression than that near LA, because of the lack of a
prominent curve nearby in the trace of the San Andreas Fault. Speed and acceleration of ellipsoidal demand for SF are similar in
profile to the LA plot; however, the traces may be shifted slightly in time (∆t ≤ 2 yrs.) with the ± of the shift depending on the
direction of polar drift at the time. The amount of time shift is due to the speed of polar drift and the longitudinal difference of
4.18° between the two locations, which alters calculations involving COS(∆long) as detailed in section [5], above. Plot excursions in
range may also increase slightly by an amount due to the product of the ratios of respective surface slopes and radfactors (SF/LA)
(0.003255/0.003120)(1.002097/1.002306) = 1.043. Thus, SF may experience excursions in range of ellipsoidal-demand aspects that
have a latent potential to be about 4% more extreme than those of LA. For concision, graphs of jerk and jounce for the SF region are
not included in this article, but are presented in Appendix C, which follows.
Figure 4
Description
[28]
Figure 4 shows calculated speed and acceleration history of the rise and fall of the crust for the SF region since 1890 and projected to
2022 based on the median of an ARIMA model. These kinematic aspects of ellipsoidal demand are calculated by using geographic
coordinates of the estimated city center: 37.78° N, 122.42° W. For this analysis, quakes of M ≥ 5.5 during 1890–2014.05 were
selected from an area within ± 2.5° latitude and ± 2.5° longitude of city coordinates. The record of quakes in the SF region is plotted
on the smoothed acceleration trace (black) with categorically sized markers. Quake markers are positioned along the acceleration
trace by time of occurrence. No other shared attribute is implied by marker superposition. Markers for 6.0 ≤ M ≤ 6.9 quakes that
are hidden by other markers are noted. Markers for smaller quakes that are hidden are not noted. The start (S) and finish (F) of a
series of volcanic eruptions (1914.4–1917.5) of Lassen Peak are indicated on the speed trace (maroon) with the largest event marked
as VEI = 3, which occurred on 1915.4. Lassen Peak is actually 0.21° (23.3 km) north of the SF region, as defined, but is included
because of its close proximity.
Interpretation
[29]
General behavior of the speed and acceleration traces and the distribution of quakes along the smoothed acceleration trace appear
to fit the framework, similar to that in Figure 1, of three periods of different influence patterns of triggering (i.e. 1890–1966.5,
1966.5–1994.5, and 2001–2014.17). (See Interpretation of Figure 1 for details.) For the first period (1890–1966.5) of 76.5 years, the
temporal proximity of occurrences of M ≥ 6.0 quakes to times of great jerk and jounce—such as near points that are either high or
low turning points or points of inflection on, respectfully, the speed trace and the smoothed acceleration trace—appear not as wellmatched as with the LA quakes in Figure 1. This impression should be tested for accuracy with a statistical analysis. Because
ellipsoidal demand imparts vertical components of time-derivative aspects of crustal motion, the energetic impetus for triggering is
dilatational. In the SF region, where the general tectonics has a relatively larger component of strike-slip (lateral shear stress/strain)
than near LA, it is speculated that quake triggering may be less controlled by such vertical components of added stress impetus.
During the 76.5-year period, speed of ellipsoidal demand was usually negative (descending crust), which added a component of
negative dilatation (compression) that constructively added to the region’s varying tectonic compression. Within the 76.5-year
period, the sub-period of 1935–1948.5 is an exception that experienced positive speed (rising crust), which, instead, added a small
component of positive dilatation (extension) into the mix. The effect of this addition should have been a minor reduction in general
compression and, possibly in support of this scenario, there are no quakes of M > 5.7 during these 13.5 years.
[30]
The only quake of M ≥ 7.0 for the SF region is the famous M7.8 San Francisco quake of 1906.3. This disastrous event occurred
notably at a time of great ellipsoidal-demand jounce and jerk as indicated by the close proximity in time (∆t ≈ 1 year) before the
sharp tip of the preeminent excursion peak on the smoothed acceleration trace.
[31]
Lassen Peak (40.488 N, 121.505 W) is the southernmost active volcano in the Cascade Range. The Lassen Peak eruption series of
1914.4–1917.5 occurred on a portion of the ellipsoidal-demand speed trace that shows low speed and little change in speed for
about six years (1914-1920). A later portion of the same trace shows somewhat similar behavior with a period of low speed and
little change in speed for about eight years (1972–1980). Curiously, another Cascade volcano, Mt. St. Helens (46.191 N, 122.194 W),
experienced the now famous VEI = 5 eruption on May 18, 1980 (1980.38). Mt. St. Helens is located 8.4° N and 0.23° E of SF, so the
speed and acceleration traces of ellipsoidal demand for St. Helens would be similar to SF, but generally shifted in time by ∆t ≤ 1 year
and the range of plot excursions may also slightly increase by an amount due to the product of the ratios of respective (St. Helens /
SF) surface slopes and radfactors (0.003355/0.003255)(1.001607/1.002097) = 1.0302. Thus, Mt. St. Helens may experience
excursions in range of ellipsoidal-demand aspects that have a latent potential to be about 3% more extreme than SF.
[32]
For the 28 years after 1966.5, more quakes of 6.0 ≤ M ≤ 6.9 occurred during intervals between jounce and jerk extrema—with
events episodically dispersed. Apparently, the necessity of jounce and jerk for triggering had diminished, relative to other factor(s).
Rationale for the apparent change in triggering response for this period are speculated in sections [11] through [14] of the
Interpretation for Figure 1.
[33]
After 2001, speed of the crust is positive (rising crust) and steeply increasing, and this movement should apply an increasing regional
component of positive dilatation, helping to reduce accumulated compressive strain in the SF region. The great speed that is
projected to be reached by 2022 (0.98 mm/yr.) is 3.6 times the next highest positive speed since 1890 (0.27 mm/yr. in 1942.3).
Sense of Ellipsoidal-Demand Speed (SEDS)
[34]
Similar to those of LA, SEDS transitions occur three times on the SF graph (i.e. 1935.0, 1948.7, and 2000.5), as detailed in Table 2.
The SF region shows little effective triggering by SEDS transitions—the transitions are matched by only two quakes of 6.0 ≤ M ≤ 6.9.
Note: the M6.0 quake of 1934.43 is common to both SF and LA lists, because both defined regions share a small overlap in territory,
which hosted the quake.
