Earthquake Reaearch in China  2018, Vol. 32 Issue (3): 400-411
Application of the Levenberg-Marquardt Method in Improving Technology of the Seismic Influence Field
Yang Yanming1, Jiang Lixin2, Wang Zhenxiang1     
1. Earthquake Agency of Inner Mongolia Autonomous Region, Hohhot 010010, China;
2. China Earthquake Networks Center, Beijing 100045, China
Abstract: Attenuation relation of seismic intensity is fitted by using 152 complete isoseismic lines from 65 earthquakes which were greater than MS4.0 from 1940 to 2015 in Inner Mongolia and neighboring regions. Meanwhile, based on the difference of land form and geological structure characteristics, the study area is the divided into eastern, western and central. The intensity attenuation relationships are established separately by using the LM (Levenberg-Marquardt) method and elliptic attenuation model. Comparisons are made by using the earthquake affecting field model of North China and computed results. The analytical study shows that the relation obtained represents the features of earthquake damage distribution in the areas well, and has positive meaning to guide loss assessments immediately after a destroyed earthquake.
Key words: Seismic Intensity     Earthquake affecting field     Inner Mongolia     Levenberg-Marquardt     Attenuation relationship    

INTRODUCTION

Seismic intensity influence field is the reflection of area coverage affected by different intensities (Sun Yanping et al., 2014), and rapid estimation of seismic influence field of a destructive earthquake is the basis for accurate assessment of damage caused by the earthquake. Determination of the seismic influence field directly depends on seismic intensity attenuation relationship in the region.In the process of earthquake damage assessments using the rapid evaluation system for earthquake damage loss, model parameters of seismic influence fields are directly obtained by intensity attenuation laws (Zhou Zhonghong et al., 2011). In addition, seismic intensity attenuation relationships also play an important role in engineering seismology and seismic hazard assessments.

At present, there are few studies carried out by relevant scholars on the law of intensity attenuation in Inner Mongolia. Owing to the absence of relative research achievements, the seismic influence field model of the evaluation system of earthquake damage in Inner Mongolia adopts the empirical formula of intensity attenuation relationship of North China, the calculation results from which are quite different from the actual situation of historical earthquakes. There are few strong earthquake records in the region, so it is also necessary to explore the intensity attenuation relationship in areas where there are not effective strong earthquake records. The intensity attenuation relationship varied with epicentral distance can be traced back to 1975, when it is deduced by Howell and Schultz from the perspective of seismology (Howell B.F. et al., 1975). Subsequently, according to historical seismic data and on the basis of isoseismal maps, Hanks and Hileman (1975) carried out the study of intensity attenuation relationships and proposed the method of using the empirical relationship between area and magnitude of earthquake intensity zones to determine earthquake magnitude (Hanks T.C. et al., 1975). Because different geological structures in different regions, seismic intensity and ground motion parameter attenuation have strong regional characteristics (Hu Yuxian, 1999), it is important to use appropriate partitioning of seismicity features, collect historical seismic intensity data, and to take into account the statistical rationality and regional applicability of attenuation relationships, and to establish seismic intensity attenuation relationships of the partitions which fully reflects near-field and far-field characteristics of seismic intensity distribution (Wang Xiaojun et al., 2012). Chen Dasheng et al. (1989) selected 77 isoseismic lines from 27 historical earthquakes, and built the intensity attenuation model of North China by using the method of non-steering long axis and steering long axis. Analysis shows that the adoption of data of non-steering and steering long axis makes minor differences between the results, however, the number of earthquake cases is small and the study area is large, which cannot reflect the difference of impact of geological structures and seismogenic structures on intensity attenuation relationships in local areas. The seismic intensity attenuation relationship of North China was studied by Sha Haijun et al. (2004) by using isoseismic lines of 48 earthquakes, and because sea-area earthquake cases were used, the accuracy of the curve obtained by fitting was low. On the premise of reasonable partitioning, Wang Xiaojun et al. (2012), selected seismic data of 59 earthquakes with M≥5.0 in 9 areas during 1900-2006, covering Shaanxi, Inner Mongolia, Ningxia and Gansu etc., and obtained seismic intensity attenuation relationship of the partitions of the study area by using the multiple robust linear regression method. Zhang Yang et al. (2009), using data of 10 historical earthquakes, obtained seismic intensity attenuation models of North China by fitting, which cannot reflect the intensity attenuation characteristics of North China on the whole due to limited sample data and territorial restrictions. In recent years, more and more scholars have studied the regionalization of intensity characteristics in Chinese mainland (Wang Suyun et al., 2000; Yu Yanxiang et al., 2004; Lü Jian et al., 2009; Xiao Liang et al., 2011).

