10:00 AM - 10:15 AM
[SSS11-11] A scaling relation of crustal earthquakes assuming constant stress drop - in a case when fault width is used as a parameter -
Keywords:Crustal earthquake, Scaling relation, Stress drop, Fault size, Fault width
- Outline -
In Japan, the Irikura and Miyake formula (Irikura and Miyake, 2001) and 3 stage model (Murotani et al, 2015) are the key of the “recipe for the strong motion prediction”. Those formulas take into account the limitation of crustal thickness and are basically considered physically reasonable. However, some defects are pointed out especially for long fault. In our previous study (Hikima and Shimmura, 2020; BSSA), we proposed the similar MO-S scaling relation, which realizes constant stress drop from small to large fault models. Although the relation was consistent with existing parameters, it was necessary to examine using newly analyzed data.
In recent studies, fault parameters are estimated by inversion analyses, which are performed mostly assuming larger fault model than true slip area. Therefore, we have to extract major fault area from the inversion results. Somerville et al. (1999) proposed a method in that marginal rows and columns having relatively small slips are trimmed and rectangular main area is extracted. However, it seems that the physically clear criteria are not proposed and it is difficult automatically select the main target. Therefore,
we proposed a procedure to extract an effective rectangular area based on wavenumber spectra of slip distribution (Hikima and Shimmura, 2020; JpGU).
In this study, we shortly explain the proposed trimming method again, because the explanation at the time of presentation was somewhat different from the published abstract. Then, we discuss scaling relations using the determined source parameters.
- Trimming method -
First, a slip distribution at 1 or 2 km intervals for length (X-direction) and for width (Y-direction) is made from an original slip model. Then, zero values are added outside of the fault area, and amplitude wavenumber spectra of these data are calculated using two-dimensional FFT. For these spectra, we consider that the spectrum at ky=0 (ky: wavenumber for Y-direction) is in length direction and the spectrum at kx=0 (kx: wavenumber for X-direction) is in width direction, respectively.
On the side, Fourier transform of a rectangular function having width as L is f(ν)=Lsin(πLν)/(πLν) [ν=1/λ; λ: wavelength]. This sinc function crosses with zero line at ν=n/L [n=1, 2, …], and these correspond to trough of the spectrum in the case of amplitude spectrum.
We adopted following criteria to extract almost equal fault areas as those using Somerville et al. (1999). We focused on lower part than first trough (it corresponds to n=1) of sinc function, and we determined maximum L length not to exceed observed spectra from lower wavenumber side. We applied the same process to the spectrum of width direction. In the previous presentation, we confirmed that the determined fault areas are almost same as those by Somerville’s procedure.
- Scaling relation for crustal earthquakes -
We applied these analyses for the fault models in the SRCMOD (Mai and Thingbaijam, 2014) in last year. Following discussions are based on determined parameters.
First, we checked the relation between the fault length and the fault width (Fig. 1). Conventionally, it has been recognized that the width saturates for long faults. Although such a trend can be seen among the data, it is also observed that the fault width increased in the long fault. Such a tendency is particularly remarkable for longitudinal slip faults. On the other hand, when the stress drop was calculated considering the formulas of the rectangular fault, there was no obvious correlation between the earthquake size and the stress drop (Fig. 2).
Considering these relations, we thought that the data could be explained more appropriately by treating the stress drop as a constant value and the fault width as a parameter. In Fig. 3, the fault widths of each earthquake are denoted according to the color table, and the curves show the MO-S relations, assuming a constant stress drop of 2.5 MPa and with four saturation fault widths (Wmax). The observed fault width becomes larger in larger earthquakes, and they are consistent with the relation with larger Wmax.
These results may be affected by uniform treatment without considering various tectonic conditions in the world. However, it is shown that the relation for the crustal earthquakes can be basically scaled with constant stress drop condition. In addition, such as the area around Japan, where the tectonic environment is clear, Wmax can be considered as almost constant in areas. Therefore, it may become possible to consider scaling with a single formula.
