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M-Synchronizations: The B-Megasignal and Large Earthquakes
C H A P T E R
24 M-Synchronizations: The B-Megasignal and Large Earthquakes O U T L I N E The Magnitude-Synchronization Function
347
B-Megasignal: The Papua New Guinea Case Study 347 Results and Interpretation: The B-Megasignal 349 The Southern California Case Study
Results and Interpretation: The Minimum Magnitude Rule 351 References
351
349
In the previous chapter, we illustrated that time-synchronizations (or T-synchronizations) are insufficient to cause an earthquake. An additional condition was the equality between at least two B-magnitude signals belonging to two different Ω-sequences. This also implies equality between at least two Cosserat characteristic lengths. The existence of M-synchronizations implies that the Ω-sequences are not just phase synchronized (T-synchronized), but their synchronization and/or coupling is even more extensive (see also Pecora and Carroll, 1990, 1991; Carroll and Pecora, 1991, 1993; Boccaletti et al., 2002; Arenas et al., 2008; Manzano et al. 2013, for additional definitions of synchronizations). Whenever two Cosserat characteristic lengths Lk and Lm are equal, the corresponding B-magnitude signals will also be equal. The B-magnitude signals associated with Lk and Lm are according to Eq. (13) in Chapter 20: Lk , K Lm Mm ¼ 2 log : K Mk ¼ 2 log
(1) (2)
It follows: Lk ¼ Lm , Mk ¼ Mm :
(3)
An earthquake will happen when at least two Ω-sequences can produce the same magnitude earthquake at the same time. An example of this principle is illustrated in Test 2 of the previous chapter. The above rule will here be termed magnitude synchronization (or M-synchronization). In the Ω-sequences the magnitude of each earthquake can be defined by the interaction with all previous earthquakes. Let mi be the magnitude of the ith earthquake in the sequence. According to Båth’s law the magnitude of the next event would be: mi + 1 ¼ mi + Δm:
(4)
In the DSR Ω-sequences, this corresponds to the B1-magnitude signal, whilst in the ISR Ω-sequences the above equality corresponds to the B2-magnitude signal. However, the above equation will only be valid when the earthquake number i + 1 interacts with the ith earthquake. When the earthquake number i + 1 interacts with the earthquake number i 1, the magnitude will be:
The Omega-Theory https://doi.org/10.1016/B978-0-12-814580-7.00024-1
345
© 2018 Elsevier Inc. All rights reserved.
346
24. M-SYNCHRONIZATIONS: THE B-MEGASIGNAL AND LARGE EARTHQUAKES
FIG. 24.1 Ω-sequences can be defined by several B1- and B2signals. The example in this figure illustrates how interactions between different sequence events according to the B2magnitude signal define several levels of freedom for the magnitudes m1, m2, m3 and m4 of the clustered seismic events in the last sequence event CE5 of this ISR Ω-sequence. The example illustrated in this figure is frequently encountered in the seismic catalogs.
mi + 1 ¼ mi1 + 2Δm:
(5)
This creates and defines several levels of freedom for the future magnitudes of earthquakes. In Fig. 24.1, these levels of freedom are marked as m1, m2, m3, and m4. In Fig. 24.1 the magnitudes of earthquakes in the last sequence event CE5 are defined by interactions with all previous earthquakes defined by the B1-magnitude signal. The differences between the magnitudes of these earthquakes (within CE5) are defined by Båth’s law: mi + 1 mi ¼ Δm:
(6)
Because earthquakes can interact with all previous earthquakes the B-magnitude signals can rarely be reconstructed in the real Ω-sequences. The Ω-sequences are always geometric series of earthquakes based on the B-spectral theorem; however the magnitudes of earthquakes which build these sequences are not always in accordance with the B-magnitude signals defined for those particular sequences. It is therefore convenient to approximate the magnitude of the next (extrapolated) event in the Ω-sequence according to equation: mi + 1 ¼ mi + kΔm; k ¼ 1, 2,3, …
(7)
Here, k is a free parameter. According to Fig. 24.1, we should take: mCE5 ¼ mCE4 + 4Δm; k ¼ 4:
(8)
This equation then defines the maximum possible magnitude mmax that the Ω-sequence can produce. However, the above equation will only be valid when the magnitude mCE4 of the CE4 event in the ISR Ω-sequence is defined by the B1-magnitude signal of that particular Ω-sequence. When the magnitude of the CE4 event is defined by an interaction with some other past earthquake, then it will be more convenient to take: mðTn Þ ¼ m + kΔm; k ¼ 1,2,3, …