The Inner Mongolia Autonomous Region, located in the north boundary of China, spans northeast China, north China and northwest China, where the geological structure is complex, and active faults crisscross and landform vary greatly. There have been several strong earthquakes and moderate-strong earthquakes in the autonomous region recorded by instruments since 1923. Earthquakes in the area are distributed mostly in two big areas, the eastern and the western region. The eastern region is affected by deep-focus earthquakes in northeast China, and the earthquake zone in central and western area is mainly affected by the Yinshan, Yanshan tectonic belt and Ordos platform, forming the main area of strong earthquake activities in the Inner Mongolia Autonomous Region (Yang Yanming et al., 2016, 2016; Dai Yong et al., 2012). At present, no scholars have systematically studied the data of historical earthquakes in Inner Mongolia. There is little data of earthquake damage, so its seismic influence field model has been using the seismic intensity attenuation relationship of North China. It is therefore urgent that we study seismic data on the basis of isoseismic lines of historical earthquakes in Inner Mongolia and its adjacent areas, in order to meet the needs of engineering seismology and earthquake emergency hazard assessments. This paper intends to make a statistical analysis of historical earthquakes in Inner Mongolia and its adjacent areas by using the elliptic model formula and Levenberg-Marquardt non-linear inversion method (the Levenberg-Marquardt method, hereinafter referred as LM), to establish seismic intensity attenuation relationships of the partitions in the Inner Mongolia Autonomous Region and its adjacent areas.

1 RESEARCH DATA 1.1 Intensity Data

Data of historical earthquake damage used in this article are cited from Earthquakes Cases in China (China Earthquake Administration, 1988, 1990a, 1990b, 1999, 2000, 2002a, 2002b, 2003, 2008, 2013), Catalog of Earthquakes in China (Gu Gongxu et al., 1983; Office of the Central Earthquake Working Group, 1971), Compilation of Assessment of Earthquake Damage Losses in Chinese Mainland (China Earthquake Administration et al., 1996; Department of Earthquake Monitoring and Prediction, CEA, 2001; Department of Earthquake Disaster Emergency Response and Relief, CEA, 2010), document literature (Chen Dasheng et al., 1989; Sha Haijun et al., 2004; Xiao Liang et al., 2011) and the network database2 (official website of China Earthquake Administration). Historical earthquake cases with more detailed documentary records, complete isoseismogram information, regular shapes and distinct directions of long short axes are selected as objects of study. Because intensity anomaly areas are related to local geological conditions, which shows regional characteristics of a small area, and not the intensity changes of the whole area with epicentral distances (Zhang Yang et al., 2009), scattered intensity anomaly areas are therefore not selected. For main shock and strong aftershocks of the same seismic sequence with isoseismic lines, only isoseismic lines of the main shock are selected, which are of greater reliability (Yang Yanming et al., 2016). According to the above rules, 65 earthquakes during 1940-2015 are selected as data for this study.

