# Keyboard Model of Seismic Cycle of Great Earthquakes in Subduction Zones: Simulation Results and Further Generalization

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## Abstract

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## 1. Introduction

## 2. Materials and Methods

#### 2.1. Keyboard Model of Seismic Cycles of Great Subduction-Associated Earthquakes

^{−4}$\frac{1}{yr})$, $\kappa $ = 0.2–0.5 yr is characteristic time required for the contact layer defects to be healed, and ${T}_{ps}$ is the time passed since the last earthquake in a given block.

#### 2.2. GNSS-Measured Motions and Stress-Deformation Cycle in Subduction Regions

## 3. Results

_{j}($j=1,\dots ,L)$ fall within the frontal block, while those with $j=L+1,\dots ,N$ correspond to the rear block. In discrete formulation, the problem takes form.

^{19}Pa·s. Lower viscosity (less than 10

^{18}Pa·s) causes numerical instability of the solution. In the case of higher viscosity (of the order of 10

^{20}Pa·s), the slow (at around 1 cm/year) oceanward motion persists all over the cycle, and the rear block does not experience any continent-ward motion at all. The obtained estimates for model viscous parameters are consistent with the earlier results lying in the range 5 × 10

^{17}–5 × 10

^{19}Pa·s [35,36,37,38,39,40].

## 4. Discussion

## 5. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 1.**(

**a**) Keyboard model geometry; (

**b**) seismic deformation cycle phases. A is the fixed undeformable continental margin; B is the seismogenic block; C is the block-margin interface; D is the subducting plate; (1) 2-D geometry for the no deformation state; (2) stress accumulation (preseismic) phase; (3) coseismic phase (destruction of the mechanical coupling in the contact layer); (4) postseismic (aftershock) phase with continued oceanward motion of the blocks; (5) spring imitating the elastic interaction between blocks and margin.

**Figure 2.**Flowchart showing calculation scheme of the original keyboard model. (

**a**) Model input parameters; (

**b**) block thickness; (

**c**) 1-D elastic model; (

**d**) time-dependent viscosity of the contact layer; (

**e**) stresses acting on lateral sides of the block; (

**f**) stress acting at the block’s bottom; (

**g**) stress equilibrium equation; (

**h**) equation governing displacement evolution; (

**i**) boundary conditions applied at the frontal (x = 0) and rear (x = l) edges of the block; (

**j**) calculating incremental displacements; (

**k**) converting displacements into elastic energy accumulated by the block; (

**l**) condition to identify the instant of the earthquake; (

**m**) stress dumping conditions; (

**n**) output dataset.

**Figure 3.**Displacement variations W(x,t), simulated with an original single-segment model for an outer edge and midpoint of blocks. Panel (

**a**) shows an overall cycle pattern for 2 different blocks (assuming 7-block structure), while panel (

**b**) provides a detailed picture showing the postseismic oceanward motion, observed at the outer region of the block.

**Figure 4.**GNSS-observed motions in Kuril subduction zone after large Simushir earthquakes modified from [16]. Upper panel shows velocity vectors measured by the Kuril GNSS stations during time intervals from May 2007 to May 2011 (

**a**) and from May 2011 to May 2015 (

**b**). (1) Sources of strongest earthquakes with M ≥ 8; (2) main shocks of earthquakes in 2006 and 2007; (3) subduction velocity equal to 80 mm/yr. The velocities indicated here are relative to the North American lithospheric plate. Lower panel shows northern (

**c**) and eastern (

**d**) components of the displacements of the Kuril network observation points. Dashed vertical lines denote the moments: (1) Simushir earthquake in 2006; (2) Simushir earthquake in 2007; (3) Eruption of Sarychev peak in 2009; (4) deep earthquake of the Sea of Okhotsk in 2013.

**Figure 5.**GNSS-observed motions in Chile subduction zone before (

**a**) and after (

**b**,

**d**) large 2010 Maule earthquake modified from [17]. Map plots show the velocity vectors estimated from the Chilean and Argentinean regional satellite geodesy networks data for the time intervals of 27 February 2009–26 February 2010 (

**a**); 28 February 2010–27 February 2011 (

**b**); 28 February 2013–27 February 2014 (

**c**); and 28 February 2016–27 February 2017 (

**d**). Lower panel shows eastward displacement timeseries measured at CONT (

**e**) and PCLM (

**f**) GNSS stations. Displacements are relative to the South American lithospheric plate.

**Figure 6.**Surface motions in Japan before (

**a**), during (

**b**) and after (

**c**–

**f**) 2011 Tohoku earthquake estimated from GEONET GNSS data, modified from [18]. Red arrows show velocity distributions for time intervals as following: 11 March 2010–10 March 2011 (

**a**); 11 March 2011 (

**b**); 12 March 2011–10 March 2012 (

**c**); 11 March 2012–10 March 2013 (

**d**); 11 March 2013–10 March 2014 (

**e**), and 11 March 2014–10 March 2015 (

**f**). Velocities are relative to the South American lithospheric plate.

**Figure 7.**Timeseries for the northern (

**a**) and eastern (

**b**) displacement components recorded at GEONET GNSS station 0033. Dashed vertical line indicates 2011 Tohoku earthquake.

**Figure 8.**(

**a**) Three-dimensional geometry of the two-segment keyboard-block model, (

**b**) seismic deformation cycle phases, and (

**c**) vertical plane section for the particular block along x-direction. A is the fixed undeformable continental margin; B is the rear-segment block; C is the frontal-segment block; D is the subducting plate; E is the crustal asthenosphere. (1) Two-dimensional geometry for the no deformation state; (2) stress accumulation (preseismic) phase; (3) coseismic phase (destruction of the crust); (4) postseismic (aftershock) phase with continued oceanward motion of the blocks.