(9)
Here, m is the average magnitude of the earthquakes that already belong to the analyzed Ω-sequence, and k is a free parameter. Tn is the time of the extrapolated earthquake. For the DSR Ω-sequence the time Tn is defined by Eq. (22) in Chapter 20: Tn ¼ Tn1 + BΔTn1,n2 :
(10)
Here, ΔTn1, n2 is the time interval between the sequence event numbers n 1 and n 2. For the ISR Ω-sequence, time Tn is defined by Eq. (23) in Chapter 20: 1 Tn ¼ Tn1 + ΔTn1, n2 : B
IV. SEISMIC CHAOS SYNCHRONIZATIONS
(11)
B-MEGASIGNAL: THE PAPUA NEW GUINEA CASE STUDY
347
To predict the maximum possible magnitude mmax the Ω-sequence can produce, we usually take: k ¼ N ðΩÞ or k ¼ N ðΩÞ 1:
(12)
N(Ω) is the number of sequence events in the Ω-sequence. If future earthquakes were predicted based on the Ω-sequences that at the moment of prediction contained 4 sequence events, then the next event would be the CE5 event. This means N(Ω) ¼ 5 and k ¼ 5 (or k ¼ 4). If future earthquakes were predicted based on the Ω-sequences that at the moment of prediction contained 3 sequence events, then the next event would be CE4 event. This means N(Ω) ¼ 4 and k ¼ 4 (or k ¼ 3).
THE MAGNITUDE-SYNCHRONIZATION FUNCTION We define the magnitude function Mi(T, m) of the analyzed Ωi-sequence in the following way: 1 if T ¼ Tn and m 2 ½mmax δm, mmax + δm, Mi ðT, mÞ ¼ 0 if T 6¼ Tn and m62½mmax δm, mmax + δm:
(13)
Mi(T, m) is a 2D function, which defines the presence of the B-magnitude signal at some future time Tn in the interval of the magnitudes m 2 [mmax Δm, mmax + Δm]. It is important to emphasize that the B-magnitude signal is only present at the time Tn. In general, when the value of the Båth’s parameter is defined with the relative accuracy of δB (see Chapter 20), then time Tn will also lie within some interval Tn 2 [Tn ΔT, Tn + ΔT]. On the 2D graph the function Mi(T, m) will have a value equal to 1 when the B-magnitude signal at some time Tn around the magnitude mmax is present, but it will be equal to 0 outside this region. Definition: For a multiplicity of Ω-sequences we define the magnitude synchronization function (or M-synchronization function): N X Mi ðT,mÞ: (14) ℕM ðT,mÞ ¼ i¼1
This function defines the magnitude-synchronization between the B-magnitude signals that belong to N Ω-sequences in the seismic catalog. We will demonstrate the M-synchronization function ℕM(T, m) in the following subchapters.
B-MEGASIGNAL: THE PAPUA NEW GUINEA CASE STUDY During the Nepal (2015) experiment described in the previous chapter, several large earthquakes occurred in the region of Papua New Guinea. Fig. 24.2 illustrates the USGS seismic catalog that was available on 08/05/2015, at the time of our Nepal (2015) experiment. This figure illustrates the seismic catalog between 02/04/2015 and 08/05/2015. The lower cutoff magnitude was MC ¼ 4. Most earthquakes occurred in the New Britain Trench, while some also occurred near the Bougaiville Island. These two regions represented the vertices of the Ω-cell active during the selected time period. In the test illustrated in Fig. 24.3, we used the series of earthquakes between 02/04/2015 and 23/05/2015 as the system SM2 M1[T1, T2] to predict the M-synchronization in the future. In Fig. 24.3, this system is illustrated with yellow columns. Later earthquakes that were not used to calculate the M-synchronization function ℕM(T, m) are marked with red columns. Fig. 24.3 also illustrates the M-synchronization function ℕM(T, m) by using a full-color spectrum. The M-synchronization function was calculated for Ω-sequences containing three sequence events on 22/04/2015. The fourth sequence event was extrapolated to the future, so the total number of sequence events in the analyzed Ω-sequences was N(Ω) ¼ 4. Therefore the parameter k in Eq. (12) was k ¼ 4. In Fig. 24.3 the blue color represents the total absence of any B-magnitude signal. The red color, on the other hand, represents the maximum density of B-magnitude signals. In other words the red color represents the maximum value of the M-synchronization function ℕM(T, m). Note that the M-synchronization is patchy due to the fact that it also contains information on the T-synchronizations. This is very important.
