The attenuation relationship of ground motion based on seismology has always been a front subject of engineering earthquake. Among them, the regional finite-fault source model is very important. In view of this point, the general characteristics of regional seism-tectonics, including the dip and depth of the fault plane, are emphasized. According to the statistics of regional seism-tectonics and focal mechanisms in Sichuan, China, and the sensitivity of estimated peak ground acceleration (PGA) attenuation is analyzed, and the dip angle is taken as an average of 70°. Based the statistics of the upper crustal structure and the focal depth of regional earthquakes, the bottom boundary of the sedimentary cover can be used as the upper limit for estimating the depth of upper-edge. The analysis shows that this value is sensitive to PGA. Based on the analysis of geometric relations, the corresponding calculation formula is used, and a set of concepts and steps for building the regional finite-fault source model is proposed. The estimation of source parameters takes into account the uncertainty, the geometric relationship among parameters and the total energy conservation. Meanwhile, a set of reasonable models is developed, which lay a foundation for the further study of regional ground motion attenuation based on seismology.

The near-field strong ground motions of synthetic large earthquakes are mostly based on the source model of finite faults, which need to be predicted by empirical parameter scaling law of statistical source parameters of the occurred earthquakes. The empirical attenuation relationship cannot be established directly in the areas without strong earthquake records. Two methods are commonly used to establish the regional attenuation relationship, which are based on seismology (simplified random vibration method for ground motion estimation) [

Earthquakes mainly occur in the deformation accumulation area of plate boundary and the release of huge kinetic energy. In order to express the influence of source spatial scale on ground motion, it is necessary to establish a focal model based on finite fault. There are two kinds of source parameters need to be estimated in the process of model establishment. The first is global parameters reflecting the macro characteristics of the seism-genic rupture, including the location, occurrence and burial depth of the rupture plane, in which the size and the average slip, are all related to the magnitude. The higher is the magnitude, and the greater is the parameter values. The relationship between the area, length, width, average slip and moment magnitude of the rupture plane is called the global parameter scaling law. Thingbaijam et al. [^{–2} model for the randomness of the slip distribution on the fracture surface. Ruiz et al. [^{−2} source model. Irikura et al. [^{−2} model is used to estimate the short ones. Combining these two methods, the spatial wavenumber spectrum of source slip is obtained, and the spatial distribution of slip on the fracture surface is generated. The local source parameters from the source slip distribution data of 211 global earthquakes collected by SRCMOD [

In addition to using the scaling law of source parameters to estimate global and local values, the location of regional source model is very important. The trend is restricted by the range and extension direction of potential source area. The dip angle and buried depth play a controlling role, which also has a global impact on the seismic attenuation relationship. Although these two kinds of parameters have certain relationship with magnitude, they are mainly controlled by regional crustal structure and regional seismic tectonic stress field, thus special research and analysis are needed. According to the present seismic zoning method, the attenuation relationship of regional ground motion should be adopted for every potential source area and various magnitudes in the attenuation zoning of a country or region. Accordingly, the dip angle and buried depth of the source model for establishing the regional attenuation relationship should reflect the overall characteristics of the regional seism-tectonics, and parameters selection should comprehensively reflect the representative values of regional characteristics.

In the attenuation relationship of zoning map of the Central and Eastern US in 2014, the dip angle of the earthquake is counted, or the median value of the range is taken [

There are nine main fault zones in Sichuan, including Anning River (Q_{4} in the latest active age), Litang (Q_{3}), Longquan Mountain (Q_{3}), Daliang Mountain (Q_{4}), Yingjing-Mabian-Yanjin (Q_{3}), Yunongxi (Q_{4}), Lijiang-xiaojinhe (Q_{4}), Longmenshan fault zone (Q_{4}) in the northeast and Xianshuihe fault zone (Q_{4}) in the northwest [

No. | Name of fault zone | Rupture type | Dip angle | Length and width of fault |
---|---|---|---|---|