2 http://www.cea.gov.cn/publish/dizhenj/468/553/101802/101804/20150417170022088284333/index.html

In this article, the method of steering long axis is used to measure radius of long and short axes of isoseismic lines, namely, the longest radius of each isoseismic line is regarded as the radius of the long axis, and the radius in the perpendicular direction is the radius of the short axis (Chen Dasheng et al., 1989; Yang Yanming et al., 2016). The measurement accuracy is influenced by measuring scale and positioning accuracy of isoseismic lines, which is subjective, in order to improve data accuracy and reliability, the ellipse fitting method (Sha Haijun et al., 2004) is used in this article to vectorize samples, to digitize the collected isoseismic map and convert it into a set of point dataset in the form of latitude and longitude. It then converts point data into line data format, and makes an ellipse fitting for each isoseismic line, and finally, the radius of long and short axes is derived from the elliptic equation obtained by fitting. By using the above methods, totally data of 152 isoseismic lines are obtained. Study of earthquake cases adopts surface wave magnitude MS and the Chinese seismic intensity scaleⅠ, with a magnitude range of 4.7-7.8 and seismic intensity of Ⅳ-Ⅺ degrees, incorporating 10 earthquakes with magnitude of 4.5-4.9, 40 earthquakes with magnitudes of 5.0-5.9, 12 earthquakes with magnitudes of 6.0-6.9 and 3 earthquakes with M≥7.0.

1.2 Partition Basis

Source rupture process, propagation of seismic waves and geological conditions of shallow ground are the three main factors that influence intensity, among which, source rupture process is mainly affected by magnitude, source location, rupture direction, fault length and width. The effect of seismic wave propagation path is reflected in epicentral distance and refraction, reflection, scattering and attenuation of seismic waves, and geological conditions of shallow ground incorporate crustal structure, activity intensity, and thickness of surface material, topographic relief and mechanical properties. In the study of intensity attenuation relationships, there are two important factors: the source rupture process and propagation of seismic waves, namely magnitude and epicentral distance are taken into account (Ma Gan et al., 2010). However, there are differences in geological structures, seismic source characteristics, propagation medium and site conditions in different regions, and seismic intensity attenuation relationship has strong regional characteristics (Hu Yuxian, 1999; Wang Xiaojun et al., 2012). Therefore, the influence of geological conditions on intensity in different tectonic regions is an essential part of reasonably determining the extent and distribution of earthquake influence fields. According to geological conditions in Inner Mongolia and its adjacent areas, the study of seismic intensity attenuation relationships based on partitioning is more in line with the regional characteristics of earthquake damage and influence scope in the study area.

The difference in the ratio of the length of long and short axes of the inner ring (meizoseismal zone) of isoseismic lines in different regions reflects regional differences of the seismic intensity attenuation relationship, which is the basis for the partitioning of seismic intensity attenuation relationship (Wang Suyun et al., 2000). In addition, there are two main areas of moderate-strong earthquake activity in east and west of Inner Mongolia, which is another basis for partitioning. The Pacific plate dives deep into northeastern China in the Japanese Trench, which is the dynamic source for tectonic movements and deep-focus seismic zones in the northeast China region. The seismically active zone in eastern Inner Mongolia is mainly affected by deep-focus earthquakes in northeast China, and the earthquake zone in the central and western region is mainly affected by peripheral fractures around the Yinshan, Yanshan tectonic belts and the Ordos block. Based on the above analysis, the method proposed by Wang Suyun et al. (2000) in the study of seismic intensity attenuation relationship is firstly used to divide the study area into two regions, the western region and the central and eastern region, bounded by 105°E, and then subdivides the central and eastern region into the central region and the eastern region, bounded by 115°E (Fig. 1).

Fig. 1 Epicenter distribution map of selected earthquakes The blue circle represent earthquake events, M is the central region, E is the eastern region, W is the western region. The interpolated illustration shows the location of the study areas

According to the above partitions, the ratios of radii of long and short axes of the innermost rings of earthquake isoseismic lines used in this study are calculated (Table 1). The mean value of the ratios of the length of long and short axes of inner rings of earthquake isoseismic lines is 2.46 in the western region, 1.95 in the central region and 2.12 in the eastern region. The mean values of the ratios of long and short axes are obviously different in the three partitions, indicating that the partition of this study is comparatively reasonable.