IV. SEISMIC CHAOS SYNCHRONIZATIONS
348 24. M-SYNCHRONIZATIONS: THE B-MEGASIGNAL AND LARGE EARTHQUAKES
IV. SEISMIC CHAOS SYNCHRONIZATIONS
FIG. 24.2
(A) Seismic catalog (USGS) of the Papua New Guinea region between 02/04/2015 and 12/05/2015 for the lower cutoff magnitude 4. (B) Epicenters (blue circles) of the earthquakes illustrated on (A). The radius of the circles represents the relative amount of energy released by these earthquakes.
349
THE SOUTHERN CALIFORNIA CASE STUDY
FIG. 24.3
M-synchronization for the earthquakes in the Papua New Guinea region between 02/04/2015 and 26/04/2015 (marked with yellow columns) above the lower cutoff magnitude 4. Blue color represents the absence of the B-magnitude signals. The M-synchronization between the B-magnitude signals is illustrated with light blue to red colors. At the time of the earthquakes marked with red columns the M-synchronization increased in magnitude. The B-megasignal is clearly visible, defining the magnitude of the largest earthquakes in the region.
Results and Interpretation: The B-Megasignal Results illustrated on Fig. 24.3 show that M-synchronization after 23/05/2015 occurred at higher and higher magnitudes (red and yellow colors) along the hypothetical line (marked with a dashed line) that was increasing with time. We will define this as the B-megasignal. This describes the time development of the M-synchronization. From Fig. 24.3, it is obvious that the B-megasignal defined the magnitude of the large earthquakes that occurred after 30/04/2015.
THE SOUTHERN CALIFORNIA CASE STUDY A similar test as described in the previous subchapter was also performed in the Southern California region, which is important for the fact that the area is highly populated and seismically highly active. Fig. 24.4 illustrates the SCEDC seismic catalog between 19/05/1990 and 31/08/2015 for the lower cutoff magnitude MC ¼ 4.5. Several large earthquakes hit the region in this time period. These large earthquakes were constrained to the San Andreas Fault zone between the Los Angeles and Palm Springs regions. In the test illustrated in Fig. 24.5, we used the series of earthquakes between 19/05/1990 and 01/01/1999 as the system SM2 M1[T1, T2] to predict the M-synchronization in the future. In Fig. 24.5, this system is illustrated by dark red columns. Later earthquakes that were not used to calculate the M-synchronization function ℕM(T, m) are marked by red columns. Fig. 24.5 also illustrates the M-synchronization function ℕM(T, m) by using a full-color spectrum. The M-synchronization function was calculated for Ω-sequences containing three sequence events within the system SM2 M1[T1, T2]. The fourth sequence event was extrapolated to the future, so the total number of sequence events in the analyzed Ω-sequences was N(Ω) ¼ 4. Therefore the parameter k in Eq. (12) was k ¼ 4. In Fig. 24.5 blue marks the total absence of any B-magnitude signal. Red represents the maximum density of B-magnitude signals or the maximum value of the M-synchronization function ℕM(T, m). Fig. 24.5 also illustrates the T-synchronizations (black and gray).
IV. SEISMIC CHAOS SYNCHRONIZATIONS
350 24. M-SYNCHRONIZATIONS: THE B-MEGASIGNAL AND LARGE EARTHQUAKES
IV. SEISMIC CHAOS SYNCHRONIZATIONS
FIG. 24.4 (A) Seismic catalog (SCEDC) of the Southern California region between 19/05/1990 and 31/08/2004, for the lower cutoff magnitude 4.5. (B) Epicenters (brown circles) of the earthquakes illustrated on (A). The radius of the circles represents the relative amount of energy released by these earthquakes.
REFERENCES
351 FIG. 24.5 M-synchronization in the Southern California region calculated for the earthquakes marked in dark red that occurred between 19/05/1990 and 01/01/ 1999 above the lower magnitude cutoff 4.5 (see also Fig. 24.4). Blue represents the absence of the B-magnitude signals. The M-synchronization between the B-magnitude signals is illustrated with light blue to red colors. Several B-megasignals are visible. Three of them are marked with black dashed lines. The yellow dashed line represents the minimum magnitude value of the B-megasignals at some chosen time. This line defines the magnitude of the second- and third-order events according to the minimum magnitude rule. This figure also illustrates the T-synchronizations for δB ¼ 5% (gray) and δB ¼ 3% (black).