**Figure 9.**Flowchart showing calculation scheme of the generalized (two-element) keyboard model. (

**a**) Schematic representation of rear (left) and frontal (right) modeling domains (blocks); (

**b**) model input parameters for the rear (left) and frontal (right) blocks; (

**c**) block thickness; (

**d**) 1-D elastic model; (

**e**) viscosity as function of x and t: constant viscosity beneath the rear block (left), time-dependent viscosity of the contact layer beneath the frontal block (right); (

**f**) stresses acting on lateral sides of the blocks; (

**g**) stress acting at the blocks’ bottom; (

**h**) stress equilibrium equation; (

**i**) equation governing displacement evolution; (

**j**) boundary conditions applied at the frontal edge of the frontal block (x = 0), contact between the frontal and rear blocks (x = l), and rear (x = r) edge of the rear block; (

**k**) stress dumping conditions; (

**l**) output dataset.

**Figure 10.**Displacement variations W(x,t), simulated with a two-segment model for a series of points within the rear-segment blocks (x = 150, 160, 175, and 225 km), calculated for a typical set of model parameters (see Table 1).

**Figure 11.**Model of the mechanical coupling distribution in the source zone of the 2006 Simushir earthquake. Black arrow displays magnitude and direction of the plate convergence vector [41].

**Figure 12.**Simulation of several seismic deformation cycles in Kuril subduction zone for generalized keyboard model. Left column of images shows the displacements of two boundary and one middle points of frontal block, right column shows the displacements of the same points of rear block. Stochastic condition for amount of relaxed and accumulated elastic energy was applied. Initial large displacement is caused by stabilization of the numerical scheme and is ignored in further analysis.

**Figure 13.**East-west displacement timeseries measured by Kuril GNSS network following the 2006–2007 Simushir earthquakes (black curves) and modeled displacement data (light blue line) calculated within the interior part of the rear block using the two-segment model. Positive values correspond to eastward (i.e., oceanward) motion.

**Table 1.**Main characteristics of the structure and seismic cycle in Kuril-Kamchatka, Japan, and Chile subduction zones, according to our studies and available data.

Subduction Zone | Block Width/Length, km | Block Length (Frontal/Rear Segment), km | Cycle Period, yrs | Horizontal Coseismic/Aftershock Displacement at Rear Segment, cm |
---|---|---|---|---|

Kuril-Kamchatka | 30–100 [15] | 100/100 [15] | 140 ± 60 [23] | 50/50 [19] |

Japan | 100–150 [24,25] | 200/300 [24,25] | 100–1000 [21,26] | 50–200/10–50 [18] |

Chile | 100–200 [27,28,29,30] | 100/300 [27,28,29,30] | 63–176 [31] | 5–50/50–80 [17] |

**Table 2.**Typical values for parameters of viscous elements of the model according to [20].

Contact Layer Thickness, $\mathit{h}$ | Stationary Viscosity of the Contact Layer, ${\mathit{\eta}}_{\mathit{f}}^{}$ | Postseismic Viscosity of the Contact Layer, ${\mathit{\eta}}_{\mathit{a}}^{}$ | Interblock Fault Thickness, ${\mathit{h}}_{\mathit{g}}$ | Interblock Viscosity, $\mathit{\mu}$ |
---|---|---|---|---|

~0.5 km | ~10^{19} Pa·s | ~10^{18} Pa·s | ~1 km | ~4 × 10^{18} Pa·s |

**Table 3.**Main characteristics of the seismic deformation cycle in the Kuril-Kamchatka subduction zone.

Geological+ Geophysical Data [32] | Continuous Medium Model [19] | Mechanical Keyboard Model | Generalized Mechanical Keyboard Model | |
---|---|---|---|---|

Seismic cycle duration | 140$\pm $60 years | 159 years | 60–267 years (av. 136 years) | 40–338 years (av. 162 years) |

Postseismic stage duration | Up to 35–50 years | 0.5 years (afterslip) up to 10 years (viscoe-lastic relaxation) | 2–7 years | 1–5 years |

Coseismic horizontal surface displacements | ─ | 55 × 10^{−2} m (GNSS observations) | 10 × 10^{−2}–1.8 m (depends on distance from trench and magnitude) |

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**MDPI and ACS Style**

Lobkovsky, L.I.; Vladimirova, I.S.; Gabsatarov, Y.V.; Alekseev, D.A.
Keyboard Model of Seismic Cycle of Great Earthquakes in Subduction Zones: Simulation Results and Further Generalization. *Appl. Sci.* **2021**, *11*, 9350.
https://doi.org/10.3390/app11199350

**AMA Style**

Lobkovsky LI, Vladimirova IS, Gabsatarov YV, Alekseev DA.
Keyboard Model of Seismic Cycle of Great Earthquakes in Subduction Zones: Simulation Results and Further Generalization. *Applied Sciences*. 2021; 11(19):9350.
https://doi.org/10.3390/app11199350

**Chicago/Turabian Style**

Lobkovsky, Leopold I., Irina S. Vladimirova, Yurii V. Gabsatarov, and Dmitry A. Alekseev.
2021. "Keyboard Model of Seismic Cycle of Great Earthquakes in Subduction Zones: Simulation Results and Further Generalization" *Applied Sciences* 11, no. 19: 9350.
https://doi.org/10.3390/app11199350