SF: SEDS TRANSITIONS
↑ = Comp. to Tens. ↓ = T to C
1935.0 ↑
1948.7 ↓
2000.5 ↑
QUAKE
DATE
1934.43 (LA repeat)
1948.99
QUAKE
MAGNITUDE
M6.0
M6.0
Table 2
Discussion
Energetic Potential of Ellipsoidal Demand for Triggering
[35]
As shown by the speed traces of ellipsoidal demand in the LA and SF graphs, the amount of ellipsoidal demand that has accumulated
in these regions during typical ten-year samples of maximal peak or valley excursions is only about ± 5 mm; however, the global
expanse similarly affected is large and the net sense of adjustment for many regions within the expanse may trend consistently for
millennia at a minimum. Thus, the amount of accumulated energy available to help energize and trigger seismic events in any
included region could be immense. Considering the global symmetry of Earth, equivalent crustal adjustment is concurrently made to
an equally large expanse that is antipodal to the region of reference. Similarly, by symmetrical reflection about the equator, two
globally antipodal expanses that are equal to the first pair, concurrently experience ellipsoidal demand of opposite sense. Thus,
ellipsoidal demand, which is driven by polar drift, represents an enormous amount of energetic potential by region and a
stupendous amount globally. A fundamentally related and supporting observation is that total global seismic energy appears to
track commensurately with the speed of polar drift. This relationship is made apparent by comparing a graph that shows the
minimum strain energy drop in global great quakes (M ≥ 7.75) as a function of year for 1904–1976 (Kanamori, 1977) to a graph of
the speed of polar drift that covers the same period (Zbikowski, 2012).
Notes:
Kanamori, H. (1977). Energy release in great earthquakes. J. Geophys. Res. Vol. 82 No. 20, p. 2984, Figure 2.
Zbikowski, D. W. (2012). Global View of Great Earthquakes and Large Volcanic Eruptions Matched to Polar Drift and its Time Derivatives. Institute
for Celestial Geodynamics, Figure 3. Latest draft available at: http://www.celestialgeodynamics.org/content/global-view
Conceptualizing Influences to Triggering
[36]
Earthquakes along pre-existing faults are conventionally thought to occur when fault stresses build to levels that exceed a critical
threshold for fault rupture and slip. Under the action of tectonic loading, a stress threshold is created by the roughness of fault
surfaces—including attributes that may vary from rough asperities that must be overcome, involving the strength of that material, to
fine textures where resistance to relative movement is dominated by static friction. Most quakes take place along locked faults
where strain slowly builds from relative tectonic motions. When stress-gaps-to-criticality are exceeded, quakes follow, and then the
cycle of building strain and stress to seismic release repeats. Stated simply, the current consensus in the seismological community
seems to be that quake triggering is primarily determined by a slow, regular increase of strain and stress until a critical threshold is
reached, at which time seismic release occurs.
[37]
Qualitative analysis of the LA and SF regions, herein, suggests that (1) stress from tectonic loading is being augmented significantly
by stress induced from ellipsoidal demand, resulting in the triggering of quakes and (2) the temporally in-depth manner in which
stress components, generated by time-derivative aspects of ellipsoidal demand, are applied appears to influence triggering—
possibly preempting the loading cycle to an otherwise critical threshold. That is, the magnitudes of terms within the composite
profile of time-derivative aspects of ellipsoidal demand appear to influence quake triggering and affect quake size.
Importance of, and Calculating Time-Derivative Aspects of Earth Tides
[38]
Numerous scientific studies have attempted to relate earthquake triggering to tidal interaction of Earth with the Moon or with both
the Moon and Sun. For during more than 117 years (Schuster, 1897), results have been mixed and seem to average inconclusive
with correlations ranging from slightly positive to no correlation. The inconsistency between many study results may be explained
by the importance of time-derivative aspects of earth tides to quake triggering, because, to be materially consistent with ellipsoidal
demand, strong triggering impetus should accompany extrema of tidal jerk and jounce—which occur just before and just after
maximum tidal deformation during both the bulge and depression phases of displacement. Thus, the times of potential triggering
within the daily tidal displacement cycle may be multiple and more complexly arranged than has previously been anticipated and
tested for. No published work has been identified that investigated the possibility of this subtle nuance.
[39]
An interesting focus of study recently was an investigation to relate earthquake occurrences with phases of both the diurnal (~24
hour periodicity) and semidiurnal (~12 hour periodicity) components of earth-tide displacement (Métivier et al., 2008). Tidal phases
were determined at the time of each seismic event using an earth-tide model based on the HW95 tidal potential catalogue
(Hartmann and Wenzel, 1995). An intriguing application of the Métivier et al. work, relevant to the study herein, is that such a
model may also be used to produce a time-series array of total tidal displacement for any global location at nearly any time—past,
present, or future. Tidal displacement is vertical (georadial) and so implies deformation that is analogous to that from ellipsoidaldemand adjustment. Therefore, time-derivative aspects of motion could be calculated from a time-series array of total tidal
displacement, similar to the procedure for calculating time-derivative aspects of ellipsoidal demand that is described in [6], above. A
significant benefit resulting from a usefulness of kinematic aspects of tidal displacement for forecasting quakes would be that tidal
aspects have much finer temporal resolution than kinematic aspects of ellipsoidal demand (minutes/hours vs. months/years), which
may prove additionally useful for identifying short periods of increased seismic hazard.
Notes:
Schuster, A. (1897). On lunar and solar periodicities of earthquakes. Proc. R. Soc. Lond. 61, pp. 455–465.
Métivier, L., O. Viron, C. Conrad, S. Renault, M. Diament, G. Patau (2008). Evidence of earthquake triggering by the solid earth tides. Earth and
Planetary Science Letters 278 (2009), pp. 370–375.
Hartmann, T., H.-G. Wenzel (1995). The HW95 tidal potential catalogue. Geophys. Res. Let. 22, pp. 3553–3556.
Criticality by Constructive Combination of Time-Derivative Aspects of Motion - Hypothesis B
[40]
From the above work, it appears that time-derivative aspects of ellipsoidal demand are influential to the triggering of M ≥ 6.0 quakes
in the LA and SF regions. Additionally, time-derivative aspects of lunisolar-tidal displacement may also be important in that they
may work together constructively with aspects of ellipsoidal demand to reach a threshold of impetus necessary to achieve criticality.