Results and Interpretation: The Minimum Magnitude Rule The results in Fig. 24.5 indicate the presence of several B-megasignals. The large first-order earthquake on 16/10/ 1999 north of the Palm Springs region was clearly defined by one of this B-megasignals. On the other hand the secondand third-order earthquakes were defined by the minimum value of the B-megasignals. Their magnitudes are always below the region, in which the M-synchronizations occur. In Fig. 24.5 the lower boundary of this region is marked with a dashed yellow curve. From this test, we see that the dashed yellow curve represents the envelope for the magnitudes of the second- and third-order earthquakes. This empirical rule will be termed here the minimum magnitude rule.
References Arenas, A., Díaz-Guilera, A., Kurths, J., Moreno, Y., Zhou, C., 2008. Synchronization in complex networks. Phys. Rep. 469 (3), 93–153. Boccaletti, S., Kurths, J., Osipov, G., Valladaras, D.L., Zhou, C.S., 2002. The synchronization of chaotic systems. Phys. Rep. 366, 1–101. Carroll, T.L., Pecora, L.M., 1991. Synchronizing chaotic circuits. IEEE Trans. CAS 38, 453. Carroll, T.L., Pecora, L.M., 1993. Cascading synchronized chaotic systems. Physica D 67, 126–140. Manzano, G., Galve, F., Giorgi, G.L., Hernández-Garcia, E., Zambrini, R., 2013. Synchronization, quantum correlations and entanglement in oscillator networks. Sci. Rep. 3, 1439. https://doi.org/10.1038/srep01439. Pecora, L.M., Carroll, T.L., 1990. Synchronization in chaotic systems. Phys. Rev. Lett. 64, 821. Pecora, L.M., Carroll, T.L., 1991. Driving systems with chaotic signals. Phys. Rev. A 44, 2374.
IV. SEISMIC CHAOS SYNCHRONIZATIONS
24 M-Synchronizations: The B-Megasignal and Large Earthquakes O U T L I N E The Magnitude-Synchronization Function
347
B-Megasignal: The Papua New Guinea Case Study 347 Results and Interpretation: The B-Megasignal 349 The Southern California Case Study
Results and Interpretation: The Minimum Magnitude Rule 351 References
351
349
In the previous chapter, we illustrated that time-synchronizations (or T-synchronizations) are insufficient to cause an earthquake. An additional condition was the equality between at least two B-magnitude signals belonging to two different Ω-sequences. This also implies equality between at least two Cosserat characteristic lengths. The existence of M-synchronizations implies that the Ω-sequences are not just phase synchronized (T-synchronized), but their synchronization and/or coupling is even more extensive (see also Pecora and Carroll, 1990, 1991; Carroll and Pecora, 1991, 1993; Boccaletti et al., 2002; Arenas et al., 2008; Manzano et al. 2013, for additional definitions of synchronizations). Whenever two Cosserat characteristic lengths Lk and Lm are equal, the corresponding B-magnitude signals will also be equal. The B-magnitude signals associated with Lk and Lm are according to Eq. (13) in Chapter 20: Lk , K Lm Mm ¼ 2 log : K Mk ¼ 2 log
(1) (2)
It follows: Lk ¼ Lm , Mk ¼ Mm :
(3)
An earthquake will happen when at least two Ω-sequences can produce the same magnitude earthquake at the same time. An example of this principle is illustrated in Test 2 of the previous chapter. The above rule will here be termed magnitude synchronization (or M-synchronization). In the Ω-sequences the magnitude of each earthquake can be defined by the interaction with all previous earthquakes. Let mi be the magnitude of the ith earthquake in the sequence. According to Båth’s law the magnitude of the next event would be: mi + 1 ¼ mi + Δm:
(4)
In the DSR Ω-sequences, this corresponds to the B1-magnitude signal, whilst in the ISR Ω-sequences the above equality corresponds to the B2-magnitude signal. However, the above equation will only be valid when the earthquake number i + 1 interacts with the ith earthquake. When the earthquake number i + 1 interacts with the earthquake number i 1, the magnitude will be:
The Omega-Theory https://doi.org/10.1016/B978-0-12-814580-7.00024-1
345
© 2018 Elsevier Inc. All rights reserved.
