Moment Tensor Components with Their Corresponding Eigen Values and Eigen Vectors for Earthquakes Mw ≥ 6.0 in Northern and Southwestern of Pakistan

Journal homepage: http://ejournal.upi.edu/index.php/AJSE/ © 2021 Universitas Pendidikan Indonesia ASEAN Journal of Science and Engineering 1 (2) (2021) 79-96


INTRODUCTION
Pakistan is located on the dynamic seismic belt where folds and faults have been observed due to the intra-plate collision between the Indian and Eurasian plates (Kazmi & Jan, 1997;Khalid et al., 2016). These plates are continuously moving towards north at the speed of 15 to 20 cm/year (Patriat & Achache, 1984;Scotese et al., 1988). The Himalayan Fold and Thrust Belt, Kohistan Magmatic Arc, Suleiman Fold and Thrust Belts are the evident features generated under the action of this collision (Powell, 1979). The seismo-genic stresses are acting on the northern side of Pakistan whereas on the western side these two plates slide pass with each other producing Chamman Fault at Quetta. In the southern part the continental and oceanic plates are colliding with each other to form Makran triple junction called Makran Seismic Zone (MSZ). Transform faulting has dynamic position in this compressional region (Sercombe et al., 1998;Mona Lisa et al., 2004). The Himalayas Mountain Range and dynamic convergent plate boundary in this area is tectonically highly under stress (Kazmi & Jan, 1977). Therefore, Kohistan Ladakh, Nanga Parbat contact shows the flexible shear zone (Treloar et al., 1991). The main tectonic components in this area are Main Mantle thrust (MMT), Main Karakorum thrust (MKT), Dasu-Sassi Fault, Upper Hunza Fault, Krakoram Fault, Trichmir Fault, Nanga Parbat and Raikot Sassi Fault as shown in tectonic map of Pakistan (Figure 1). MKT and MMT represent by Shyok Suture and Indus Suture zones respectively (Yeast & Lawarence, 1984).
The seismic energy accumulates into the earth's crust and released from faults during an earthquake. Stress analysis and fault plane solution (Presti et al., 2013;Khalid et al., 2016) are tools in practice to analyze the orientation and geometry of the earthquake related features. The fault plane solution describes by the first P wave motion from the nearest station and remaining far stations measure the solution using waveform analysis (Pondrelli et al., 2006;Scognamiglio et al., 2009). This solution is based on the stress moment tensor either it is single couple or double couple stress tensor. All orientations of earth are being formed after earthquake geometrically shown on the earth surface by beach ball solution. The forces applied vertically produces normal faulting, which is due to tensional stress and behaves like a single couple moment tensor. On the other hand, when the forces acting on both sides of the block, horizontally reverse faulting produced under the compressional regime. Transform faulting as well as strike slip faulting produced the double couple moment. Their stress values produced the nine independent stress orientations. In this paper, we will discuss the tectonic stress orientation as well as fault geometry in Pakistan and adjacent area with respect to the significant earthquakes.
Moment tensor analysis has been utilized for decades as a tool for understanding the faulting process during earthquakes (Guilhem et al., 2014). This analysis provides unique information regarding the fault plane orientations (fault plane solution) and the size of the seismic events of all magnitudes by assuming that these events can be represented by a point source for long wavelengths and relatively larger distances between the epicenters and the recording stations.

STRESS ANALYSIS AND MOMENT TENSOR
The stress produced under the action of applied forces has nine directional stress components. Three stress components are called principal or normal stresses denoted by σx, σy, σy and remaining six components are called shear stresses denoted by τxy, τyx, τyz, τzy, τzx, τxz. The generalized double couple phenomenon to nine possible couple of forces is demonstrated in Figure 2 described by (Aki & Richard, 2002).