Table 1 Comparison of the ratios of radii of long and short axes in different regions
1.3 Near-field and Far-field Point Interpolating

Generally, isoseismal of seismic intensity has the following two characteristics : any point in the meizoseismal zone has the same epicentral intensity, and the shape of the isoseismic line tends to be circular because of the vanish of the effect of the seismogenic structure in far-field regions (Wang Xiaojun et al., 2012). In order to make the intensity attenuation curve meet the above two requirements, the near-field and far-field interpolating point needs to be determined. The research method proposed by Wang Suyun et al.(2000) is adopted in this article, and in the condition of epicenter intensity I0≥7 and radius of the innermost ring of isoseismic lines R > 5km, proper data points are added at different distances in the meizoseismal area. In far-field regions, according to empirical formulas of earthquake magnitude and radius of earthquake-felt range lgR=0.161+0.289M(Wang Suyun et al., 2000), the radius of the felt range is taken as a far-field control point, which is as the far-field interpolating point, with corresponding intensity of Ⅲ-Ⅳ degrees.

2 MODEL AND RESEARCH METHODS

There are two kinds of seismic intensity attenuation relation models that are internationally in common use, namely, the circular intensity attenuation model and elliptic intensity attenuation model (Yang Yanming et al., 2016). According to the methods proposed by Chen Dasheng et al. (1989) and Wang Suyun et al. (2000) in the study of seismic intensity attenuation relationships, earthquake focus is used as the point source, and the starting points of attenuation curves of the long and short axes are overlapped. In far-field regions, the influence of seismogenic structure has disappeared, intensity isoseismic lines tend to be round, and attenuation curves along the long and short axes tend to overlap each other. However, there are differences in the length of long and short axes at an intermediate distance (Wang Suyun et al., 2000). Moreover, isoseismic lines are mainly elliptical in tectonic zones of China. To sum up, the elliptic intensity attenuation model is adopted in this article.

The mathematical expression (Howell B.F. et al., 1975; Yang Yanming et al., 2016) of elliptic intensity attenuation model is as below:

$ I = A + BM - {C_1}\ln \left( {{R_{\rm{a}}} + {R_{{\rm{a0}}}}} \right) - {C_2}\ln \left( {{R_{\rm{b}}} + {R_{{\rm{b0}}}}} \right) - {D_1}{R_{\rm{a}}} - {D_2}{R_{\rm{b}}} + \xi $ (1)

where, I denotes earthquake intensity, R epicentral distance, M earthquake magnitude, R0 near-field saturation factor, parameters A, B, C1, C2, D1 and D2 represent regression constants, among which, C1 and C2 denote the influence of geometric diffusion damping, and D1 and D2 the influence of medium damping. The latter only has an effect on the far-field region, and is often negligible. ξ is a random variable that represents uncertainty in a regression analysis, which is usually assumed to follow a normal distribution with a mean value of zero and standard deviation of σ. Formula(1) can be further simplified to:

$ I = A + BM - {C_1}\ln \left( {{R_{\rm{a}}} + {R_{{\rm{a0}}}}} \right) - {C_2}\ln \left( {{R_{\rm{b}}} + {R_{{\rm{b0}}}}} \right) + \xi $ (2)

According to the non-linear regression theory proposed by Aster R.C. et al., (2013), any inversion can be expressed as below,

$ \mathit{\boldsymbol{G}}\left( \mathit{\boldsymbol{m}} \right) = \mathit{\boldsymbol{d}} $ (3)

where, m stands for model parameter, d observation data and G is construct operator, which is constructed according to m and d. The inversion can be expressed as given the observation data d, and using optimal algorithm to find model parameter m. LM non-linear inversion method is used in this article to find model parameters A, B, C1, Ra0, C2 and Rb0 under minimum weighted residual norms (formula (4)).