A dominant role of ellipsoidal demand to the triggering of large quakes may explain, in part, the failure of predicting such quakes
using lunisolar configurations alone—a process which produces many false positives (optimal configurations that result in no
quakes). In any case, to qualitatively consider the possible relative contributions from both modes of deformation, data for the six
quakes of M ≥ 7.0 from the LA and SF regions are listed in Table 3, below.
Proposed Components of Triggering Impetus
for 7.0 ≤ M ≤ 7.8 Quakes in LA and SF Regions during 1890–2014.17
Jerk
of
Ellipsoidal
Demand
3
(mm/yr )
Jounce
of
Ellipsoidal
Demand
4
(mm/yr )
7.8
7.0
0.0068
-0.0075
-0.0068
0.0015
7.0
-0.0141
-0.0060
7.3
0.0078
0.0061
7.3
0.0081
0.0006
7.1
-0.0106
0.0078
Quake
Date
& UTC Time
(decimal)
Area /
Distance of
epicenter to
coast (km)
Size
(M)
1906.29445
1934.99870
SF / 0
Colorado
River Delta,
Baja / 119
Imperial
Valley / 144
Kern County /
82
Landers / 131
Hector Mine /
165
1940.38108
1952.55439
1992.49143
1999.78960
Lunisolar Aspects for Earth Tides
including configurations ± 30° or
± 2.46 days of lunar syzygy
± 2.0 hrs. from max compression
by solar E-tide (day’s SC)
Full or New Moon = FM / NM
Day’s SC
1.7 days (1.68°) before annual
perihelion (Jan 2)
Earth-Tide
Amplitude
Envelope
for day of
quake
(cm)
Jerk
of
Lunisolar
E-tide
3
(cm/hr )
Jounce
of
Lunisolar
E-tide
4
(cm/hr )
45
0.16
-0.84
19
-0.5
0.2
2.37 days before FM, 0.38 day
after lunar perigee, day’s SC
0.49 day before NM
7 days (6.7°) after summer solstice,
2.0 days before NM & total solar
eclipse, day’s SC
2 hrs. before day’s max extension
Table 3
[41]
The M ≥ 7.0 quakes in Table 3 have values listed for ellipsoidal-demand jerk and jounce that were estimated from the respective
graphs. Because the positive and negative senses of these values are not presently known to have different triggering efficacies, this
first reasoning will posit the absolute value of jerk or jounce as the quantity commensurate with triggering impetus. Column six,
Lunisolar Aspects for Earth Tides, contains the orbital aspects present during each quake, which act as factors increasing the amount
of annual, monthly, and daily components of the earth-tide generating effect. The combined strength of the earth-tide generating
effect of these various combinations of orbital aspects should be indicated by the relative height of the corresponding earth-tide
amplitude envelope (column seven). If ellipsoidal-demand jerk and jounce are posited roughly equivalent as triggering impetuses,
the weakest numerical contribution from ellipsoidal demand may be for the Landers quake—for which tidal potential from its
lunisolar configuration seems relatively strong. Supporting the relative strength of the tidal potential for the Landers event is its
quite large earth-tide amplitude envelope of 45 cm. Therefore, for the Landers quake, these measures seem to suggest that the
combined ellipsoidal-demand components were relatively small and the lunisolar-tidal components were quite large and probably
helped significantly to supply the impetus necessary for criticality. Conversely, by numerical values, the Imperial Valley and Hector
Mine quakes were strongly valued in combined ellipsoidal-demand components—and the Hector Mine event appears to have
triggered with little contributing impetus from lunisolar-tidal components. Further, the San Francisco quake shows a large combined
value of its ellipsoidal-demand components, and its triggering appears to have required little contributing impetus from lunisolartidal components. Clearly, the strain from tectonic loading and corresponding stress-gaps-to-criticality along the faults of these
quakes probably varied considerably, so one should not take these qualitative comparisons as indicating the complete context of
such triggering. Also, for pre-1980 quakes, the lunisolar-tidal amplitude, jerk, and jounce values are not yet known and such
calculations require earth-tidal software of more comprehensive capability. These values will be posted as they become available—
as this living article evolves to its next draft revision.
[42]
The above observations add support to the apparent merit of investigating the development of a polynomial with terms of time
derivatives of tidal displacement that equates to total seismic energy—analogous to the previously described polynomial with terms
of time derivatives of ellipsoidal demand that equates to total seismic energy. Originating from two different modes of
displacement, the terms of the two polynomial equations might be explored individually or combined in some manner. Particular
time periods that appear to exhibit strong influence by either mode—such as the period 1890-1966.5 for the LA region illustrates
strong influence from ellipsoidal demand—could provide quake data that when analyzed mathematically, consistent with either
mode, might yield information about characteristic tendencies for triggering. Ultimately, a system of equations, which includes both
ellipsoidal-demand terms and earth-tidal terms, as supplied by a complete data set of quakes for a particular region, may yield a
general solution of coefficients that defines a mathematical tool useful in forecasting events in that region.
Expanded Criticality and Slip by Energetically Bridging Asperities - Hypothesis C
[43]
Seismic slippage along faults is not a smoothly continuous spatial process. As indicated by slip amplitude distributions obtained by
mathematical inversion models, mosaic patches of widely varying slip occur along a rupturing fault (Udias et al., 2014). After a main
quake and adjustment period of area stress transfer and dissipation (aftershocks), the slipped fault is left with corresponding zones
of varying stress with which to begin the next loading cycle. Therefore, during the next cycle, fault stress-gaps-to-criticality will
continue to reflect some residual irregularities as tectonic loading advances.
[44]
Pulsed stress components of time-derivative aspects of motion of ellipsoidal demand and earth tides have very long wavelengths.
The greater the combined energetic impetus for triggering from a complete time-derivative profile, the more likely that more stressgap irregularities may be bridged, rupture initiated, and slip advanced—thus, the more probable a rupture of greater length and slip
will propagate. Therefore, the larger the triggering impetus applied, the greater the size of the quake that may result and,
conversely, the larger the future quake, the more clearly it can be anticipated and its potential forecast, because more and greater
components of energetic impetus are usually necessary for its triggering.