Anderson model of stress distribution
To study the directional stresses in lithosphere the model proposed by Anderson (1951) is used (Figure 3). This model shows these directional stresses act on the body in the form of Sv, SHmax and SHmin at depth of S1, S2 and S3 respectively. Where Sv is vertical stress due to overburden, SHmax the maximum principal horizontal stress and SHmin is the minimum principal horizontal stress.
In the Anderson model, horizontal principal stresses are always unequal. It is either be less or greater than the vertical stress. When the vertical stress is more in divergent movement area (S1=Sv), the gravity promotes the normal faulting. Whereas when SHmax and SHmin go over the Sv (S3=Sv) the compression regime (thinning) is entertained the reverse faulting. (Barba et al., 2010) Finally, the transform faulting shows the intermediate stress behavior (S2=Sv). This is the stage where horizontal stress is greater than the overburden vertical stress. So mathematically, expression for the reverse faulting described as SHmx≥Sv≥SHmin. The overburden stresses along the fault line inside the earth at any location also a major cause of earthquake. After the earthquake, the fault can be express geometrically on the earth surface using beach ball. To find out the earthquake focal mechanism, study of stress analysis and seismogenic forces operating in the region is the essential part (Montone et al., 2004;Zoback & Zoback, 1980). The fault divides into two planes. These planes are formed by three major components i.e., strike lies between 0 0 to 360 0 , dip ranges from 0 0 to 90 0 and rake/slip lies in -180 0 to 180 0 . Nodal Plane 1 and Nodal Plane 2 also called fault plane and auxiliary plane respectively. These planes contain two major axes: one is "P" pressure axis and other "T" tensional axis. The third axes M shows the movement plane. A stress map was drawn using world stress map (WSM) database that indicates the directional forces and their fault DOI: http://dx.doi.org/10.17509/xxxx.vxix p-ISSN e-ISSN mechanism. Only A, B and C quality (add refer) data was used in stress map of Pakistan bounded by coordinates from (24 0 -38 0 latitude) and (60 0 -78 0 longitude). The CASMO stress map is showing the stress indication of faults. In the stress map, different colors were used for different regimes identification. Lambert projection was used for preparing the stress map with their focal mechanism inversion.

METHODOLOGY AND DATA SET
To study the moment tensor inversion and focal mechanism earthquake stress orientation 14 number of events were selected for self-generated earthquake catalogue (SGEC) from the local and international earthquake databases like Pakistan meteorological department (PMD), United State Geological Survey (USGS) and centroid moment tensor (CMT). The epicenter, focal depth and magnitude of each event with their unique ID are presented in Table 1.
Each event contains the information about the seismic moment, rupture length, fault length, magnitude and displacement (Wells & Coppersmith, 1994). The moment tensor solution describes the nine couple forces in the Cartesian coordinate system as given in the equations (1) -(7). In the double couple source, rectangular or Cartesian component can be written in the form of direction cosine using strike, dip and rake of the fault plane and the scalar seismic moment Mo (Aki & Richards, 2002). We are familiar about fault parameters i.e., strike (φ), dips (δ) and rake/slip (λ). The numerical values source parameters can be found out by using moment tensor directional components (Aki & Richards, 2002).
where Mo is the seismic moment defined as: here μ is modulus of rigidity typically taken as 3×10 10 Nm -2 for crust and 7×10 10 Nm -2 mantle, D is displacement of fault and A is rupture area of the fault zone measures in (m 2 ). Mo can also be calculated from moment magnitude Mw by (Hiroo & Kanamori, 1979).
The source parameters of the significant earthquake are describing using inversion of moment tensor (Bock, 2002). Focal mechanism or fault plane solution, explain by beach ball using polarity of P wave first motion at the observatory station which produce the strong relation between earthquake and seismic faults geometry (Anderson et al., 1993). The fault parameters of nodal plane 1 and Nodal plane 2, azimuth and plunge values determine by MATLAB code for seismic inversion (Efthimios & Sokosa, 2008).