F1 | Xianshuihe | SS | 60°~80° | 400 km × 40 km |

F2 | Anning River | SS | 50°~80° | 200 km × 30 km |

F3 | Longmenshan | DS | 35°~80° | 500 km × 40 km |

F4 | Litang | SS | 60°~90° | 65 km × 30 km |

F5 | Longquan Mountain | DS | 28°~82° | 200 km × 20 km |

F6 | Daliang Mountain | SS | 60°~80° | 280 km × 30 km |

F7, F8 | Yingjing-Mabian-Yanjin | DS | 50°~80° | 250 km × 30 km |

F9 | Yunongxi | DS | 50°~85° | 170 km × 35 km |

F10 | Lijiang-xiaojinhe | SS | 60°~90° | 360 km × 30 km |

From _{1}, F_{2}, F_{4}, F_{6}, and F_{10}) in Sichuan are mainly SS and others are DS rupture. Most fault zones have a dip angle of 50°−80°, and DS is small.

Using the moment tensors in CMT catalogue, the focal mechanism solutions of 99 major earthquakes [_{S} ≥ 5.0 in Sichuan from 1850 to 2019 are collected. According to the distribution direction of aftershocks and the occurrence of regional seism-tectonics, the rupture planes are selected from the two nodes of the mechanism solutions to determine the dip angle; the rupture types of each earthquake are determined according to the classification standard in

Angle of stress axis | Rupture type | ||
---|---|---|---|

p1 < 40 | p1 ≥ 45 | p1 ≤ 20 | SS |

p1 ≤ 20 | p1 ≥ 45 | p1 < 40 | SS |

Other | DS |

In order to deeply analyze the rupture types and change rules of dip angle, the relationship between dip angle and magnitude of two rupture types in Sichuan is drawn as shown in

From

Further distinguish the magnitude segment and make statistics on the proportion of two rupture types, which are shown in

Magnitude range | Earthquake number | Ratio (%) |
---|---|---|

5.0 ≤ Ms ≤ 6.0 | 28/46 | 37.8/62.2 |

6.0 < Ms ≤ 7.0 | 7/11 | 38.9/61.1 |

Ms > 7.0 | 4/3 | 57.1/42.9 |

Sum | 39/60 | 39.4/60.6 |

From

Distinguishing SS and DS to establish regional attenuation relationship will increase the complexity of application, and makes it more convenient to use the attenuation relationship of all earthquakes with different rupture types. This kind of attenuation relation expresses the overall characteristics of regional strong ground motion, and it has a wide application range. It is very hard to specify dip angles for different fault zones, rupture types and earthquake magnitudes. Thus the average dip angle of 70° is selected for the calculation of all earthquakes. The feasibility of this idea will be demonstrated through the sensitivity analysis in

The thickness of the weak sedimentary cover on the crustal surface in Sichuan is about several kilometers, and earthquakes rarely occur. The crystalline basement under the cover is mainly composed of hard and brittle rocks [

Site | Northeast [ |
South | West | Sichuan basin [ |
Northwest plateau basin |
---|---|---|---|---|---|

Thickness of sedimentary cover (km) | 2.0~3.0 | 3.0~5.0 | 4.0 | 5.0~6.5 | 2.5~3.0 |

The total number of earthquakes (including main shocks and aftershocks) with M ≥ 5.0 occurred in Sichuan before November 25, 2019 is 229 [

The focal depth of more than 90% earthquakes is in 5–35 km, with an average of 16 km. Compared with

According to the location diagram of rupture plane, the geometric relationship between the three parameters of focal depth, rupture width and dip angle and the upper-edge depth of rupture plane is analyzed, as shown in

where, _{T} is the buried depth of the upper edge; _{D} is the focal depth, taking the average focal depth of 16 km; _{s} is the distance of the source to its upper edge along dip, which is about half of the rupture width (

Mw | 6.0 | 6.5 | 7.0 | 7.5 | 8.0 | 8.5 |
---|---|---|---|---|---|---|

Rupture width W (km) | 9.5 | 17.0 | 20.0 | 27.5 | 27.5 | 27.5 |

Buried depth of the upper edge _{T} (km) |
12 | 8 | 7 | 3 | 3 | 3 |

1. Establishing finite fault source models are shown as below:

(1) The moment magnitude is substituted into the global parameter scaling law to estimate the expected values of global parameters, and then the global parameters are substituted into the corresponding equations of local parameter scaling law to estimate the expected values of local parameters.