[45]
For global seismic reality to be consistent with both hypotheses, B and C, the greatest quakes (M ≥ 8.8) worldwide will statistically
require major contributions of triggering impetus from most or all of the time-derivative aspects of both ellipsoidal demand and
earth tides. One astronomical aspect that, on an annual basis, affects daily tidal displacement amplitudes is the seasonal translation
of the apparent arcs that the Sun and Moon follow across the sky every Earth rotation. In either the northern or southern
hemisphere, the arcs are more directly overhead in summer and closer to the horizon in winter. When the Sun and Moon are more
directly overhead, tidal amplitudes are greater at that latitude. Therefore, to investigate the effect on triggering of M ≥ 8.8 quakes
by this seasonal change in tidal amplitudes, the relationship between quake latitude and quake time-of-year is illustrated in Figure 5,
below.
Figure 5
Description
[46]
Figure 5 shows a roughly annular loop of ‘seasonality’ for the epicenters of M ≥ 8.8 quakes, worldwide. Statistically, with a data set
of only 13 (+2 NM) events, the pattern is suggestive only. Smaller great quakes (8.4 ≤ M ≤ 8.7) were filtered from this version of the
graph for clarity of the loop. Those quakes showed a rough adherence to the loop, but extended the pattern to make a thicker
form—much like a doughnut with the same perimeter, but with a smaller hole. Quakes occurring before 1900 are included if the
estimated maximum is at least M8.8, except for both New Madrid events (labeled), which have been estimated at M8.1 maximum
for NM1 (16 Dec 1811) and M8.3 maximum for NM2 (07 Feb 1812) (Schulte and Mooney, 2005).
Interpretation
[47]
The most recent quake of the pre-1900 group was an event in 1868, so all these quakes occurred before seismometers were
practicable. Quake magnitudes determined without seismometers are far less certain, because estimates have to be made by
interpreting limited information from historic records of effects and modern field investigations using the methods of
paleoseismology. For a deeply historic event, one can never find and examine all the evidence and so estimates of magnitude may
not reflect all the energy that was expended. The post-1900 quakes started with an event in 1906, when a global distribution of
somewhat crude seismometers was operating and so magnitudes for this group have been more accurately determined. Of course,
the more recent the event the more accurate the estimate of magnitude, due to the increasing sophistication of seismometers over
time and the growing global network supplying seismic information.
[48]
The annular loop appears to indicate that a seasonal effect of the Sun and Moon is remarkably influential for triggering M ≥ 8.8
quakes. This seems especially true after considering how unequal the landmasses and plate margins are between the northern and
southern hemispheres. The seasonality of quake triggering is speculated to result from two direct lunisolar-tidal effects and one
indirect insolation effect. (1) On days when the Sun and Moon pass more directly overhead, average tide-generating effects are
greater at that latitude. So, in either the northern or southern hemisphere—spring will see epicenters move poleward and fall will
see them move toward the equator, with spring and fall being reversed on the opposite hemisphere. (2) Sun is closest to Earth in
the first week in January, making the Sun’s maximum tide-generating effect for the year occur then, which may help to explain the
two equatorial quakes. (3) An influx of groundwater in spring from both snowmelt and ground thaw probably helps to lubricate
faults and reduce stress-gaps-to-criticality, which may also help to explain the progression of epicenters toward higher latitudes as
spring trips poleward in either hemisphere. The pattern shown provides evidence that triggering of the greatest quakes requires
lunisolar-tidal assistance, therefore, supporting Hypothesis C and further suggesting that, triggering of the greatest quakes probably
requires nearly all supportive factors to be near optimal.
Notes:
Udias, A., R. Madariaga, E. Buforn (2014). Source Mechanics of Earthquakes: Theory and Practice, Cambridge: Cambridge University Press; pp.
181–188, 270–273.
Schulte S.M., W.D Mooney (2005). An updated global earthquake catalogue for stable continental regions: reassessing the correlation with
ancient rifts. Geophysical Journal International, 161, pp. 707-721.
Invariance of Spatial Scale for Jerk and Jounce in Rock Mechanics?
[49]
Many natural physical systems appear to exhibit an operational invariance of spatial scale. For example, the roughness of a coastline
may be similarly characterized (fractal dimension) for a wide range of distance scales. Materially consistent rock mechanics, such as
that herein proposed, temporally related between time-derivative aspects of ellipsoidal demand and earth tides, would be advanced
by such scale invariance and could be considered in a spatially invariant context by asking, “Because relatively large magnitudes of
jerk and jounce of vertical crustal motions appear to be energetic impetus for (large-scale) fault rupture and, thus, quakes; would
the magnitude of jerk and jounce aspects in a loading cycle prove important to rock fracture when applied repetitively to (smallscale) specimens in the laboratory?” The capability of synthesizing loading functions with varied time-derivative aspects has
certainly existed for decades with the development of computer-controlled electrohydraulic servo valves with stress-strain feedback
loops. Although textbook mention of the common effect of strain history (in loading/unloading cycles) on strain softening of rock
was found (Jaeger et al., 2007), no result has yet been identified that describes the importance to rock fracture of time-derivative
aspects of loading during repetitive cycles in laboratory testing of specimens.
Earthquake Forecasting and Earthquake Prediction
[50]
Considering the obvious complexity of quake triggering, anyone presently using the term prediction in the context of earthquakes
should be spanked with a Richter scale. Responsible forecasting of earthquakes may become the task of creating a time series of
regional warning levels (e.g. scale of 0–100) normalized to profiles of time-derivative aspects of crustal motion that are determined
by combining both the indirect and direct modes of tidal forcing—ellipsoidal demand and earth tides. Such composite profiles from
both modes may be adaptively analyzed mathematically and used to project potentials for quakes of particular size. Responsible
prediction of earthquakes may result from those forecasts directing the attention of seismologists to a locale or fault to develop a
probable scenario for quake nucleation, rupture, and slip using knowledge of historic seismicity, local crustal structure, and current
conditions informed by instrumentally monitoring precursory indicators (e.g. micro-earthquakes (sensitivity to M0.0), surface
deformation (GPS and tiltmeters), and crustal strain fluctuations (borehole strainmeters, radon detectors, and sensors of water level
in boreholes)). Overcoming the challenges of quake forecasting and, eventually, quake prediction will require a formidable,
cooperative enterprise.