Eigen vectors and eigen values of moment tensor solution
Stress tensor is a linear operator which provide the traction of vector t from normal vector n^  n . In the seismology, seismologist always used stress tensor in the form of Cartesian geometry as 3x3 matrix. This is fully description of stress tensor in Cartesian coordinate system. To describe the fully state of stresses, stress tensor contained six independent components at any points in medium. In the field of science and technology eigenvectors and eigenvalues are frequently used. Specifically, in the field of matric transformation, system of simultaneously equations, mathematical differential expression, tensor analysis, stress orientation and axis transformation, eigenvectors and eigenvalues are imperative. In physics, earth rotation model (geodesy), geophysical simulation of earth motion belongs to cubic polynomial called characteristic polynomial which have root. These roots are called eigen values and their vectors are called eigen vectors. in coordinate system rigid bodies and their orientation with respect to their axis transform one axis to another called tensor transformation. The complicated shapes of rigid bodies definitely have some rotational direction. This phenomenon is called axis of inertia and these are determined by obtaining eigen values and eigen vectors. This inertia tensor also called moment of inertia. In any medium for any stress tensor it is always possible to determine the direction of n vector such as there is no shear stress across the plane to normal n. In rock mechanics, +ive sign represents the compressional forces and tensional forces denoted by -ive. Traction of any vectors defined by  t (Figure 4).
n t n t t is normal to the plane called normal stress and its parallel stress called shear stress. Thus, according to the Shearer (2009), where I is identity matrix and λ is scalar quantity. This has non-trivial solution and give the eigen values and their corresponding eigen vectors when it is a nonsingular matrix i.e., det[A]=0. Such that This give the cubic characteristics polynomial and have three solutions in which λ1, λ2, λ3 are the eigen values. λ1, λ2, λ3 are simply the principal stresses. Eigen matrices/Eigen vectors used to determine the numerical values of individual component. These vectors problems are usually solved by MATLAB or Mathematica software or simply solve the mathematical expression of Eq. 12. These source parameters are utilizing to find the moment tensor values, which can be written in 3x3 matrixes as shown in matrix A. Seismic moment tensor is always symmetric. Therefore, the Mxy and Myx are always equal due to the same magnitude but opposite direction (Aki & Richard, 2002). ) where and τ are the shear and normal stresses. Moment magnitude (M w ) was consider in whole analysis yet the other magnitude can be found using conversion by different author.

RESULT AND DISCUSSION
Tectonically Pakistan seized by hidden and visible forces which based on the velocity of plate motion of Indian and Eurasian. Although the situation is concerned with the seismogenic behavior however this may a lead to stress dispersion inside crust.
In this work, time domain moment tensor inversion is applied to significant earthquakes (MW ≥ 6.0) occurred in the northern and southwestern parts of Pakistan. A self-generated earthquake catalogue (SGEC) was prepared, which constitutes 14 significant seismic events. The stress tensor transformation in the form of components was obtained to determine the Eigen values and Eigen vectors of direction of each event. The seismo-tectonic map of Pakistan and its surrounding region was drawn in Figure 5 which shows that seismicity distribution of selected events and local tectonic setting. Shallow earthquake can be seen in northern and northwestern side whereas the depth particularly in Hindukush region and Gilgit Balitistan. Earthquakes with depth more than 270 km were also observed in the area extreme towards China, Uzbekistan, and Afghanistan.

Stress orientation and focal plane solution
The stress orientation under the action of various tectonic activities in and around the Pakistan presented in stress map (Figure 6). The faults produced under the action of these seismic sources which classified into strike slip, thrust and normal faults described by different color schemes. Strike slip (SS) and thrust faults (TF) are dominant tectonic features in the southwestern part of Pakistan. Normal faults (NF) and TF are dominant fault systems in the northern part of the country, indicate the continuous collision between the Indian and Eurasian Plates, thus compressional regime is present. In southern part, the Arabian plate slides pass to the Indian plate, thus producing the strike slip faulting on axial belt. The most prominent fault in this region is the Chamman Transform Fault. Stress map shows the maximum horizontal stress (SHmax), acting in the compressional zone, generated the reverse faulting, the condition SHmax>SHmin will satisfy whereas no fault system is observed in the eastern side of Pakistan due to far distance from the collision zone thus, SHmax< SHmin holds correctly.   The focal plane solution of each seismic event is drawn in the form of beach ball (Figure 7). The parameters used for focal plane solution of each seismic event are presented in Table 2. The green beach balls represent the reverse fault at compressional regime due to the collision in the north side. Orange color beach ball shows the transform boundary due the Chamman Fault on the axial belt on the western side of the country and yellow balls show the behavior of normal faulting in the region of Pak-Iran border (Iran block) and the Makran subduction zone between the Arabian and Eurasian plates.