(2) Considering the discreteness of the statistical relation of source data, the expected values of source parameters and the corresponding standard deviations obtained in Step 1 are substituted into the probability density function of the truncated normal distribution to generate 30 random numbers.

where, _{max} is the upper limit of the parameter, i.e.,

(3) By substituting each random number into

where, _{i} is the _{i} is the

Due to the control of the total slip of each earthquake, the rupture area and the average slip also have a potential relationship. With the increase of rupture area, the average slip will decrease in accordance with

where, _{i} is the average slip of the _{S} is the expected value of the rupture area, and _{i} is the rupture area of the

(4) For ^{−2} model, the rupture plane is discretized into 2^{M} * 2^{N} small micro-grids. With the same method, the asperity slip is allocated into each micro-grid, and the slip of each micro-grid is obtained by interpolation and smoothness.

(5) By means of Fourier transform, the slip distribution of the deterministic part is transformed from the spatial domain to the wavenumber domain, that is, the long-wavelength (low wavenumber) slip feature, which is called deterministic part.

(6) The numerical value of spatial corner wave in two directions is substituted into ^{−2} model, i.e., short-wavelength (high wavenumber) slip feature, which is called random part.

where, _{x} and _{y} are the spatial wavenumbers along the strike and down dip of the rupture plane; _{cx} and _{cy} are the corresponding corner wave numbers, respectively [

(7) The combination of deterministic part and random part forms the total wavenumber spectrum.

(8) By means of inverse Fourier transform, the slip distribution on the rupture plane can be transformed from the wavenumber domain to the spatial domain.

2. Establishing the ground motion attenuation relationship:

Each sub-source in the finite fault source model can be regarded as a point source. Based on seismology, the Fourier amplitude spectrum of ground motion caused by point source can be calculated by

where, _{0}^{2}.

Regional parameters in Sichuan are from Jiang [

The main range of fault dip angle in Sichuan is from 50° to 80°. It is difficult to establish and use attenuation relationship by multiple dip angles. In order to analyze the influence of dip angle on the attenuation of ground motion, the burial depth of the upper edge of rupture plane is temporarily taken as 5 km, and the same values of shear wave velocity and medium density of the crust media in Sichuan are selected [

It can be seen from

Mw | Dip angle | |||
---|---|---|---|---|

50° | 60° | 70° | 80° | |

6.0 | 386 | 377 | 371 | 369 |

6.5 | 472 | 465 | 451 | 446 |

7.0 | 522 | 509 | 497 | 493 |

7.5 | 615 | 605 | 599 | 595 |

When the buried depth of the upper edge of rupture plane is taken as a fixed value, the change of dip angle of the finite fault source model will make the upper edge of the whole rupture plane rotate as an axis. The steeper the dip angle is, the deeper the whole rupture plane is, and the smaller the surface PGA is, but the relative difference with 70° is less than 5%. It can be seen that the change of dip angle is not sensitive to the synthetic PGA, and it is feasible to select the mean value of 70° to establish the source model of regional finite fault in Sichuan, which can meet the needs of engineering.

The dip angle is 70° and the buried depth of the upper edge is 4, 5, 6, 7 and 8 km, respectively. The regional source parameters, magnitude and synthetic ground motion method are the same as the above. The average PGA attenuation characteristics are studied, and PGA attenuation curves corresponding to four magnitudes and five different buried depths are obtained, as shown in

Mw | Buried depth of the upper edge of the rupture plane (km) | ||||
---|---|---|---|---|---|