346
24. M-SYNCHRONIZATIONS: THE B-MEGASIGNAL AND LARGE EARTHQUAKES
FIG. 24.1 Ω-sequences can be defined by several B1- and B2signals. The example in this figure illustrates how interactions between different sequence events according to the B2magnitude signal define several levels of freedom for the magnitudes m1, m2, m3 and m4 of the clustered seismic events in the last sequence event CE5 of this ISR Ω-sequence. The example illustrated in this figure is frequently encountered in the seismic catalogs.
mi + 1 ¼ mi1 + 2Δm:
(5)
This creates and defines several levels of freedom for the future magnitudes of earthquakes. In Fig. 24.1, these levels of freedom are marked as m1, m2, m3, and m4. In Fig. 24.1 the magnitudes of earthquakes in the last sequence event CE5 are defined by interactions with all previous earthquakes defined by the B1-magnitude signal. The differences between the magnitudes of these earthquakes (within CE5) are defined by Båth’s law: mi + 1 mi ¼ Δm:
(6)
Because earthquakes can interact with all previous earthquakes the B-magnitude signals can rarely be reconstructed in the real Ω-sequences. The Ω-sequences are always geometric series of earthquakes based on the B-spectral theorem; however the magnitudes of earthquakes which build these sequences are not always in accordance with the B-magnitude signals defined for those particular sequences. It is therefore convenient to approximate the magnitude of the next (extrapolated) event in the Ω-sequence according to equation: mi + 1 ¼ mi + kΔm; k ¼ 1, 2,3, …
(7)
Here, k is a free parameter. According to Fig. 24.1, we should take: mCE5 ¼ mCE4 + 4Δm; k ¼ 4:
(8)
This equation then defines the maximum possible magnitude mmax that the Ω-sequence can produce. However, the above equation will only be valid when the magnitude mCE4 of the CE4 event in the ISR Ω-sequence is defined by the B1-magnitude signal of that particular Ω-sequence. When the magnitude of the CE4 event is defined by an interaction with some other past earthquake, then it will be more convenient to take: mðTn Þ ¼ m + kΔm; k ¼ 1,2,3, …
(9)
Here, m is the average magnitude of the earthquakes that already belong to the analyzed Ω-sequence, and k is a free parameter. Tn is the time of the extrapolated earthquake. For the DSR Ω-sequence the time Tn is defined by Eq. (22) in Chapter 20: Tn ¼ Tn1 + BΔTn1,n2 :
(10)
Here, ΔTn1, n2 is the time interval between the sequence event numbers n 1 and n 2. For the ISR Ω-sequence, time Tn is defined by Eq. (23) in Chapter 20: 1 Tn ¼ Tn1 + ΔTn1, n2 : B
IV. SEISMIC CHAOS SYNCHRONIZATIONS
(11)
B-MEGASIGNAL: THE PAPUA NEW GUINEA CASE STUDY
347
To predict the maximum possible magnitude mmax the Ω-sequence can produce, we usually take: k ¼ N ðΩÞ or k ¼ N ðΩÞ 1:
(12)
N(Ω) is the number of sequence events in the Ω-sequence. If future earthquakes were predicted based on the Ω-sequences that at the moment of prediction contained 4 sequence events, then the next event would be the CE5 event. This means N(Ω) ¼ 5 and k ¼ 5 (or k ¼ 4). If future earthquakes were predicted based on the Ω-sequences that at the moment of prediction contained 3 sequence events, then the next event would be CE4 event. This means N(Ω) ¼ 4 and k ¼ 4 (or k ¼ 3).