Example for conversion of tensor components to eigen values.
Stress tensor components can be described by 3x3 matrix as mention in the matrix A. All obtained values of stress tensor can be written in square matrix of order 3. These square matrixes have non trivial solution and their solution are real and their values are in numerical form. Eigen values and vectors are Cartesian representation of characteristic polynomial. To obtained the values and their corresponding vectors, we take ID Event 1A as example. The eigen values and eigen vectors were determined using mathematical and their characteristics polynomial can also be derived on piece of paper. As we know all stress tensor components can be written in 3x3 matric therefore the matrix of the event ID 1A will be expressed as which is non-singular matrix and has non trivial solution. Thus, using Mathematica 7.0 The solution of this matrix in the form of eigen values their corresponding eigen vector is given below using solving the matrix (3x3) by system of linear equations. Therefore, augmented matrix will be form of order (3x1) matrix. Eigen values and eigen vectors of event 1A explain below: For real eigenvalue λ1 = −603.57, the eigenvector is Similarly, all the remaining matrixes can be transform to their eigen values and eigen vectors. Point to be noted that all matrixes are non-singular, therefore solution of all stress tensor components are exist in the form of eigen vectors and eigen values given in Appendix A at the end.

Crustal thickness
The thickness of the earth crust varies largely in Pakistan from 10 -70 km in the northern side of the country whereas in the southern side its thickness is merely 5 -8 km. The devastating earthquakes occur there. This is highly vulnerable area for Tsunami and their triggering process. The lithosphere thickness varied from southwestern to North as oceanic towards continental crust thickness. The earth crustal model showing in Figure 8 Pakistan region in the northern part continental crust is thicker i-e 60-70 km than the oceanic crust thickness varied from 0 to 10 km with Arabian sea. Therefore, deep earth causes low damage and low intensity whereas shallow earthquakes have very dangerous and their consequences are very unpleasant. Seismic activity in the northern part is very intense due to forceful impact of tectonic plates causes formation of Himalayan, Hindu Kush and Karakorum mountain ranges. Thinner crust causes to produce the volcanic activity which is immense process due to seafloor spreading. This phenomenon causes to generate the Tsunami "inside disturbance of oceanic crust".

CONCLUSION
Tectonically Pakistan comprised by the compressional and extensional regime. The stresses around and inside the Pakistan due to tectonic boundary passing inside Pakistan. To observe the direction / orientation of these hidden forces, a self-generated earthquake catalog (SGEC) was prepared contained 14 events Mw≥6.0. Most of these events were on the Afghanistan side. The direction of beach ball expresses the azimuth towards North to west mostly were strike slip (SS) and normal faults (N) and these areas lies on Main Boundary Thrust (MBT). On the northern due to thrust faulting, all faults are reverse and beach ball direction shows towards north, which lies between the Main Mantle Thrust (MMT) and Main Karakoram Thrust (MKT). Pakistan is situated where continental and oceanic plates are moving into different direction. CASMO stress map was used to observe the stresses orientation of hidden forces which describe fault orientation easily. According to stress map, SHmax stress is maximum at plate boundary or collision junction of tectonic plates at shallow depth and SHmin where stresses reduce away from the collision but vertical principal stress will be all around due to overburden pressure of rocks here depth and vertical stress will be equal i-e S3=SV. So, the condition is Shmax>Smin>SV and SHmax>SV. we have enabled to analyzed where the maximum and minimum directional stress is applied in the compressional and extension tectonic regimes. A matlab code was used to drive the values of fault parameters such as Azimuth, Plunge strike, dip and rake. Using azimuth and plunge values ArcGIS tools was used to draw the beach ball or fault plane solution to understand the stress orientation. Using Aki and Richard (2002) mathematic equations obtained all stress components of shear and normal stresses numerical values in the form of 3x3 matrix. The numerical values of stress components evaluated using equations 2 to 7 written in the form of 3x3 matrixes. After determine the non-singularity of these matrix, their eigen values and their corresponding vectors are also calculated using Mathematica 7.0 given in the Appendix A. The earth crustal thickness showing Pakistan region in the northern part continental crust is thicker i-e 60-70 km than the oceanic crust thickness varied from 0 to 10 km with Arabian sea. Therefore, deep earth causes low damage and low intensity whereas shallow earthquakes have very dangerous and their consequences are very unpleasant. Seismic activity in the northern part is very intense due to forceful impact of tectonic plates causes formation of Himalayan, Hindu Kush and Karakorum mountain ranges.

ACKNOWLEDGEMENTS
Catalog was prepared using sources such as Pakistan meteorology department (PMD), united states geological survey (USGS), International seismological center (ISC), IRIS and ETH Zurich.

AUTHORS' NOTE
The authors declare that there is no conflict of interest regarding the publication of this article. The authors confirmed that the paper was free of plagiarism.

APPENDIX A
The computed stress tensor components written in the form of 3x3 matrixes given below with their eigen values and their corresponding eigen vectors.