4 | 5 | 6 | 7 | 8 | |

6.0 | 435 | 371 | 329 | 288 | 265 |

6.5 | 504 | 451 | 416 | 388 | 371 |

7.0 | 520 | 497 | 442 | 410 | 395 |

7.5 | 633 | 599 | 550 | 530 | 491 |

It can be seen from

From the Figures and Tables, the composite epicenter and short-distance PGA are sensitive to the buried depth of the upper edge of rupture plane, and the difference between the epicenter area is 20%–40%, thus the selection of fixed value cannot meet the engineering requirements. Therefore, the buried depth of the upper edge of the regional finite-fault model established is reasonable according to

Aiming at 26 magnitudes of 6.0 ≤ Mw ≤ 8.5 and 0.1 intervals, 30 regional finite-fault source models of each magnitude in Sichuan are studied and established based on the FORTRAN program developed by us, and the specific steps are as follows:

(1) According to the regional crustal structure and focal depth distribution, the average value of the regional focal depth is determined, and then the rupture width corresponding to each magnitude is estimated according to the scaling law with the dip angle of 70°, and the burial depth of the upper edge is calculated by

(2) For each magnitude, the expected values of the global parameters of the three sources are estimated according to the global parameter scaling law, which are substituted into the local parameter scaling law to estimate the expected values of the asperity parameters and the coordinates of the rupture starting point. They are replaced with the corresponding standard deviation and substituted into the truncated normal distribution to generate 30 groups of source parameter values, which are placed in the determined part of the slip distribution of the rupture plane.

(3) The spatial corner wave numbers _{cx} and _{cy} of each group are estimated. The high wave number sliding wave number spectrum is generated by k^{−2} two-dimensional sliding model formula. The slip distribution is generated by transforming the part together with the determined part back to the space domain [

According to the above steps, the expected values of some magnitude global parameters of the regional source model obtained are shown in

Mw | 6.0 | 6.5 | 7.0 | 7.5 | 8.0 | 8.5 | |
---|---|---|---|---|---|---|---|

Rupture area ^{2}) |
120 | 374 | 900 | 2133 | 6750 | 21384 | |

0.13 | 0.15 | ||||||

Rupture width |
10 | 17 | 20 | 27 | 27 | 27 | |

0.08 | 0.11 | 0.17 | 0.23 | ||||

Rupture length |
12 | 22 | 45 | 79 | 250 | 792 | |

Average slip |
32 | 56 | 135 | 316 | 562 | 1000 | |

0.13 | 0.18 | 0.13 |

In order to express the discreteness of source parameter data, 30 random numbers are generated by substituting

Referring to the local parameter scaling law, the maximum asperity parameter can only be estimated when Mw ≤ 6.5, and other asperity parameters can be estimated when Mw > 6.5 [_{m}, width _{m}, length _{m}, average slip _{m}, center coordinate along strike _{m} and dip _{m}, other asperity area _{0}, width _{0}, length _{0}, average slip _{0}, center coordinate along strike _{0} and dip _{0}, the rupture starting point coordinate along strike _{s} and dip _{s}. The wave numbers of spatial corners are shown in

Local parameter | ||||||||
---|---|---|---|---|---|---|---|---|

M_{w}6.0 |
M_{w}6.5 |
M_{w}7.0 |
M_{w}7.5 |
M_{w}8.0 |
M_{w}8.5 |
|||

Maximum asperity | Area _{m}(km^{2}) |
20 | 56 | 135 | 338 | 1079 | 3406 | 0.08 |

Width _{m}(km) |
5 | 8 | 9 | 13 | 13 | 13 | – | |

Length _{m}(km) |
4 | 7 | 15 | 26 | 83 | 262 | 0.11 | |

Average slip _{m}(cm) |
77 | 134 | 324 | 758 | 1348 | 2399 | 0.08 | |

The center along strike _{m}(km) |
6 | 11 | 22 | 38 | 120 | 379 | 0.13 | |

The center down-dip _{m}(km) |
4 | 8 | 9 | 12 | 12 | 12 | 0.10 | |

Other asperity | Area _{0}(km^{2}) |
– | – | 63 | 144 | 480 | 1520 | 0.13 |

Width _{0}(km) |
– | – | 7 | 9 | 8 | 8 | – | |

Length _{0}(km) |
– | – | 9 | 16 | 60 | 190 | 0.10 | |

Average slip _{0}(cm) |
– | – | 276 | 645 | 1147 | 2042 | 0.06 | |

The center along strike _{0}(km) |
– | – | 7/36 | 12/63 | 38/200 | 122/634 | – | |

