Recurrence time analysis of the 1883 (Mw 8.6 ~ 9.1) paleo megathrust earthquake in West Bengkulu, Sumatra, Indonesia, using plate kinematic model for supporting mitigation program

The great paleo-megathrust earthquake occurred in west Bengkulu-Sumatra on November 25, 1833 and it was estimated to have the magnitude of about Mw 8.6–9.1. This earthquake triggered a tsunami that has 3-4 m height in the Bengkulu coastal and its vicinity. The fundamental earthquake parameter was not known exactly, thus it was difficult to conduct the analysis. We tried to analyze it using scaling law relations of earthquake parameter and magnitude scenarios to estimate the recurrence time based on the plate kinematic model to support the long-term mitigation program. The recurrence time was estimated at 179 years, 250 years, and 314 years if the earthquake is assumed to have a dip-slip mechanism, plate slip rate about 7 cm/year, and magnitude scenarios of 8.6, 8.8, and 9.1 respectively. Then, the recurrence time was estimated at 314 years, 438 years, and 550 years respectively if the earthquake is assumed to have a plate slip rate about 4 cm/year. The other estimation was used the strain rate value of about 1.19 x 10 -14 /s for Sumatra region and the rigidity modulus of 3.10 11 dyne/cm 2 . We found the recurrence estimation of about 161 years (Mw 8.6), 222 years (Mw 8.8), and 220 years (Mw 9.1) for the dip-slip model. Meanwhile, if this study used all rupture models, the recurrence time of the Bengkulu paleo megathrust earthquake would be about 370 years (Mw 8.6), 558 years (Mw 8.8), and 553 years (Mw 9.1).


Introduction
The great earthquake occurred in 1833 in west Bengkulu-Sumatra, Indonesia, and this earthquake triggered a tsunami to stuck the coastal area and its vicinity (Pararas, 2000). A seismograph did not exist yet in this era and the earthquake was only noted in the form of the macroseismic data. Therefore, the fundamental earthquake parameters were not known exactly, such as the location of the epicenter, focal depth, magnitude, or moment magnitude (Mw).
There were only simple reports of the earthquake impact based on the macroseismic data, including the tsunami impact data. Some scientists estimated that Bengkulu paleo megathrust 1833 has a varying magnitude of Mw 8.8 (Prawirodirdjo et al. 2010), Mw 9.0 (Subarya et al. 2006), Mw 8 ¾ (Newcomb and McCann 1987) and Mw 8.8-9.1 (Zachariasen et al. 1999). The magnitude was still not clear exactly, so several magnitude moment scenarios were used to analyze the recurrence time of the Bengkulu earthquake 1833. The moment magnitude is a basic parameter to calculate a seismic moment (Mo), rupture length (L), area (S), and static stress drop (Δσ) using scaling law earthquake relations. After considering the magnitude uncertainty in this study, we used the magnitude scenarios of 8.6 ≤ Mw ≤ 9.1 to calculate its recurrence time. The epicenter location could not be determined exactly (see Figure 1)  To minimize the uncertainty of Mo value, we presented earthquakes which has the same (or almost) magnitude, such as Chili Bio-Bio earthquake Mw 8.8, Japan Tohoku earthquake Mw 9.1, and outer rise earthquake in western Aceh Mw 8.6. Meanwhile, L and W of fault were used to do the scaling law relations of earthquake formula proposed by (Wells and Coppersmith 1994). This is one of the reconstructions of the paleo earthquake which limits the earthquake parameters data to estimate the recurrence time (Tr) using a plate kinematic model. In this paper, we determined the recurrence time estimation using the slip rate of the Bengkulu arc segment of about 5-7 cm/year (Bock et al. 2003) and the strain rate value for Sumatra region of about 1.19 x 10 -14 /s (Murjaya, 2011).
The recurrence is dependent upon stress drops and in their study, the static stress drop is calculated using seismic moment measured from seismogram and estimated rupture area ((Beeler et al. 2001) (Kanamori and Allen 1986). The recurrence time of earthquakes within one segment is controlled by the tectonic load as a slip rate for a long time, stress accumulation, rheology rocks, and release mechanism on the fault plane. The recurrence time or return period of the tectonic earthquake will last longer if it has a larger static stress drop and the return period of the earthquake will paradoxically last shorter in time if it has a smaller static stress drop ( (Kanamori and Allen 1986). This parameter is important to support earthquake mitigation planning in the future.
The tectonic of the Sumatra region in the western part of Indonesian tectonic is characterized by a complex tectonic region. The western part of Sumatra is a convergence or joint land between Eurasian and Australian plates characterized by oblique subduction. Also, there is a right-lateral large Sumatran fault along almost 1600 km from north to the south (southeast).
Mentawai islands (see Figure1) is one of the effects of oblique convergence plates between Eurasian and Australian plates (Hall, 1998) (Murjaya, 2011). According to Handayani (1993) devided three earthquake distribution zones in Sumatra and there are earthquake zones in its vicinity along the trench region (subduction zone), Mentawai fault, and Large Sumatran Fault (LSF).
The slip rate value of convergence is estimated between 5 -7 cm/year (Bock et al. 2003) and 4 cm/year (Irsyam et al. 2017). According to the large earthquakes historical data, it has occurred in Sumatra and its vicinity, such as earthquakes that occurred in 1797, 1833, 1861, 2000, 2004, 2005, and 2007(Newcomb and McCann 1987. The three earthquakes event of 1797, 1833, and 1861 occurred before the seismograph existed.

