CHARACTERIZATION OF HISTORICAL MEGATHRUST EARTHQUAKE RUPTURES IN SOUTH CENTRAL CHILE USING LOGIC TREE ANALYSIS

Characterizing the spatial distribution of ruptures from historical and recent earthquakes is key to understanding the seismic cycle of large earthquakes in subduction zones, and thus to assessing the potential risks associated with future earthquakes. The southern portion of Central Chile (35 ◦ S-38 ◦ S) has been continuously aﬀected by large earthquakes, such as the 2010 Maule (Mw 8.8) and the 1835 earthquakes witnessed by Robert Fitzroy (HMS Beagle captain). Our goal is to identify the rupture pattern and tsunami propagation of the 1570, 1657, 1751, 1835, and 2010 mega-earthquakes, events that overlapped in central Chile, by compiling historical records and applying robust statistical tools. We used an adaptation of a logic tree methodology to generate random sources of slip distribution for each event, constrained by tsunami and deformation data. We ﬁnd that the three events studied have diﬀerent slip peaks. The 1751 earthquake has the largest slip with a maximum patch of ∼ 26 m, while the 2010 and 1835 earthquakes reach slips of ∼ 16 m and ∼ 10 m, respectively. Our results show that a part of the segment between 36 ◦ S and 37 ◦ S was consistently aﬀected by large earthquakes, but with diﬀerent slip and depth. The northern part of the segment accumulated energy for at least 300 years and was released by the 2010 earthquake. This work provides important information for identifying rupture patterns between historical and recent earthquakes, and highlights the importance of extending the time scale of earthquake slip distribution analyses to multiple cycles to describe both earthquake characteristics and their spatial relationship, and thus gain a better understanding of seismic hazard.


Generation
The seismic cycle in subduction zones is a period of time that can last from decades to hundreds of years (Ruiz et al. 2018), where there is an accumulation of energy commonly caused by a locking at the interface of tectonic plates, which in turn depends on structural and frictional factors. This locking is known as plate coupling (Vigny et al. 2009;Moreno et al. 2010) and controls the characteristics of the deformation field in the area, causing uplift or subsidence movements on the coast, which occur mainly when there is an abrupt release of energy during an earthquake.
The seismic cycle is described by the different stages that compose it (see Table   1.2.1) and the changes in the tectonic configuration or deformation field where the earthquake occurs.
The great variability of earthquakes denote a hallmark of each seismic cycle; depending not only on geometric factors (such as the depth and size of the event) but also on the structural and frictional control of the zone in which they occur (Shi et al., 2020). In addition, the rupture length of an earthquake determines an important factor in variability, depending on whether it ruptures the entire plate

Period Description Time
Interseismic Pre-earthquake loading period corresponds to the accumulation of stresses at the plate interface as a result of plate locking or coupling and subduction of the oceanic plate beneath the continental plate.
Decades to hundreds of years.

Coseismic
Period in which the release of energy from the seismic event occurs.

Seconds to minutes.
Post seismic This period is characterized by the gradual release of energy that was not released during the co-seismic.
Decades of years. When an earthquake occurs, the seafloor deforms greatly above the fault source.

Propagation
The shallow water equations describe a number of physical characteristics,

Where
• η : Vertical displacement of the water surface on an equipotential surface.
• U, V: Horizontal and vertical component of the water surface.
The combination of these equations gives us the expression: Deriving the expression for the one-dimensional case: The above equation has the form of the wave equation, therefore, we can propose the following solution: The equation 1.3.6 will be a solution if and only if: Using surface wave theory, the dispersion relation will be, in effect: The relationship 1.3.8 shows that the waves generated by a tsunami must be considerably longer than the ocean depth for the consideration of shallow waves to be valid. In the case of the Pacific Ocean, the average depth is about 4 km; while tsunami wavelengths are about 300-400 km, so the above equations are valid

Logic tree approach
The logic tree approach corresponds to a machine learning algorithm used to solve regression and classification problems. This technique is used to predict the value of a target variable by combining parameters. Annaka et al. 2007 used this methodology for the development of a probabilistic analysis of tsunami risk in the coasts of Japan, thus obtaining a long-term projection that allows knowing the associated risk, for example, through the recurrence periods of an event. In this work, an adaptation of such methodology is proposed, with the difference that it will not be used to project tsunami risks in the future but will be used to know the probable characteristics of the source of past earthquakes (and respective tsunamis), characterized through their magnitude, slip distribution and geographical limits.  As shown in 1.4.1, the tree is composed of nodes and branches. The nodes represent the different parameters to be used: earthquake magnitude, northern limit, southern limit, complexity and aspect ratio. The last two being a measure of the amount of slip variation between contiguous faults and the ratio between the large and width of the fault. The branches represent the alternatives of possible values that each node can take, therefore each branch is assigned a probability of occurrence. Once the different sources have been obtained, the models must be restricted and selected according to the constraints addressed in the next subsection. An example of these are the observations of tsunamis for events prior to the 20th century, which despite being scarce, can be found in historical records, e.g. the letters kept by the General Archive of the Indies, Spain, which describe through accounts some effects and consequences of earthquakes and tsunamis in the 18th century.