Note:
th
Jaeger, J.C., Cook, N.G.W., Zimmerman, R. W. (2007). Fundamentals of Rock Mechanics, 4 ed., Blackwell Publishing; pp. 80–85, 260–270.
Summary Conclusions
[51]
In the LA region, fluctuating values of kinematic, time-derivative aspects of ellipsoidal demand, both individually and when
combined, appear to be slightly influencing the triggering of 6.0 ≤ M ≤ 6.9 quakes and significantly influencing the triggering of M ≥
7.0 quakes. Although not as completely presented herein, the SF region appears to be similar in its triggering responsiveness, when
allowance is made for its less compressive stress regime—an allowance presuming an influence and its mechanism that were both
speculated in section [29].
[52]
Evidence presented herein qualitatively supports significant triggering influence of M ≥ 7.0 earthquakes in the LA and SF regions by
kinematic, time-derivative aspects of ellipsoidal demand, which is crustal displacement produced indirectly by celestial-mechanical
forcing, through the related interaction of polar drift and migration of the equatorial bulge. After considerable model development,
refined projections of the speed and direction of polar drift, as calculated using up-to-date, mean-pole data, will allow accurate
values of the kinematic aspects of ellipsoidal demand to be determined for any location, during the past century through the next
decade. Further evidence presented qualitatively suggests an auxiliary triggering influence results from kinematic, time-derivative
aspects of earth tides, which are crustal displacements produced directly by interaction of Earth with primarily lunisolar gravity.
Analogous to calculating kinematic aspects of ellipsoidal demand, kinematic aspects of earth tides may be calculated from the total
tidal displacement array that is determined for any location and time of interest. A mathematical combination of the kinematic
aspects of the two modes of crustal deformation may yield a quake forecasting algorithm of reasonable accuracy for M ≥ 6.0 quakes
and should be pursued. Additionally, creation of a state-space representation or related formalism(s) should allow use of the
relevant mathematical tools that are available to observe solution regions of greater probability of M ≥ 6.0 quakes.
[53]
Much numerical work will be required to quantitatively investigate and develop these concepts. If an understanding of the
combined triggering influence of the two modes is achieved and the effect of that influence can be removed or otherwise
compensated for, exploring models designed to investigate the additional effect of many regularly repeated, tidal bulge/depression
cycles to quake nucleation processes may also be pursued. Such repetitive dilatation of the crust may play a discernible role in
advancing nucleation and, therefore, predisposing criticality.
Acknowledgements
[54]
I wish to thank Daniel S. Helman, M.S. Geology and Ph.D. candidate at Prescott College, for seismic and volcanic research, general
comments, and originating the Mars Conjecture described in section [26]; Jason Wu, Ph.D. candidate at UC Berkeley, for statistical
coding and graphing; Amelia A. McNamara, Ph.D. candidate at UCLA, for statistical coding, graphing, and advising; Yang Fei, M.S.
candidate at UC Berkeley, for statistical coding and graphing; Dr. Debra J. Blake for non-technical editing; and Katheryn L. Schneider
for partial funding.
Dedications
[55]
This work is dedicated to the memories of Raymond A. Smith, father of our webmaster, Alan R. Smith;
and Germaine M. Blake, mother of our Executive Board member, Dr. Debra J. Blake.
Institute for Celestial Geodynamics
BY-ND- 12 Oct 2014 to 22 Nov 2015
Appendix A
Graphs of Displacement, Speed & Acceleration, and Jerk & Jounce
of the Lunisolar Earth Tide
Landers and Big Bear Quakes of June 28, 1992
Figure A1
Figure A1 shows that the M7.3 Landers quake occurred 45 minutes after maximum tidal compression of
the local crust for the day. The M6.5 Big Bear quake occurred about 34 km away, 3.13 hours later.
The lunisolar earth-tide model used here, solid, provides step-function output that approximates the Earth’s body response to the
smoothly continuous system of gravitational forces between the Earth, Moon, and Sun. Thus, solid will exhibit systemic
discontinuities in its time-derivative aspects of motion. To remedy this situation, the LOESS tool is used to smooth these
irregularities, as required, to better simulate Earth’s natural body response. Also, as described above in section [13], earth-tide
displacement is generally out-of-phase with crustal displacement from ocean-tidal loading and solid does not include an ocean-tidal
loading component in its calculation of surface displacement. Therefore, to utilize this model most accurately, we will only
investigate epicenters that are at least 100 km from the Pacific coast.
Figure A2
Figure A2 shows that the M7.3 Landers quake occurred near a daily acceleration peak of earth-tide
displacement and the M6.5 Big Bear quake occurred at the next speed peak of earth-tide displacement.
Figure A3
Figure A3 shows that the M7.3 Landers quake occurred near a daily extremum of jounce of earth-tide
displacement and the M6.5 Big Bear quake occurred near the next extremum of jerk of earth-tide
displacement.
Elmore Desert Ranch and Superstition Hills Quakes of November 24, 1987
Figure A4
Figure A4 shows that the M6.2 Elmore Desert Ranch quake occurred 15 minutes before maximum tidal
compression of the local crust for the day. The M6.6 Superstition Hills quake occurred about 9 km
away, 11.36 hours later. (The Elmore epicenter is about 143 km from the Landers epicenter.)
Figure A5
Figure A5 shows that the M6.2 Elmore Desert Ranch quake occurred just before a daily acceleration
peak of earth-tide displacement and the M6.6 Superstition Hills quake occurred at the second
subsequent speed extremum of earth-tide displacement.
Figure A6
Figure A6 shows that the M6.2 Elmore Desert Ranch quake occurred somewhat before a daily
extremum of jounce of earth-tide displacement and the M6.6 Superstition Hills quake occurred just
after the second subsequent extremum of jerk of earth-tide displacement.