Data and Methods
The magnitude data of the Bengkulu paleo earthquake occurred in 1833 was taken from the secondary data which has been published with a magnitude scenario (Mw 8.6 -9.1) and it has been checked in the Indonesia Agency for Meteorology, Climatology, and Geophysics (BMKG) catalog. Seismic Moment (Mo), Length of fault (L), and rupture area (A) was determined using scaling law relations. We used the slip rate of Bengkulu arc segment of about 7 cm/year (Bock et al. 2003), the minimum is about 4 cm/year (Irsyam et al. 2017), the strain rate value for Sumatra region 1.19 x 10 -14 /s (Murjaya, 2011).

a. Fault Length and Rupture Area
The Bengkulu paleo megathrust earthquake occurred in 1833 had limited information only about the earthquake impacts. Therefore, we did not have the aftershocks distribution, geodetic data to describe the fault length and/or the rupture area. We assumed that the focal mechanism of the earthquake is dip fault and/or all rupture models, then the L and A value is calculated using (Wells and Coppersmith 1994) formula in equation (1) and (2).  (1) and (2).

b. Static Stress Drop Theory
Several tectonic energies will release during a co-seismic event after the critical stage of rock is passed over. This energy is accumulated along the interseimic period since the last earthquake event has occurred. Figure 2 shows the illustration of the built-up stress and released stress. It is begun from the initial stress (σo), the build-up stress along interseismic period until it achieves the critical stress (σ1) stage. After the critical stress stage passes over, it will release several stresses called a stress drop. Furthermore, to describe the stress release effects, we used one type only, i.e. a static stress drop. Then, we used the (Kanamori and Anderson 1975) formula as follows: µ is rigidity (SI), ̅ ̅ is strain change, and C is a non-dimensional shape factor. To determine the shape factor (C) and L, we referred to shallow infinite transverse shear (dip-slip) faults. (Kanamori and Anderson 1975) , where λ is the Lame constant, then a seismic moment is: Mo is a static seismic moment (Nm -1 ) and S is a rupture area (m 2 ). Based on the substitute equation (3) and (4), then the static stress drop formula for strike-slip fault is: Equation (5) is similar to (Beeler et al. 2001) which reveals that static stress drop is calculated using seismic moment measured from seismogram and estimated rupture area, while the recurrence interval is estimated from historic records.

c. Seismic Energy and Moment Magnitude Relation
The relation between the surface wave magnitude (Ms) and the total energy of seismic waves (Es) has been studied by (Gutenberg and Richter 1955) which is revealed by Kanamori and Anderson (1975). They obtained the relation as: Log Es = 1.5 Ms + 11.8 Then, we used the conversion Mw to Ms with the formula as: The equation (7) is used for magnitude interval 6.2 ≤ Ms ≤8.7 with R 2 = 0.8183.

