11:00 〜 11:15
[SSS06-08] 地震アスペリティの階層モデル
キーワード：地震、断層、固着面、階層、自己相似
Introduction
Asperity models (Lay and Kanamori, 1981) have contributed greatly to understanding of such phenomena as regional characteristics of great earthquakes (Lay et al., 1982) and repeating earthquakes on plate boundaries (Matsuzawa et al., 2002). Hierarchical rupture models, on one hand, have been developed by remarking the self-similar characteristics of seismic rupture growth (Smalley et al., 1985; Fukao and Furumoto, 1985). The self-similarity implies unpredictability of the eventual size of rupture growth by observing seismic radiation. Recent careful analyses of seismic waves seem to support this implication (Maier et al., 2016; Okuda and Ide, 2018; Ide, 2019). The self-similar behavior is a phenomenon expected at a critical state of the stress system (Allégre et al., 1982; Turcotte, 1997). The power law of the event-size distribution known as the Gutenberg-Richter relation (Gutenberg and Richter, 1954) is regarded as an expression of this critical state (Narteau, 2007). Here, we explore a hierarchical asperity model in which asperity patches are distributed to allow their progressive unlocking from smaller to larger scales at a critical state.
Basic idea of the model
The elastic upper plate and the rigid lower plate are in frictional contact. The lower plate is kept at rest. The elasticity of the upper plate is modeled by an array of springs connecting the rigid surface at the top and a fractal set of asperities at the bottom (Fig.1). Uniform shear traction across the top surface is transmitted to the asperities through the equally deflected springs. Fig.1 shows our one-dimensional asperity model (Cantor set) and two-dimensional asperity model (Sierpinski carpet). The basic slip-system consists of the central major asperity and the surrounding minor asperities mutually isolated by already slipped patches. The major and minor asperities have the same frictional strength but their sizes are different, so stresses concentrate more on the minor asperities (Stage A). All the minor asperities are eventually unlocked and the imposed force is now supported only by the major asperity (Stage B). This state is understood as the rebirth of Stage A at the next larger scale (Stage C). The subsequent process is then the rebirth of Stage-B at this larger scale (Stage D). Rupture develops self-similarly from the smallest to larger scales, if the process A→B is appropriately renormalized to the process C→D.
Renormalization of the system
The process A→B involves the stress-release from the surrounding minor asperities and the consequent stress-accumulation on the central major asperity. Such stress transfer is guaranteed in our model by imposing the scale-independent condition that the probability for all the surrounding minor asperities to be unlocked equals the probability for the central asperity to be unlocked. These two probabilities can be calculated as functions of po(σ/σo), where po(σ/σo) describes the probability for an asperity to be unlocked by relative applied stress σ/σo (Turcotte, 1997). For the one-dimensional asperity model, the equality of the two probabilities occurs at po = 0.42565 or σ/σo = 0.74466. For the two-dimensional asperity model, the equality occurs at po = 0.55831 or σ/σo = 0.90396. If the system is tuned to this critical state, the basic system at any scale grows self-similarly into one at the next larger scale with probability po.
Characteristics of the two-dimensional hierarchical asperity model
(1)The asperity distribution in Sierpinski carpet → Fault plane with a fractal dimension D of 1.8928. (2)Non-asperity patches = Already unlocked asperities → GR law with a b-value of 1 (b ≠ D/2). (3)Stresses on minor asperities 9 times higher than one applied on the top surface → Frictional strength of asperity much higher than the overall stress drop. (4)Successively upscaled radiations from surrounding minor asperities and from the central major asperity → Unpredictbility of final earthquake size by observing seismic radiation.
Asperity models (Lay and Kanamori, 1981) have contributed greatly to understanding of such phenomena as regional characteristics of great earthquakes (Lay et al., 1982) and repeating earthquakes on plate boundaries (Matsuzawa et al., 2002). Hierarchical rupture models, on one hand, have been developed by remarking the self-similar characteristics of seismic rupture growth (Smalley et al., 1985; Fukao and Furumoto, 1985). The self-similarity implies unpredictability of the eventual size of rupture growth by observing seismic radiation. Recent careful analyses of seismic waves seem to support this implication (Maier et al., 2016; Okuda and Ide, 2018; Ide, 2019). The self-similar behavior is a phenomenon expected at a critical state of the stress system (Allégre et al., 1982; Turcotte, 1997). The power law of the event-size distribution known as the Gutenberg-Richter relation (Gutenberg and Richter, 1954) is regarded as an expression of this critical state (Narteau, 2007). Here, we explore a hierarchical asperity model in which asperity patches are distributed to allow their progressive unlocking from smaller to larger scales at a critical state.
Basic idea of the model
The elastic upper plate and the rigid lower plate are in frictional contact. The lower plate is kept at rest. The elasticity of the upper plate is modeled by an array of springs connecting the rigid surface at the top and a fractal set of asperities at the bottom (Fig.1). Uniform shear traction across the top surface is transmitted to the asperities through the equally deflected springs. Fig.1 shows our one-dimensional asperity model (Cantor set) and two-dimensional asperity model (Sierpinski carpet). The basic slip-system consists of the central major asperity and the surrounding minor asperities mutually isolated by already slipped patches. The major and minor asperities have the same frictional strength but their sizes are different, so stresses concentrate more on the minor asperities (Stage A). All the minor asperities are eventually unlocked and the imposed force is now supported only by the major asperity (Stage B). This state is understood as the rebirth of Stage A at the next larger scale (Stage C). The subsequent process is then the rebirth of Stage-B at this larger scale (Stage D). Rupture develops self-similarly from the smallest to larger scales, if the process A→B is appropriately renormalized to the process C→D.
Renormalization of the system
The process A→B involves the stress-release from the surrounding minor asperities and the consequent stress-accumulation on the central major asperity. Such stress transfer is guaranteed in our model by imposing the scale-independent condition that the probability for all the surrounding minor asperities to be unlocked equals the probability for the central asperity to be unlocked. These two probabilities can be calculated as functions of po(σ/σo), where po(σ/σo) describes the probability for an asperity to be unlocked by relative applied stress σ/σo (Turcotte, 1997). For the one-dimensional asperity model, the equality of the two probabilities occurs at po = 0.42565 or σ/σo = 0.74466. For the two-dimensional asperity model, the equality occurs at po = 0.55831 or σ/σo = 0.90396. If the system is tuned to this critical state, the basic system at any scale grows self-similarly into one at the next larger scale with probability po.
Characteristics of the two-dimensional hierarchical asperity model
(1)The asperity distribution in Sierpinski carpet → Fault plane with a fractal dimension D of 1.8928. (2)Non-asperity patches = Already unlocked asperities → GR law with a b-value of 1 (b ≠ D/2). (3)Stresses on minor asperities 9 times higher than one applied on the top surface → Frictional strength of asperity much higher than the overall stress drop. (4)Successively upscaled radiations from surrounding minor asperities and from the central major asperity → Unpredictbility of final earthquake size by observing seismic radiation.