<Acknowledgment> We appreciate the SRCMOD and its related parties.
In Japan, the Irikura and Miyake formula (Irikura and Miyake, 2001) and 3 stage model (Murotani et al, 2015) are the key of the “recipe for the strong motion prediction”. Those formulas take into account the limitation of crustal thickness and are basically considered physically reasonable. However, some defects are pointed out especially for long fault. In our previous study (Hikima and Shimmura, 2020; BSSA), we proposed the similar MO-S scaling relation, which realizes constant stress drop from small to large fault models. Although the relation was consistent with existing parameters, it was necessary to examine using newly analyzed data.
In recent studies, fault parameters are estimated by inversion analyses, which are performed mostly assuming larger fault model than true slip area. Therefore, we have to extract major fault area from the inversion results. Somerville et al. (1999) proposed a method in that marginal rows and columns having relatively small slips are trimmed and rectangular main area is extracted. However, it seems that the physically clear criteria are not proposed and it is difficult automatically select the main target. Therefore,
we proposed a procedure to extract an effective rectangular area based on wavenumber spectra of slip distribution (Hikima and Shimmura, 2020; JpGU).
In this study, we shortly explain the proposed trimming method again, because the explanation at the time of presentation was somewhat different from the published abstract. Then, we discuss scaling relations using the determined source parameters.
- Trimming method -
First, a slip distribution at 1 or 2 km intervals for length (X-direction) and for width (Y-direction) is made from an original slip model. Then, zero values are added outside of the fault area, and amplitude wavenumber spectra of these data are calculated using two-dimensional FFT. For these spectra, we consider that the spectrum at ky=0 (ky: wavenumber for Y-direction) is in length direction and the spectrum at kx=0 (kx: wavenumber for X-direction) is in width direction, respectively.
On the side, Fourier transform of a rectangular function having width as L is f(ν)=Lsin(πLν)/(πLν) [ν=1/λ; λ: wavelength]. This sinc function crosses with zero line at ν=n/L [n=1, 2, …], and these correspond to trough of the spectrum in the case of amplitude spectrum.
We adopted following criteria to extract almost equal fault areas as those using Somerville et al. (1999). We focused on lower part than first trough (it corresponds to n=1) of sinc function, and we determined maximum L length not to exceed observed spectra from lower wavenumber side. We applied the same process to the spectrum of width direction. In the previous presentation, we confirmed that the determined fault areas are almost same as those by Somerville’s procedure.
- Scaling relation for crustal earthquakes -
We applied these analyses for the fault models in the SRCMOD (Mai and Thingbaijam, 2014) in last year. Following discussions are based on determined parameters.
First, we checked the relation between the fault length and the fault width (Fig. 1). Conventionally, it has been recognized that the width saturates for long faults. Although such a trend can be seen among the data, it is also observed that the fault width increased in the long fault. Such a tendency is particularly remarkable for longitudinal slip faults. On the other hand, when the stress drop was calculated considering the formulas of the rectangular fault, there was no obvious correlation between the earthquake size and the stress drop (Fig. 2).
Considering these relations, we thought that the data could be explained more appropriately by treating the stress drop as a constant value and the fault width as a parameter. In Fig. 3, the fault widths of each earthquake are denoted according to the color table, and the curves show the MO-S relations, assuming a constant stress drop of 2.5 MPa and with four saturation fault widths (Wmax). The observed fault width becomes larger in larger earthquakes, and they are consistent with the relation with larger Wmax.
These results may be affected by uniform treatment without considering various tectonic conditions in the world. However, it is shown that the relation for the crustal earthquakes can be basically scaled with constant stress drop condition. In addition, such as the area around Japan, where the tectonic environment is clear, Wmax can be considered as almost constant in areas. Therefore, it may become possible to consider scaling with a single formula.
<Acknowledgment> We appreciate the SRCMOD and its related parties.