THE MAGNITUDE-SYNCHRONIZATION FUNCTION We define the magnitude function Mi(T, m) of the analyzed Ωi-sequence in the following way: 1 if T ¼ Tn and m 2 ½mmax δm, mmax + δm, Mi ðT, mÞ ¼ 0 if T 6¼ Tn and m62½mmax δm, mmax + δm:
(13)
Mi(T, m) is a 2D function, which defines the presence of the B-magnitude signal at some future time Tn in the interval of the magnitudes m 2 [mmax Δm, mmax + Δm]. It is important to emphasize that the B-magnitude signal is only present at the time Tn. In general, when the value of the Båth’s parameter is defined with the relative accuracy of δB (see Chapter 20), then time Tn will also lie within some interval Tn 2 [Tn ΔT, Tn + ΔT]. On the 2D graph the function Mi(T, m) will have a value equal to 1 when the B-magnitude signal at some time Tn around the magnitude mmax is present, but it will be equal to 0 outside this region. Definition: For a multiplicity of Ω-sequences we define the magnitude synchronization function (or M-synchronization function): N X Mi ðT,mÞ: (14) ℕM ðT,mÞ ¼ i¼1
This function defines the magnitude-synchronization between the B-magnitude signals that belong to N Ω-sequences in the seismic catalog. We will demonstrate the M-synchronization function ℕM(T, m) in the following subchapters.
B-MEGASIGNAL: THE PAPUA NEW GUINEA CASE STUDY During the Nepal (2015) experiment described in the previous chapter, several large earthquakes occurred in the region of Papua New Guinea. Fig. 24.2 illustrates the USGS seismic catalog that was available on 08/05/2015, at the time of our Nepal (2015) experiment. This figure illustrates the seismic catalog between 02/04/2015 and 08/05/2015. The lower cutoff magnitude was MC ¼ 4. Most earthquakes occurred in the New Britain Trench, while some also occurred near the Bougaiville Island. These two regions represented the vertices of the Ω-cell active during the selected time period. In the test illustrated in Fig. 24.3, we used the series of earthquakes between 02/04/2015 and 23/05/2015 as the system SM2 M1[T1, T2] to predict the M-synchronization in the future. In Fig. 24.3, this system is illustrated with yellow columns. Later earthquakes that were not used to calculate the M-synchronization function ℕM(T, m) are marked with red columns. Fig. 24.3 also illustrates the M-synchronization function ℕM(T, m) by using a full-color spectrum. The M-synchronization function was calculated for Ω-sequences containing three sequence events on 22/04/2015. The fourth sequence event was extrapolated to the future, so the total number of sequence events in the analyzed Ω-sequences was N(Ω) ¼ 4. Therefore the parameter k in Eq. (12) was k ¼ 4. In Fig. 24.3 the blue color represents the total absence of any B-magnitude signal. The red color, on the other hand, represents the maximum density of B-magnitude signals. In other words the red color represents the maximum value of the M-synchronization function ℕM(T, m). Note that the M-synchronization is patchy due to the fact that it also contains information on the T-synchronizations. This is very important.
IV. SEISMIC CHAOS SYNCHRONIZATIONS
348 24. M-SYNCHRONIZATIONS: THE B-MEGASIGNAL AND LARGE EARTHQUAKES
IV. SEISMIC CHAOS SYNCHRONIZATIONS
FIG. 24.2
(A) Seismic catalog (USGS) of the Papua New Guinea region between 02/04/2015 and 12/05/2015 for the lower cutoff magnitude 4. (B) Epicenters (blue circles) of the earthquakes illustrated on (A). The radius of the circles represents the relative amount of energy released by these earthquakes.
349
THE SOUTHERN CALIFORNIA CASE STUDY
FIG. 24.3
M-synchronization for the earthquakes in the Papua New Guinea region between 02/04/2015 and 26/04/2015 (marked with yellow columns) above the lower cutoff magnitude 4. Blue color represents the absence of the B-magnitude signals. The M-synchronization between the B-magnitude signals is illustrated with light blue to red colors. At the time of the earthquakes marked with red columns the M-synchronization increased in magnitude. The B-megasignal is clearly visible, defining the magnitude of the largest earthquakes in the region.
Results and Interpretation: The B-Megasignal Results illustrated on Fig. 24.3 show that M-synchronization after 23/05/2015 occurred at higher and higher magnitudes (red and yellow colors) along the hypothetical line (marked with a dashed line) that was increasing with time. We will define this as the B-megasignal. This describes the time development of the M-synchronization. From Fig. 24.3, it is obvious that the B-megasignal defined the magnitude of the large earthquakes that occurred after 30/04/2015.