$ f\left( \mathit{\boldsymbol{m}} \right) = \sum\limits_{i = 1}^m {{{\left( {\frac{{\mathit{\boldsymbol{G}}{{\left( \mathit{\boldsymbol{m}} \right)}_i} - {\mathit{\boldsymbol{d}}_i}}}{{{\sigma _i}}}} \right)}^2}} $ (4)

Scalar-valued functions and vector-valued function are defined as follows respectively

$ \begin{array}{*{20}{c}} {{f_i}\left( \mathit{\boldsymbol{m}} \right) = \frac{{\mathit{\boldsymbol{G}}{{\left( \mathit{\boldsymbol{m}} \right)}_i} - {\mathit{\boldsymbol{d}}_i}}}{{{\sigma _i}}}}&{i = 1,2, \cdots ,\mathit{\boldsymbol{m}}} \end{array} $ (5)
$ F\left( \mathit{\boldsymbol{m}} \right) = \left[ {\begin{array}{*{20}{c}} {{f_1}\left( \mathit{\boldsymbol{m}} \right)}\\ \vdots \\ {{f_\mathit{\boldsymbol{m}}}\left( \mathit{\boldsymbol{m}} \right)} \end{array}} \right] $ (6)

Gradient is calculated by formula (7)

$ \nabla f\left( \mathit{\boldsymbol{m}} \right) = 2J{\left( \mathit{\boldsymbol{m}} \right)^T}F\left( \mathit{\boldsymbol{m}} \right) $ (7)

The LM method is an iterative algorithm, Δm is calculated according to formula (8), and according to formula (10), model parameters are updated to enter the next iteration process.

$ \left( {J{{\left( {{\mathit{\boldsymbol{m}}^k}} \right)}^{T}}J\left( {{\mathit{\boldsymbol{m}}^k} + \lambda I} \right)\Delta \mathit{\boldsymbol{m}} = - J{{\left( {{\mathit{\boldsymbol{m}}^k}} \right)}^T}F\left( {{\mathit{\boldsymbol{m}}^k}} \right)} \right. $ (8)

where, I denotes unit matrix and J(m) represents Jacobian, which is defined as below:

$ J\left( \mathit{\boldsymbol{m}} \right) = \left[ {\begin{array}{*{20}{c}} {\frac{{\partial {f_1}\left( \mathit{\boldsymbol{m}} \right)}}{{\partial {\mathit{\boldsymbol{m}}_1}}}}& \cdots &{\frac{{\partial {f_1}\left( \mathit{\boldsymbol{m}} \right)}}{{\partial {\mathit{\boldsymbol{m}}_n}}}}\\ \vdots&\ddots&\vdots \\ {\frac{{\partial {f_\mathit{\boldsymbol{m}}}\left( \mathit{\boldsymbol{m}} \right)}}{{\partial {\mathit{\boldsymbol{m}}_1}}}}& \cdots &{\frac{{\partial {f_\mathit{\boldsymbol{m}}}\left( \mathit{\boldsymbol{m}} \right)}}{{\partial {\mathit{\boldsymbol{m}}_n}}}} \end{array}} \right] $ (9)
$ {\mathit{\boldsymbol{m}}^k} = {\mathit{\boldsymbol{m}}^k} + \Delta \mathit{\boldsymbol{m}} $ (10)

When formula (11) is satisfied, model parameters converge, that is, the weighted residual norm is the minimum, and model parameters at this point are the results desired.

$ \nabla f\left( \mathit{\boldsymbol{m}} \right) = 0 $ (11)

According to formula (2) and (5), scalar-valued functions of intensity attenuation model studied in this article are obtained, which is as below

$ {f_i}\left( \mathit{\boldsymbol{m}} \right) = \frac{{A + BM - {C_1}\ln \left( {{R_{{\rm{a}}i}} + {R_{{\rm{a0}}}}} \right) - {C_2}\ln \left( {{R_{{\rm{b}}i}} + {R_{{\rm{b0}}}}} \right) - {I_i}}}{{{\sigma _i}}}\;\;i = 1,2, \cdots ,n $ (12)

