Earthq Sci (2009)22: 609−614 609 Doi: 10.1007/s11589-009-0609-x Estimation of the 2001 Kunlun earthquake fault slip from GPS coseismic data using Hori’s ∗ inverse method Honglin Jin and Hui Wang Institute of Earthquake Science, China Earthquake Administration, Beijing 100036, China Abstract The Hori’s inverse method based on spectral decomposition was applied to estimate coseismic slip distribution on the rupture plane of the 14 November 2001 MS8.1 Kunlun earthquake based on GPS survey results. The inversion result shows that the six sliding models can be constrained by the coseismic GPS data. The established slips mainly concentrated along the eastern segment of the fault rupture, and the maximum magnitude is about 7 m. Slip on the eastern segment of the fault rupture represents as purely left-lateral strike-slip. Slip on the western segment of the seismic rupture represents as mainly dip-slip with the maximum dip-slip about 1 m. Total predicted scalar seismic moment is 5.196×1020 N⋅m. Our results constrained by geodetic data are consistent with seismological results. Key words: Kunlun MS8.1 earthquake; coseismic GPS data; fault slip; inversion CLC number: P315.72+5 Document code: A placements at more than 40 sites along the fault rupture, 1 Introduction the epicenter, the distribution of surface rupture and co- seismic GPS data, etc. The research fields include geo- The 14 November 2001 MS8.1 Kunlun earthquake occurred in the west of Kunlun mountain pass in Qing- logical investigation, seismometry, level survey and hai province. It is the largest earthquake that occurred on GPS survey, etc. Field investigation found that the the east Kunlun fault. The surface rupture produced by maximum surface left-lateral coseismic dislocation is − the event was the longest one among historical earth- 6 7 m, and the maximum vertical displacement reaches quakes. It spans from west of the Hoh Sai Lake to east 4 m (Xu et al, 2002; Chen et al, 2003). However, field of the Kunlun mountain pass. The western end of the investigation just provided data on several sites along fault rupture is located near the Buka Daban Peak, and the fault rupture, not giving deep slip on the rupture the eastern end of the fault rupture is located 70 km east plane. Wan et al (2004, 2008) inverted distribution of of the Kunlun mountain pass. The total length of the deep slips on fault plane basing on geodetic data by us- fault rupture is about 426 km. The maximum horizontal ing Okada’s method. In Okada’s inverse method, dis- dislocation along the fault rupture is more than 6 m. The crete observation data were directly used to estimate slip maximum vertical dislocation is less than 1 m (Xu et al, distribution with some prior information. As a result, the data conditions, such as data density, distribution and 2002). The 1973 M7.3 Mayi earthquake, the 1997 MS7.5 Mayi earthquake, the 1937 M7.5 Huashixia earthquake, accuracy, directly affect inversion results. So, their re- sults are rough and cannot constrain detailed fault slip. and the 1963 MS7.1 Dulan earthquake occurred along the east Kunlun fault. The mechanisms of these earth- The representation theorem in elastostatics relates quakes are consistent with local deformation pattern surface displacements to slip distribution on a fault sur- (Shen et al, 2003). face through a linear integral equation that uses Green’s Previous studies have provided plenty of data on function (Maruyama, 1964; Okada, 1985). Therefore we can set up an inverse problem for reconstructing the the MS8.1 Kunlun earthquake. The data include dis- static image of a seismic or aseismic source from ob- ∗ served geodetic data. The Hori’s inverse method that Received 20 July 2009; accepted in revised form 30 October 2009; published 10 December 2009. based on a spectral decomposition of the Green’s func- Corresponding author. e-mail: [email protected] 610 Earthq Sci (2009)22: 609−614 tion, is similar to the SVD (singular-value decomposi- ated with k eigenvalues. tion) analysis method (Menke, 1984; Parker, 1977, 1994; ∞ = αϕ α + Kirsch, 1996). As a continuous inverse method, the Ho- u(x) ∑u (x) noise (3) α =1 ri’s method separates the inverse into two steps: firstly ∞ α u α calculate an inverse operator based on given domains of p( y) = ψ ( y) (4) ∑ α surface and fault structure; and secondly estimate the α=1 λ α α slip distribution from observed data. Moreover, the me- In above equations, u are coefficients of ϕ (x), and k is thod does not require any prior information about the measurable number of the eigenvalues (or modes). The smoothness of fault slip distribution since it assumes