Results
Having explained above, the Bengkulu earthquake that occurred in 1833 has limited macro-seismic data, magnitude estimation (Mw 8.6 -9.1), and it has been published in some journals. We made magnitude scenarios to calculate some fundamental earthquake parameters using scaling law relations. We then referred to the earthquake which has identical magnitude with the magnitude scenario such as Mo of Nias-Sumatra earthquake for the magnitude  Table 1 shows the relations between earthquake parameters, such as Mw, Mo, L, and A. These parameters were used to estimate of the static stress drop (Δσ) value after being combined with scenarios of slip models (dip slip-DS) and all rupture-AR). The slip model scenario is needed to select a formula to calculate L and A based on (Wells and Coppersmith 1994). The Δσ estimation was verified with plotting at Log Mo-Log S diagram by (Kanamori and Anderson 1975) in Figure 3 and there are still consistent for interplate earthquake.

Recurrence Time Estimation
We used strain rate, slip rate data, and substitute the above scenarios to estimate the recurrence time. (Frederick C. Davison and Scholz 1985) and (Kanamori and Anderson 1975) calculated the recurrence time of the earthquake by dividing the moment of an earthquake by source time function or moment rate accumulation. This method needs the slip rate data of the fault or velocity rate of the plate.  Bock et al. (2003) revealed that the movement of the Australian and Eurasian plate relatively for arc Bengkulu segment with the velocity rate is about 5-7 cm/year, while (Irsyam et al. 2017) estimated that the movement is 4 cm/year only. Another method is passthrough of the physical basic approximation where we estimated the theoretical stress accumulation per unit time. This method contains the strain rate parameter value for the Sumatra region (Murjaya, 2011). Table   2 presents the estimation result of the return period or recurrence time of the Bengkulu earthquake that occurred in 1833 using the passthrough approach of the strain rate and subduction slip rate values.

Potential Shaking Impact of Earthquake Magnitude Scenario
One of the secondary effects of the earthquake is a shaking impact on infrastructures and the surrounding area. Figure 4a shows the shake map of the Bengkulu earthquake using scenario Mw 8.6, focus depth 30 km, and it does not discriminate the type of slip. If the earthquake occurs with a magnitude of more than Mw 8.6, the impact has an intensity of more than Mw 8.6. Also, Figure 4b and 4c shows the simulation of Peak Ground Acceleration (PGA) and Peak Ground Velocity (PGV) using magnitude scenario Mw 8.6.