According to formula (9), Jacobian is obtained

$ J\left( \mathit{\boldsymbol{m}} \right) = \left[ {\begin{array}{*{20}{c}} {\frac{1}{{{\sigma _1}}}}&{\frac{M}{{{\sigma _1}}}}&{\frac{{ - \ln \left( {{R_{{\rm{a1}}}} + {R_{{\rm{a0}}}}} \right)}}{{{\sigma _1}}}}&{\frac{{ - {C_1}}}{{{\sigma _1}\left( {{R_{{\rm{a1}}}} + {R_{{\rm{a0}}}}} \right)}}}&{\frac{{ - \ln \left( {{R_{{\rm{b1}}}} + {R_{{\rm{b0}}}}} \right)}}{{{\sigma _1}}}}&{\frac{{ - {C_2}}}{{{\sigma _1}\left( {{R_{{\rm{b1}}}} + {R_{{\rm{b0}}}}} \right)}}}\\ {\frac{1}{{{\sigma _2}}}}&{\frac{M}{{{\sigma _2}}}}&{\frac{{ - \ln \left( {{R_{{\rm{a2}}}} + {R_{{\rm{a0}}}}} \right)}}{{{\sigma _2}}}}&{\frac{{ - {C_1}}}{{{\sigma _2}\left( {{R_{{\rm{a2}}}} + {R_{{\rm{a0}}}}} \right)}}}&{\frac{{ - \ln \left( {{R_{{\rm{b2}}}} + {R_{{\rm{b0}}}}} \right)}}{{{\sigma _2}}}}&{\frac{{ - {C_2}}}{{{\sigma _2}\left( {{R_{{\rm{b2}}}} + {R_{{\rm{b0}}}}} \right)}}}\\ \vdots&\vdots&\vdots&\vdots&\vdots&\vdots \\ {\frac{1}{{{\sigma _n}}}}&{\frac{M}{{{\sigma _n}}}}&{\frac{{ - \ln \left( {{R_{{\rm{a}}n}} + {R_{{\rm{a0}}}}} \right)}}{{{\sigma _n}}}}&{\frac{{ - {C_1}}}{{{\sigma _n}\left( {{R_{{\rm{a}}n}} + {R_{{\rm{a0}}}}} \right)}}}&{\frac{{ - \ln \left( {{R_{{\rm{b}}n}} + {R_{{\rm{b0}}}}} \right)}}{{{\sigma _n}}}}&{\frac{{ - {C_2}}}{{{\sigma _n}\left( {{R_{{\rm{b}}n}} + {R_{{\rm{b0}}}}} \right)}}} \end{array}} \right] $ (13)

Formulas (6), (12) and (13) are substituted into formula (8) to get Δm, then the iterative process is proceeded until formula (11) is satisfied, m value at this point is the model parameter desired, and then we get the intensity attenuation relation (formula 2).

According to formula (2), when Rb=0, we get elliptic attenuation relation of seismic intensity along the long axis,

$ {I_{\rm{a}}} = A - {C_2}\ln {R_{{\rm{b0}}}} + BM - {C_1}\ln \left( {{R_{\rm{a}}} + {R_{{\rm{a0}}}}} \right) + \xi $ (14)

which is further written as

$ {I_{\rm{a}}} = \hat A' + BM - {C_1}\ln \left( {{R_{\rm{a}}} + {R_{{\rm{a0}}}}} \right) + \xi $ (15)

when Ra=0, we get elliptic attenuation relation of seismic intensity along the short axis,

$ {I_{\rm{b}}} = A - {C_1}\ln {R_{{\rm{a0}}}} + BM - {C_2}\ln \left( {{R_{\rm{b}}} + {R_{{\rm{b0}}}}} \right) + \xi $ (16)

which is further written as

$ {I_{\rm{a}}} = \hat A'' + BM - {C_2}\ln \left( {{R_{\rm{b}}} + {R_{{\rm{b0}}}}} \right) + \xi $ (17)
3 RESULT ANALYSIS 3.1 Results of the Establishment of Intensity Attenuation Relation in Partitions

In this article, based on the data of 152 isoseismic lines from 65 moderate-strong earthquakes in Inner Mongolia and its adjacent areas, the near-field and far-field data points interpolating is determined, and regression analysis is done for each region using the elliptic intensity attenuation model and LM non-linear inversion method, to establish seismic intensity attenuation relations that are applicable in Inner Mongolia and it adjacent areas (Table 2).