THE SOUTHERN CALIFORNIA CASE STUDY A similar test as described in the previous subchapter was also performed in the Southern California region, which is important for the fact that the area is highly populated and seismically highly active. Fig. 24.4 illustrates the SCEDC seismic catalog between 19/05/1990 and 31/08/2015 for the lower cutoff magnitude MC ¼ 4.5. Several large earthquakes hit the region in this time period. These large earthquakes were constrained to the San Andreas Fault zone between the Los Angeles and Palm Springs regions. In the test illustrated in Fig. 24.5, we used the series of earthquakes between 19/05/1990 and 01/01/1999 as the system SM2 M1[T1, T2] to predict the M-synchronization in the future. In Fig. 24.5, this system is illustrated by dark red columns. Later earthquakes that were not used to calculate the M-synchronization function ℕM(T, m) are marked by red columns. Fig. 24.5 also illustrates the M-synchronization function ℕM(T, m) by using a full-color spectrum. The M-synchronization function was calculated for Ω-sequences containing three sequence events within the system SM2 M1[T1, T2]. The fourth sequence event was extrapolated to the future, so the total number of sequence events in the analyzed Ω-sequences was N(Ω) ¼ 4. Therefore the parameter k in Eq. (12) was k ¼ 4. In Fig. 24.5 blue marks the total absence of any B-magnitude signal. Red represents the maximum density of B-magnitude signals or the maximum value of the M-synchronization function ℕM(T, m). Fig. 24.5 also illustrates the T-synchronizations (black and gray).
IV. SEISMIC CHAOS SYNCHRONIZATIONS
350 24. M-SYNCHRONIZATIONS: THE B-MEGASIGNAL AND LARGE EARTHQUAKES
IV. SEISMIC CHAOS SYNCHRONIZATIONS
FIG. 24.4 (A) Seismic catalog (SCEDC) of the Southern California region between 19/05/1990 and 31/08/2004, for the lower cutoff magnitude 4.5. (B) Epicenters (brown circles) of the earthquakes illustrated on (A). The radius of the circles represents the relative amount of energy released by these earthquakes.
REFERENCES
351 FIG. 24.5 M-synchronization in the Southern California region calculated for the earthquakes marked in dark red that occurred between 19/05/1990 and 01/01/ 1999 above the lower magnitude cutoff 4.5 (see also Fig. 24.4). Blue represents the absence of the B-magnitude signals. The M-synchronization between the B-magnitude signals is illustrated with light blue to red colors. Several B-megasignals are visible. Three of them are marked with black dashed lines. The yellow dashed line represents the minimum magnitude value of the B-megasignals at some chosen time. This line defines the magnitude of the second- and third-order events according to the minimum magnitude rule. This figure also illustrates the T-synchronizations for δB ¼ 5% (gray) and δB ¼ 3% (black).
Results and Interpretation: The Minimum Magnitude Rule The results in Fig. 24.5 indicate the presence of several B-megasignals. The large first-order earthquake on 16/10/ 1999 north of the Palm Springs region was clearly defined by one of this B-megasignals. On the other hand the secondand third-order earthquakes were defined by the minimum value of the B-megasignals. Their magnitudes are always below the region, in which the M-synchronizations occur. In Fig. 24.5 the lower boundary of this region is marked with a dashed yellow curve. From this test, we see that the dashed yellow curve represents the envelope for the magnitudes of the second- and third-order earthquakes. This empirical rule will be termed here the minimum magnitude rule.
References Arenas, A., Díaz-Guilera, A., Kurths, J., Moreno, Y., Zhou, C., 2008. Synchronization in complex networks. Phys. Rep. 469 (3), 93–153. Boccaletti, S., Kurths, J., Osipov, G., Valladaras, D.L., Zhou, C.S., 2002. The synchronization of chaotic systems. Phys. Rep. 366, 1–101. Carroll, T.L., Pecora, L.M., 1991. Synchronizing chaotic circuits. IEEE Trans. CAS 38, 453. Carroll, T.L., Pecora, L.M., 1993. Cascading synchronized chaotic systems. Physica D 67, 126–140. Manzano, G., Galve, F., Giorgi, G.L., Hernández-Garcia, E., Zambrini, R., 2013. Synchronization, quantum correlations and entanglement in oscillator networks. Sci. Rep. 3, 1439. https://doi.org/10.1038/srep01439. Pecora, L.M., Carroll, T.L., 1990. Synchronization in chaotic systems. Phys. Rev. Lett. 64, 821. Pecora, L.M., Carroll, T.L., 1991. Driving systems with chaotic signals. Phys. Rev. A 44, 2374.
IV. SEISMIC CHAOS SYNCHRONIZATIONS