29 Jan 2015 rev.
Appendix B
Apparent Modulation of Polar-Drift Jounce and M ≥ 8.0 Quake Triggering
By Jerk and Jounce in Earth-Mars’ Gravitational Interaction
Introduction
[1]
The foregoing article is qualitative in nature and a quantitative result awaits detailed formulation of large earthquake energies in
terms of values of sequential time-derivatives of both ellipsoidal demand (ED) and earth tides, combined in some manner, as
analyzed using a state-space representation or related formalism(s). The complex transfer function between a rotational response
of Earth’s solid, outer layers (with respect to the terrestrial frame) and celestial-mechanical forcing involves a time lag, which is due
primarily to rotational inertias (i.e. polar and equatorial) of the combined mantle and crust, and various couplings at the core-mantle
interface (e.g. inertial, electromagnetic, and viscous). This time lag limits the value of the approach for projections of large quakes
into the future. Certainly, sequential time-derivatives of polar drift (linear component of polar motion) and ED may be statistically
modeled and projected several years forward, with increasing uncertainty commensurate with increasing time of projection. In
contrast, sequential time-derivatives of earth-tide displacements are reliably projected forward, because orbits of the Earth, Moon,
and Sun are accurately modeled and such theories may be combined with proven tidal-deformation models to accurately describe
earth tides for probably more than a century into the future.
[2]
The celestial-geodynamic approach to the problem of forecasting large earthquakes decades or more into the future will require
linking geophysical measures of Earth response—polar drift and corresponding ED—to celestial-mechanical forcing by its Solar
System neighbors (Moon, Sun, Mars, and possibly Jupiter and Venus). Because orbital motions of the Earth and neighboring bodies
are accurately modeled, it seems reasonable to posit that such theories may be applied to produce quantitative forecasts of large
quakes decades into the future, if the time-derivative aspects, jerk and jounce, could be demonstrated to pass through the
governing dynamics—from celestial-mechanical origins to polar drift, then to ED, and ultimately shown to statistically enhance
triggering of the largest quakes. An example of such passing will be examined herein for the Earth-Mars system. ED response will be
presented first, only because it was noticed first in Figure 3 of the adjoining article. Figure 3 is repeated here.
Figure 3
[3]
Figure 3 is described in detail in sections [23–26] of the article, so only a brief summary is included here. The figure reveals matching
on two time scales between the jounce of ED in the LA region and the Earth-Mars’ synodic period (nearest encounter-to-nearest
encounter). The fine sawtooth waveform in the jounce trace between 1958.85 and 1995.11 exhibits a period that is calculated to be
778 Earth days. The Martian synod is 779.94 days. Also, the time series that is positioned below the jounce trace—Earth-Mars’
inverse-squared nearest separation distances—shows cyclical variation with a period of 15.78 years. The pronounced cyclicity in
distance results from an eccentricity in each orbit and a heliocentric angle of about 126° between the two perihelions. The
wavelength of this time series appears to match the jounce trace’s undulation both in period and synchronization. The Y2 axis
indicates the value and units of these data.
[4]
Because jounce of ED is calculated by using only the polar-drift time-series and geometrical constants that depend on global
location, it seems reasonable to suppose that the Earth-Mars’ encounter time-series might similarly match the jounce of polar drift,
but with even greater clarity. Figure B1 reveals this agreement.
Figure B1
Description with Background
[5]
Figure B1 shows calculated jounce of polar drift for the 1890–2011.05 portion of a polar-drift dataset supplied on 21 March 2011 by
(1)
Dr. Daniel Gambis, Director of EOP PC at the Observatoire de Paris and Board Member of the International Earth Rotation and
(2)
Reference Systems Service (IERS). The dataset is comprised of sequential mean poles of the north rotational pole for 1846–
2011.05 and consists of three consecutive and independently determined records of observations. Each record segment involves
different resolution and accuracy levels, because acquisition methods and instrumentation improved significantly during those 165
years. Once the dataset was compiled, Dr. Gambis filtered pole components to remove both the Chandler and annual wobble terms.
The most accurate segment, 1900–2011.05, contains mean poles every 0.05 year (18.26 days). When examined under scrutiny of
the fine numerical resolution provided by sequential calculation of time derivatives, this segment appears to yield the most
consistent results. Hence, the jounce trace of Figure B1 is considered here to be the most accurate during 1900–2011.05. After
2011.05, projected jounce is calculated from smoothed jerk, which is calculated from projected acceleration, as detailed in the
legend.
[6]
For the period 1958.85–2010.07 (51.22 years), a time-series trace representing Earth-Mars’ close encounters is positioned below the
jounce trace. Each data point in the series is valued as the inverse-squared of the nearest distance at the time of that encounter.
The Y2 axis indicates the value and units of these data.
Interpretation
[7]
From 1900 to about 1960, the cyclical trace of polar-drift jounce shows consistent growth in amplitude envelope at roughly equal
time intervals. Such rotational response in the jounce domain of change is hypothesized to show resonance produced by celestialmechanical forcing on an asymmetrical mass distribution of Earth. Quakes of M ≥ 9.0 have repeatedly been shown to affect the spin
dynamics of Earth by altering the global mass distribution. And from section [11] of the adjoining article:
In 1960, the greatest quake instrumentally measured (M9.5) with a fracture length of about 1000 km occurred near Chile
on a plate margin of the Pacific Rim. In 1964, the second greatest quake instrumentally measured (M9.2) with a fracture
length of nearly 800 km occurred in Prince William Sound, Alaska on a plate margin of the Pacific Rim.
Thus, because an obvious change in rotational response occurred about 1960, it seems likely that the 1960 Chilean and possibly
1964 Alaskan mega-quakes interrupted Earth’s resonant response in the jounce domain of change by altering the global mass
distribution via movements in the crust.
[8]
After 1960, the jounce of polar drift imitates the time-series trace representing Earth-Mars’ close encounters in both period and
synchronicity, while also exhibiting the fine sawtooth waveform. Apparently, regular close encounters with Mars provide pulsed
perturbations of gravitational torque, which the solid outer Earth rotationally responds to on two time scales in the jounce domain
of change.
Notes:
1. http://www.obspm.fr/ accessed 08 Oct 2015.
2. http://www.iers.org/IERS/EN/Home/home_node.html accessed 08 Oct 2015.
Figure B2
Description
[9]
Figure B2 shows the sorting of 124 great (M ≥ 8.0) quakes, worldwide, from 1958.85–2010.07 into 0.10 intervals of Earth-Mars’
synodic periods (nearest encounters). This 51.22-year period matches the time of significant gravitational influence by Mars on the
jounce of polar drift, as shown in Figure B1. Because each one of Earth-Mars’ synodic periods varies slightly in length, interval limits
have been normalized to reflect the exact fractional portion of the particular synodic period that each quake occurred in. The time
of nearest distance of each synodic period is centered on the X axis.