Table 2 Intensity attenuation relations in different research areas in Inner Mongolia and its adjacent areas

Fig. 3 provides the comparison of seismic intensity attenuation relations along the direction of long and short axes in different regions. It can be seen that the intensities of the three intensity attenuation relations are within a reasonable range at the epicenter, and there are differences between attenuation curves in different regions. Near the epicenter, the intensity in the eastern region is significantly higher than that of the central and western regions. With the increase of the epicentral distance, in the condition of the same magnitude, the intensity in the western region attenuates faster than that in the central and eastern regions, which is closer to the actual intensity distribution. The reason is related to site conditions. The western region is different from the central and eastern regions, most of which is located in Gobi and desert. Comprehensive analysis shows that this attenuation relation is more consistent with the distribution characteristics of earthquake hazard in this region, and the results are more reliable.

Fig. 2 Seismic intensity data selected in this study (a) The number of isoseismic lines at different intensity levels; (b) The number of earthquakes in different ranges of magnitude

Fig. 3 Comparison of intensity attenuation relations in different regions of Inner Mongolia and its adjacent areas Figures in the graphs represent surface wave magnitudes. (a) Comparison of intensity attenuation relations along the direction of long axes; (b) Comparison of intensity attenuation relations along the direction of short axis
3.2 Comparative Analysis

At present, few studies have been conducted on the intensity attenuation relationship in Inner Mongolia and its adjacent areas. In the absence of relative research achievements, the module for rapid determination of earthquake influence field of the evaluation system of earthquake damage in Inner Mongolia has been using the intensity attenuation relation model of North China:

$ {\rm{Long}}\;{\rm{axis}}:{I_{\rm{a}}} = 6.046 + 1.480M - 2.081\ln \left( {{R_{\rm{a}}} + 25} \right) $ (18)
$ {\rm{Short}}\;{\rm{axis}}:{I_{\rm{b}}} = 2.617 + 1.435M - 1.441{\rm{In}}\left( {{R_{\rm{b}}} + 7} \right) $ (19)

Due to the lack of historical seismic data in Northeast China, no systematic arrangement has been made before, and the published research results have always lacked local seismic intensity attenuation formulas, thus seismic intensity attenuation model of North China or eastern Chinese mainland is used (Zhang Fan et al., 2014). Besides, previous scholars have not studied seismic intensity attenuation relationships in northeastern Inner Mongolia, therefore, seismic intensity attenuation laws obtained from this study in 3 partitions (the eastern region, the central region and the western region) of Inner Mongolia and its adjacent areas are compared with the intensity attenuation relation in North China that is currently used.

Taking the eastern region as an example, Fig. 4 shows the comparison between seismic intensity attenuation relationships in North China and the eastern region. The R0 value in North China is greater than that in the eastern region, indicating that the intensity in North China attenuates faster than that in the eastern region. Near the epicenter, the intensity of qualified model in North China is significantly higher than that in the eastern region when earthquake magnitude is high (MS≥7.0). When MS=6.0, intensity obtained from this study is close to the intensity value of the attenuation relationship in North China. When earthquake magnitude is relatively low (MS≤5.0), the epicentral intensity of the attenuation relation obtained from this study is significantly higher than that of the qualified model of North China, and the maximum difference between the two is up to 0.5 degrees.