General objective
Characterize historical tsunami sources in Central Chile segment (34.0 • S -38.2 • S), by using the logic tree approach, and historical constraining models from observations found in the literature (e.g. magnitude, epicenter, fault parameters: length, width, depth, slip) and paleotsunami stratigraphic records (e.g. deformation, wave height, inundation) and historical information.

Specific objectives
1. Bibliographic compilation from different historical events in the segment and creation of the logic tree branches.
2. Use the logic tree approach to characterize historical tsunamis and obtain the source models parameters associated with each event.
3. Simulate tsunamis generated by different source models for each historical earthquake to obtain tsunami heights and compare with tsunami heights observations. 4. Identify similarities or variations between events to determine possible     Year Magnitude (Mw) Northern limit (

Model analysis
We perform the slip distribution analysis following the methodology of Cifuentes-

Historical information for data restrictions
The compilation of historical information provides observations and useful data to use as input in the deformation and tsunami wave height restrictions. Qualitative information is considered as categorical data that can indicate whether a location experienced uplift or subsidence deformation, or if a tsunami wave reached an area. Numerical data is also useful to constrain amounts of deformation or wave amplitude. Numerous efforts have been made to characterize past events, but many of them rely primarily on the qualitative description of damages and effects.
The most widely utilized documents to accomplish this task can be found in the Information about vertical deformation is not available; given the antiquity of this event, it is the least known due to the small amount of available data.

1657 earthquake and tsunami
The reports the tsunami. Its size was comparable to that of 1570 and of 1835, but the earthquake itself seems to have been slightly less severe (Lomnitz, 1970).

1570 and 1657 earthquakes and tsunamis
Despite efforts to characterize the earthquakes and tsunamis of 1570 and 1657, it was not possible to do so due to the lack of information. While we were able to gather some data to establish the logic tree parameters for each event, the absence of numerical and categorical vertical deformation observations to apply as constraints, resulted in the generation of thousands of models, indicating consistently homogeneous results. Therefore, all parameters were equally likely, and the model failed to converge.

1751 earthquake and tsunami
After evaluating 25200 candidate models, we identified 19 slip distribution models that were most likely to characterize the 1751 event. As no numerical data representing the tsunami wave height are available, no such constraint was performed, but the tsunami propagation was modeled from the final deformation model obtained for the event and compared with the available categorical data.
The predicted deformation field succeeds in reproducing categorical observations.

2010 earthquake and tsunami
We obtained a total of 8640 slip distribution models for the 2010 earthquake.
However, only 10 of these models were found to be possible after applying deformation and tsunami constraints to accurately characterize the event. reproducing deformation observations (Fig. 3.4.1). We also found that predicted  it is necessary to recognize the places where the slip patches are repeated, as well as the places that do not present significant slips and the behavior of these zones in the coupling models.
We consider that the 1835 earthquake initiated the seismic gap closed by the 2010 event, but we do not interpret it as a predecessor earthquake. We anticipate that a predecessor earthquake will not only precede the current one in time, but will also share similar characteristics with the subsequent event, such as rupture length, position of asperities, depth, and northern and southern limits.
The 1835 earthquake does not meet these conditions, as it has a smaller rupture length and a smaller and different location of the main slip patch. The 2010 earthquake repeats the slip pattern between 36 • and 37 • S. Although that is not its highest magnitude patch, it is still one of the areas with the greatest slip within the entire rupture length of the segment. aspects. By understanding the geophysical processes, including the spatio-temporal evolution of the seismic cycle, we can enhance the resilience of communities exposed to this type of natural hazard. Therefore, through this work, we can identify rupture patterns and areas that did not experience energy release as a result of the earthquake.
The methodology applied here is well-suited for numerical testing and produces model results that comply with both numerical and categorical data observations. However, in cases where data availability is limited, it tends to underestimate results. The accuracy of a slip distribution model depends on the quality and quantity of the data used to constrain the model, as well as the complexity of the geological and seismological conditions of the studied area. Nevertheless, the methodology can provide valuable information about earthquakes themselves, as well as similarities or differences with subsequent events. Regarding the 1835 earthquake, our slip distribution model was able to reproduce the tsunami propagation and deformation data, though the deformation field did not indicate subsidence in Constitucion, as categorical data suggested. We consider this event to be a special case where the southern rupture extension may have been greater than previously proposed. Future experiments may benefit from a wider range of values for southern limits. Like the 1751 earthquake, both events lacked rupture extent in the northern portion of the segment, which was eventually broken by the 2010 Maule earthquake. Our findings suggest that the northern part of the segment had been accumulating energy for at least 300 years.
Nonetheless, it is crucial to have sufficient data to reproduce the slip distribution models of precursor earthquakes, such as those in 1570 and 1657.
The primary constraint we encountered was the limited availability of data on historical earthquakes in the South Central segment, particularly for the years 1570 and 1657. Available resources provide valuable information on the 1657 earthquake (Stewart, 2021). However, they were not enough to apply the logic tree method.
Despite generating random sources, we were unable to obtain meaningful results due to insufficient deformation data, which restricted our ability to refine the computed fields. To improve this work, it is necessary to increase the amount of historical events and acquire more data to refine the models. Additionally, to gain a better understanding of the seismic cycle, we recommend incorporating other deformation and stress loading processes that also contribute to the cycle.