second term in equation (3) contains modes given by that the response function and source function are in- α ϕ (x) for α > k, but they cannot be distinguished from trinsically smooth. errors or noises due to the limited accuracy. Therefore, In order to get detailed slip distribution function on even though a response function cannot be measured, it fault plane, we will apply Hori’s method to invert the is possible to accurately estimate a continuous function coseismic slip distribution using observed GPS data. from a set of discrete data by using a set of eigenfunc- First, we introduce the inverse analysis method, then, α tions. The coefficients {u } are estimated by numerical invert the coseismic slip distribution of 2001 Kunlun computation, generally using the least squares method. earthquake using coseismic GPS data. Finally, we dis- Detailed interpretation for the formulation and the pa- cuss the tectonic implication of our results. rameters used refer to Hori (2001) and Jin et al (2007). 2 Inversion method based on a spectral 3 Estimation of back-slip distribution decomposition of Green’s function 3.1 Surface and fault structure We assume the medium to be an elastic half-space. First, we determine the domains of the surface and Let S be the surface plane and F be the fault plane, and the fault structure. Fault structure is determined from denote the points on S and F by vectors x and y, respec- geological results. Referring to previous researches tively. According to the representation theorem in elas- (Chen et al, 2003; Tan et al, 2004; Wan et al, 2004; Xu tostatics (Maruyama, 1964; Yabuki and Matsu’ura, and Chen, 2004), we take the depth range of the source 1992), the static surface displacements u(x) are ex- domain as 0.5−25 km to fully cover the rupture fault pressed as region. The fault rupture length and dip were taken as u(x) = g(x, y) p( y)dy (1) 390 km and 86°, respectively. ∫F The surface S and fault F are decomposed into N in terms of source function p( y) (slip or back slip) and and M segments. The convergence is checked by in- Green’s function. According to function analysis theory, creasing N and M which correspond to the discretization the Green’s function g(x, y) is regarded as a compact for surface and fault. It is important to examine the nu- operator that maps from source to response, and the op- merical validity of spectral decomposition. We note that erator admits the spectral decomposition (Hori, 2001) Green’s function must be discretized using the (N, M) ∞ points by employing a set of smooth functions when α α α g(x, y) = ∑λ ϕ (x)ψ ( y), (2) spectral decomposition is obtained numerically. α =1 A smoothed fault is plotted on a grid with 2 106 α α α where λ is the α-th eigenvalue, ϕ (x) and ψ ( y) are the elements for numerical computations. In this study, we associated eigenfunctions of the surface and the fault, use the point dislocations embedded in the smoothed respectively. The α-th eigenvalue is positive value that curved surface, instead of multi-rectangular fault planes α α converges to zero as α increases, and ϕ (x) and ψ ( y) as used in the previous studies. form orthonormal basis functions on S and F respec- The Earth’s surface is represented by a curved tively. plane of 870 km×610 km along the rupture fault and the u(x) is expressed as a linear combination of the surface was gridded into 3 315 elements for numerical 1 2 k measurable modes {ϕ (x), ϕ (x), ⋅⋅⋅, ϕ (x)}, and p( y) is computations. With the selected set of discrete points expressed as the linear combination of source eigen- representing the surface and fault domains (Figure 1), 1 2 k functions {ψ ( y), ψ ( y), ⋅⋅⋅, ψ ( y)}, which are associ- we calculate the Green’s function, g(x, y), and separate Earthq Sci (2009)22: 609−614 611 it into a set of surface and fault eigenfunctions by spec- α − α ⋅⋅⋅ changes ( ui+1 ui ) with different i (i =1, 2, , 25). The tral decomposition. Note that the surface deformation smaller coefficient of variation tends to more stable. caused by fault coseismic slip can be expressed by a Figure 3 shows stability of coefficients u1 and u2, from linear combination of surface eigenfunctions, and then which we can found the suitable k value with coefficient slip distribution is expressed by a linear combination of uk is 6. If k >6, large error resulted from the instable co- fault eigenfunctions. efficients will be introduced to inversion result. Once k=6 is determined, we can estimate the surface dis- placements function, as shown in Figure 2 (blue). It means the available GPS data can extract slip modes up to 6 for the given fault structure.