Interpretation
[10]
Although a larger number of great quakes would make a stronger statistical argument, Figure B2 shows a clear pattern of occurrence
with remarkable symmetry that reflects about the time zero of nearest distance of Earth-Mars’ encounters. Rationale for the overall
symmetry and the lower counts in intervals -1 and 1 are suggested, as follows. The tapering down, symmetrically away from the
center, of distribution counts in the outer intervals, hints that a forcing function of triggering impetus may share a similar profile.
The average rate of great quakes within this period is 124/51.22 = 2.42 quakes/year and the nominal synodic period is 2.135 years.
Thus, the average number of quakes within a synodic period is 2.42 x 2.135 = 5.17, or about 1 quake every two intervals. With such
a low rate of quake occurrence, a significant triggering impetus just before the two intervals of nearest distance could produce a
quake shadow within these two intervals. Supporting evidence to explain the distribution of this histogram awaits the development
of models of time derivatives of gravitational potential and factors of gravitational torque for the Earth-Mars system—Figures B3
and B4, below.
Figure B3
Description
[11]
Figure B3 presents a composite picture of gravitational jerk and jounce distance factors that are produced by Earth-Mars’
gravitational interaction during three close encounters. The histogram of Figure B2 includes great quakes from twenty-four Earth6
Mars’ synodic periods, which include close encounters that range in nearest distance from 55.76–101.32 x 10 km. Therefore, to
model dynamics with scenarios that exemplify the range of these distances, three encounter dates were selected for analysis that
included nearest distances which well-spanned this range. In this graph, for simplicity, the X axis represents time in terms of a
fraction of the nominal synodic period of 779.94 Earth days. Hence, vertical gridlines, in fractional multiples of 0.10 nominal periods,
bound 78-day periods that are essentially equivalent to the intervals in the histogram of Figure B2. The X-axis zero is the time of
nearest encounters. The traces indicate first or second time-derivatives of the inverse-squared of Earth-Mars’ separation distances.
Thus, trace values provide proportionate measures of gravitational jerk and jounce potentials throughout the synodic periods.
Interpretation
[12]
Gravitational potential during Earth-Mars’ encounters is the principal factor in calculating gravitational torque on Earth by Mars.
This proposition is supported in Figure B1 by the obvious imitation of the time series of inverse-squared nearest distances by the
jounce of polar drift. The traces of gravitational jerk and jounce distance factors shown in Figure B3 are intended for qualitative
comparison only, within and between Earth-Mars’ synodic periods, because they do not reflect relevant constants (e.g. gravitational
constant, planetary masses). Regardless, the traces clearly show times of significant gravitational jerk and jounce within the
(normalized) Earth-Mars’ synodic period. And the times of combined strength of gravitational jerk and jounce near each encounter
do appear to match the distribution of great quakes in the histogram. A computer simulation modeling the effect of these timeseries values on the triggering of great quakes should be performed to test this hypothesis.
[13]
For the Earth-Mars system, a given separation distance corresponds to a particular gravitational force, which results in a
corresponding potential to apply torque to the Earth. Torque applied to Earth results in rotational acceleration of Earth, which is
indicated by acceleration of polar drift (Zbikowski, 2012). Hence, for the Earth-Mars system, separation distance manifests
commensurately as rotational acceleration. Analogously, the first time-derivative of separation distance manifests commensurately
as rotational jerk and the second time-derivative of separation distance manifests commensurately as rotational jounce.
[14]
Traces for the three encounters of 2012.2, 2016.4, and 2018.6 exhibit values of gravitational jerk and jounce distance factors that
are both inversely and nonlinearly related to the nearest distances achieved. However, the timing of local minima and maxima of
jerk and jounce, across multiple encounters, does not change with variations of nearest distance. Also, absolute amplitudes of
extrema of jerk or jounce before an encounter are always slightly greater than those afterward. The probable reason for such
results is that Earth-Mars’ mutual gravitational attraction during an orbital approach adds constructively to the acceleration and
speed of change in separation distance, while the same attractive force reduces the relative acceleration and speed of change in
separation distance during departure. This same dynamic is likely responsible for the delay of inflection points on trace curves, until
slightly after the time zero of nearest distance (bold vertical gridline).
Note:
Zbikowski, D. W. (2012). Global View of Great Earthquakes and Large Volcanic Eruptions Matched to Polar Drift and its Time Derivatives.
Institute for Celestial Geodynamics, Figure 3. Latest draft available at: http://www.celestialgeodynamics.org/content/global-view
Figure B4
Description with Background
[15]
Figure B4 illustrates the timing of great (M ≥ 8.0) quakes, worldwide, within portions of Earth-Mars’ synodic periods that are
adjacent to the 2012.2 encounter. The graph includes gravitational jerk and jounce distance factors (dashed lines) that are taken
from Figure B3, and (approximate) gravitational torque jerk and jounce variables factors (solid lines), which are explained next.
[16]
The calculation of gravitational torque on Earth by Mars requires, in addition to a time series of inverse-squared separation
distances (and its related constants: e.g. gravitational constant, planetary masses), a time series of orientation factors and the
aspherical mass distribution of Earth. Because our result will be used solely for relative comparison within and across Earth-Mars’
synodic periods, only time-variable parameters will be utilized. Thus, similar to omitting the gravitational constant and planetary
masses, the aspherical mass distribution will not be included here, because it remains constant for the durations considered. An
approximate time series of orientation factors utilizes varying declinations of Mars with respect to Earth. The approximation used
herein:
(Approximate) Orientation Factor = OF = Sin(declinationMars) x Cos(declinationMars)
The magnitude of the orientation factor varies (-0.369–0.369 for the 2012.2 encounter) over the profiles of gravitational jerk and
jounce distance factors, and shifts in phase in a complex manner from one synodic period to the next. After factoring the
gravitational jerk and jounce distance factor time-series by the orientation factor time-series, the resulting time series of
approximate, gravitational torque jerk and jounce variables factors on Earth by Mars for the 2012.2 encounter is plotted (solid lines)
for correlation to occurrences of great quakes.
Interpretation
[17]
Figure B4 provides a qualitative illustration of Earth-Mars’ relative gravitational torque jerk and jounce variables factors, which is
valuable for visual correlation to occurrences of great quakes.