Fig. 4 Comparison of intensity attenuation relations in eastern region and North China Figures in the graphs represent surface wave magnitudes. (a) Comparison of intensity attenuation relations along the direction of long axis; (b) Comparison of intensity attenuation relations along the direction of short axis

When R > 50km, the attenuation curves of North China gradually falls beneath the attenuation curves of the eastern region. Within the distance of 50km-100km, the maximum difference between the two is up to 1 degree. With the increase of the epicentral distance, the intensity in North China attenuates significantly faster than that of the eastern region, and the difference of intensity between them increases correspondingly.

4 CONCLUSIONS

(1) Seismic intensity attenuation relation has strong regional characteristics. In statistical determination of the attenuation relationship, the size of a region and strong motion data considered are two factors that control the rationality and applicability of ground motion attenuation relationships (Wang Xiaojun et al., 2012). In this article, seismic intensity attenuation relations are established for 3 partitions in Inner Mongolia and its adjacent areas by the LM non-linear inversion method, using data of 152 isoseismic lines from 65 moderate-strong earthquakes in Inner Mongolia and its adjacent areas, which further improves the accuracy of the determination of earthquake influence field of destructive earthquakes and the reliability of the seismic damage assessment models. Because seismic data of Inner Mongolia and its adjacent areas are used, the accuracy of radius of long and short axes of isoseismic lines calculated by earthquake damage assessment software is also improved.

(2) The LM algorithm is a kind of non-linear inversion method, which is suitable for use with fewer samples, and compared with the traditional linear least square method, it has higher inversion precision and can be less affected by sample outliers. The application of the LM non-linear inversion method to the study of the intensity attenuation relationship is more suitable for regions with relatively little historical seismic data.

(3) Epicentral intensities of intensity attenuation relations in the eastern, central and western regions are all within a reasonable range, and there are differences between attenuation curves of different regions. Near the epicenter, the intensity in the eastern region is significantly higher than that in the central and western region. With the increase of epicentral distance, in the condition of the same magnitude, the intensity in the western region attenuates faster than that in the central and eastern regions, which is closer to the actual intensity distribution.

(4) Comparative analysis of the research results in this article and existing seismic intensity attenuation relation of North China shows that the intensity in the eastern region obtained from this study is higher than that in North China when earthquake magnitude is relatively lower, while the intensity in the eastern region is relatively low when earthquake magnitude is higher. On the whole, the intensity attenuation in the eastern and western regions is rather slow according to this study; therefore, seismic intensity attenuation of strong earthquakes is better reflected in this region, which is more consistent with the regional characteristics of this region.

ACKNOWLEDGEMENTS

Thanks to research professor Shuai Xianghua of China Earthquake Networks Center and teachers from the Expert Group of Key Youth Task of Earthquake Emergency Response in 2016 for their comments and suggestions given in the course of this project. Professor Yao Huajian of the University of Science and Technology of China has given guidance and help during the research process. Mr.Zheng Zhijiang of the First Monitoring and Application Center of China Earthquake Administration has provided great support for this research and reviewers have reviewed this article in detail and put forward a lot of constructive and valuable comments, here the authors express their sincere gratitude.

This paper has been published in Chinese in the journal of Earthquake, Volume 37, Number 3, 2017.

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基于Levenberg-Marquardt方法的内蒙古及邻区地震烈度影响场改进技术
杨彦明1, 姜立新2, 王祯祥1     
1. 内蒙古自治区地震局,呼和浩特 010010;
2. 中国地震台网中心,北京 100045
摘要:在收集、整理内蒙古及邻近地区1940年以来Ms 4.0以上历史震害资料的基础上,对研究区域进行合理分区,选取65次中强地震事件152条等震线数据,利用椭圆烈度衰减模型,采用Levenberg-Marquardt非线性反演方法,建立内蒙古及邻区的分区地震烈度衰减关系公式,通过与地震灾害评估系统使用的华北地区地震影响场模型进行对比分析,综合分析研究认为,建立分区烈度衰减关系更符合研究区域地震灾害的地域性特点,对于地震灾害快速评估具有重要的参考意义。
关键词烈度    地震影响场    内蒙古    Levenberg-Marquardt    衰减关系