The center down-dip _{0}(km) |
– | – | 7 | 10 | 10 | 10 | 0.13 | |

Rupture starting point coordinate | Along strike _{s}(km) |
6 | 10 | 21 | 37 | 117 | 370 | 0.14 |

Down-dip _{s}(km) |
5 | 9 | 10 | 14 | 14 | 14 | 0.11 | |

Wave numbers of spatial corners | Along strike _{cx} |
0.083 | 0.046 | 0.022 | 0.013 | 0.004 | 0.001 | – |

Down-dip _{cy} |
0.100 | 0.059 | 0.050 | 0.037 | 0.037 | 0.037 | – |

Taking the M7.0 earthquake as an example, according to a set of global and local parameters, the process and steps of source model generation can be simply summarized into the following six steps: Put the maximum and other asperities on the rupture plane, assign the corresponding fault slip; further divide the rupture plane into 2^{9} × 2^{9} grids, interpolate and smooth to obtain the corresponding fault slip; transform it into wavenumber domain through Fourier transform; combined with the wavenumber spectrum of k^{−2} model to form the total wavenumber spectrum; transform it from the wavenumber domain to the space domain. Then the staggered distribution is generated as shown in

According to the above ideas and steps, 30 source model parameters of 26 magnitudes are obtained, and the slip distribution of Mw6.0, 7.0 and 8.0 are shown in

^{−2} model, the original rectangular asperity boundary becomes very complex.

^{−2} model.

In this paper, a set of regional finite-fault source model of Sichuan, China is established for the research and development of regional ground motion attenuation relationship. The main results and work novelties are as follows:

(1) Different from the source model of given magnitude on active fault, in order to establish a finite fault source model with regional attenuation relationship, this paper emphasizes that the dip angle and buried depth of the source rupture plane should reflect the overall characteristics of regional seism-tectonics. The comprehensive representative values of the parameters are selected, and the results of regional seism-tectonics and focal mechanism solution are summarized. It is pointed out that the overall average value of dip angle is 70°. Based on the sensitivity analysis of the dip angle, the attenuation characteristics of PGA are estimated. A fixed value of 70° is determined, which is simpler and more practical than the zoning map method of the United States in 2014. According to the statistical analysis of the upper crustal structure and the focal depth of Sichuan earthquake, the thickness of the sedimentary cover restricts the burial depth of the upper edge of rupture plane. Through the sensitivity analysis of PGA attenuation characteristics of buried depth, it is pointed out that the sensitivity is high. Based on the analysis of geometric relationship, the concept and calculation formula of buried depth can be calculated. It avoids the instability of PGA attenuation characteristics caused by the random value of buried depth on the upper edge of the zoning map of the United States.

(2) In order to develop the attenuation relationship of regional ground motion in Sichuan, the general characteristics of regional seism-tectonics are emphasized. Considering the characteristics of regional crustal structure and focal depth, the maximum value of rupture plane width is calculated, and a set of concepts and steps for establishing the source model of regional finite fault are put forward. The specific estimation methods of two kinds of source parameters and the problems that should be paid attention to are introduced. In the process of 30 groups of source parameters generation for each magnitude, the coordination of geometric relations among parameters, such as the area, length and width, is emphasized. In order to ensure the total energy conservation of each magnitude, the relationship between the asperity area and the average slip is coordinated by formulas. The expected values of source parameters, standard deviations and the slip distribution of some source models of several earthquakes are listed. These are all new concrete processing methods, which are practical and make up for the possible loopholes in logic. The regional finite-fault source models established lay a foundation for the further study of regional ground motion attenuation based on seismology.

This work was supported by the

The authors declared that there is no conflict of interest in the paper.

^{−2}source model for generating physical slip velocity functions