[18]
The Earth-Mars’ orientation factor time-series appears to critically modulate both the first and second time-derivative, inversesquared separation distance factor time-series. An example is the torque variables factor for jerk (blue solid line), where the
maximum absolute extremum is changed from occurring before the time of nearest distance to about 30 days afterward. The three
great quakes of 11 April occurred about one week after that jerk maximum, which was the greatest jerk from Earth-Mars’
gravitational interaction since at least the similar part of the previous encounter (about 111 weeks before). The absolute value of
torque jerk factor on the day of the quakes was still about 93% of the recent maximum. The torque jounce factor on the day of the
quakes was about 34% of its subsequent local maximum. Triggering of the 6 February quake was clearly not enhanced by EarthMars’ gravitational interaction, but its occurrence certainly now begs for similar torque analyses of Earth with its other Solar System
neighbors. Relatedly, it is important to not consider the quake triggering impetus due to Earth-Mars’ gravitational interaction as the
only triggering impetus, but rather, only one component in a time-varying mix of celestial-geodynamic contributions.
Figure B5
Description
[19]
Figure B5 is a log2 (n) scaled energy graph of M 5.5–7.3 quakes in the Los Angeles region for 1958.85–1995.11. The equation used
for magnitude conversion to energy is: E (joules) = 10^(1.5M + 4.8). This is the same duration of 36.26 years, shown in Figure 3,
when the jounce of ED in the LA region and the Earth-Mars’ synodic period (nearest encounter-to-nearest encounter) appears to
match on two time scales. This uncommon degree of agreement suggests that the Earth-Mars’ synodic period might graphically
exhibit a noticeable amount of influence on the triggering of these quakes. The graph design of Figure B5 predates the above graphs
and the synodic periods are not displayed with the times of nearest distances centered on the graph, but with the end points (0.00
and 1.00) representing times of successive nearest distances, which bound each synodic period. Normalized phase angle is
calculated as the time of each quake positioned within the exact length of its particular synodic period.
Interpretation
[20]
Although Figure B5 displays quake energies distinctively within Earth-Mars’ synodic periods, no pattern is visually apparent. Also,
the sample size is too small for a statistical analysis to be of much value. The log 2 (n) scaled energy axis appears to display quakes in
a clear and physically intuitive manner, and seems to be an appropriate design option when a sample of quakes has a magnitude
range of three or less.
Figure B6
Description
[21]
Figure B6 is a log2 (n) scaled energy graph of M 5.5–7.3 quakes in the Los Angeles region for 1900–2014.17 (114.17 years). In this
graph, 94 quake clusters (136 quakes) are plotted with their energies determined by the magnitude of the largest quake in each
cluster. The equation used for magnitude conversion to energy is: E (joules) = 10^(1.5M + 4.8). As in Figure B5, synodic periods are
not displayed with the times of nearest distances centered on the graph, but with the end points (0.00 and 1.00) representing times,
in this case, of successive planetary oppositions, which bound each synodic period. Because planetary oppositions determine
synodic periods in this graph, a small shift in phase angle may result in the position of a quake from that in Figure B5, as is noticeable
for the M 7.3 quake at X = 0.764. Normalized phase angle is calculated as the time of each quake positioned within the exact length
of its particular synodic period.
Interpretation
[22]
Similar to Figure B5, this graph clearly displays quake energies within Earth-Mars’ synodic periods. However, with a much larger
sample size of 94 clusters (136 quakes), there are five quakes of M ≥ 7.0, of which four occur at times that correspond to intervals of
potentially significant gravitational torque jerk or jounce variables factors. Further graphical analysis, similar to that shown in Figure
B4, might help to mechanistically explain these occurrences. One M 7.0 quake occurrence (X = 0.394) corresponds to a time of
minimal torque variables factors and so does not suggest enhanced triggering by Earth-Mars’ gravitational interaction. In Figure B6,
no pattern is visually apparent for events of smaller magnitude. Hence, in the LA region for these 114.17 years, triggering of M ≥ 7.0
quakes may positively correlate moderately to periods of significant Earth-Mars’ gravitational torque jerk or jounce variables factors.
Further work with supporting evidence is needed to advance this proposition.
Summary Conclusions
[23]
The Earth-Mars’ quake triggering effect seems to be relatively small, but is graphically demonstrable for adequate sample sizes of M
≥ 8.0 quakes, worldwide. The behavior of small triggering influences in evidence for only the largest quakes is consistent with
Hypothesis C in the adjoining article. Under statistical analysis, samples of M ≥ 7.0 quakes from the LA region may show triggering by
the Earth-Mars’ gravitational influence; however, an adequate sample size may require centuries of record. These intriguing results
call strongly for continued related research on the Earth-Mars system. Ultimately, a compilation of time series of gravitational
torque jerk and jounce that are applied to Earth by all of its Solar System neighbors will likely provide valuable insights into both past
and future occurrences of the largest earthquakes.
Acknowledgments
[24]
I wish to thank Dr. Daniel Gambis for polar drift data; Dr. Ivan N. Tikhonov, Head of Laboratory, Institute of Marine Geology and
Geophysics, Far Eastern Branch of Russian Academy of Sciences, for assistance with concepts, methods, and calculations; Daniel S.
Helman, M.S. Geology and Ph.D. candidate at Prescott College, for seismic research, comments, and originating the Mars Conjecture
described in section [26] of the main article; Yang Fei, M.S. Statistics, for statistical coding and graphing; Jason Wu, Ph.D. candidate
at UC Berkeley, for statistical coding and graphing; Amelia A. McNamara, Ph.D. Statistics, for advising; and Dr. Debra J. Blake for nontechnical editing.
Institute for Celestial Geodynamics
BY-ND- 21 November 2015
Appendix C
San Francisco Region: Graphs of Jerk and Jounce of Ellipsoidal Demand
Figure C1
Figure C1 shows the jerk that is intrinsic within SF’s speed and acceleration of ellipsoidal demand, which
are plotted in Figure 4 and described in sections [27–34]. The quakes are the same as in Figure 4. All
calculations and descriptions are analogous to those for Figure 2, which shows LA’s jerk of ellipsoidal
demand.
Figure C2
Figure C2 shows the jounce that is intrinsic within SF’s jerk of ellipsoidal demand, which is plotted in
Figure C1. The quakes are the same as in Figure C1. All calculations and descriptions are analogous to
those for Figure 3, which shows LA’s jounce of ellipsoidal demand.
14 Nov 2015 rev.