Discussion
Bengkulu paleo megathrust earthquake occurred in 1833 in western Bengkulu coastal zone, the location between subduction zone and Mentawai strait in Bengkulu area (Figure1). We tried to determine the recurrence time for the Bengkulu paleo megathrust earthquake that occurred in 1833 by choosing the magnitude moment scenarios Mw 8.6 -9.1 and assuming that the faults are dip-slip and all rupture models. We considered that the strain energy was accumulated along the inter-seismic period for several years until a hundred years without breaking this area. The strain energy or stress accumulation will release during co-seismic events after the critical stage of rock passes over or called a stress drop. Table 1 shows the relations among Mw, Mo, L, A, and Δσ by scaling law relations of the earthquake. Δσ is an important variable in the dynamic model as the basic parameter to estimate the recurrence time of an earthquake. (Dębski 2018) explains that to describe the stress release effects, seismology uses three basic parameters namely the static stress drop, the Coulomb stress transfer, and the dynamic stress drop.
Furthermore, the Δσ result can be compared to some earthquakes as showed in Figure 3. All results of static stress drop value have the range around 36-125 bars and it is located in the fixed-line. (Kanamori and Anderson 1975) demonstrate that the average stress drop is higher for intraplate than interplate earthquakes and since intraplate events have generally longer repeat times than interplate events. Kanamori (1994) revealed that for a typical sequence along active plate boundaries, Δσ ≈ 30 to 100 bars, and Tr ≈ 300 years, and it is agree with the Figure 3. The other it also presents the largest earthquake at active plate margins that has a relatively short repeat time of 30-200 years (Kanamori and Allen 1986), while some intraplate events have repeat times of thousand years, even if they are relatively close to a plate boundary. Meanwhile Natawidjaja et al. (2006) investigated the growth patterns and U-Th dating of the coral microatolls vertical deformation associated with the 1797 and 1833 earthquakes. They revealed also that paleoseismic evidence indicates that over the past millennium, the island has risen during giant earthquakes or earthquakes couplets about every 230 years. Subarya et al. (2006) revealed that if all slip was released only by the repetition of event like the Sumatra-Andaman earthquake 2004 would occur on average every 230-600 years. If we refer to their research result related of the recurrence time of earthquakes (Kanamori and Allen 1986;Kanamori 1994;Subarya et al. 2006;Natawidjaja et al. 2006), our result for the static stress drop value have a range around 36-125 bars it is still a good result. Also, if it is related with the estimation of the recurrence time of large earthquake in Western Bengkulu 1833 as showed on table 2. Furthermore, we used two approximations to estimate the recurrence time of earthquake. First, we calculated the recurrence time based on the plate kinematic model (Frederick C. Davison and Scholz 1985) by using the slip rate of the convergence plate of Australian and Eurasian plates. The maximum values of the slip rate data about 5-7 cm/year (Bock et al. 2003), and minimum is about 4 cm/year (Irsyam et al. 2017) as well as the Mw scenario for arc Bengkulu segment 8.6-9.1. The maximum value of convergence plate of the Australian-Eurasian is about 7 cm/year has an almost identical with the data by McCaffery (McCaffrey et al. 2000) from the NUVEL-1.
Second, we used the strain rate value in Sumatra (Murjaya, 2011) and assumed an average value for the modulus of rigidity 3.10 11 dyne/cm 2 , and the result is presented in Table 2. If it is assumed that the source has a dip-slip (DS) fault, a slip convergence plate rate value of 4 cm/year (Irsyam et al. 2017)  Although it uses the strain rate approach, it still has an ambiguity like what has been revealed by White (1994) that the calculated strain rates are not constant, but it varies between 10 −18 /s until 10 −15 /s, such as a surface deformation strain rate near the Alpine arc in Switzerland of around 16.3x10 -16 /s (Houlié et al. 2018). In this paper, we used the strain rate for Sumatra area is around 1.19 x 10 -14 /s for λ buckling ~ 300 km and the thick crust (h) in the Western Bengkulu segment based on the distribution of earthquake hypocenter (from the surface to depth) of around 70 km (Murjaya, 2011). Meanwhile, the strain rate in the Indian Ocean is around 2 x 10 -16 /s (Gerbault 2000). Furthermore, the total energy released for all earthquakes based on term 5.3 is a small value if it is compared with the seismic energy based on the magnitude scenarios.

Conclusion
Based on Table 1, the Δσ result is still consistent (Figure 3) to reconstruct the Bengkulu paleo earthquake that occurred in 1833 using magnitude scenarios 8.6-9.  (Bock et al. 2003) and the magnitude scenarios 8.6, 8.8, and 9.1, the recurrence time of the earthquake are 179 years, 250 years, and 314 years respectively. This recurrence time of earthquake with a slip rate of 7 cm/year is faster than the slip rate of 4 cm/year.
When we used the strain rate value for Sumatra earth crust 1.19 x 10 -14 /s and the mechanism source is dip-slip, we found the recurrence time of about 161 years (Mw 8.6), 222 years (Mw 8.8), and 220 years (Mw 9.1) respectively. Meanwhile, we assumed that the mechanism source is all rupture models, the strain rate value for Sumatra earth crust 1.19 x 10 -14 /s, and we found have influenced the performance or presentation of the work described in this manuscript.