Study on Some Recent Earthquakes of Sikkim Himalayan Region and Construction of Suitable Seismic Model: A Mathematical Approach

Abstract

Sikkim Himalayan region lies between Nepal–India border in the west and the Bhutan Himalaya in the east. The region is known to be characterized by strike-slip motion on certain deep-rooted faults. In the past, the region has experienced several devastating earthquakes, namely April 25, 2015 Nepal earthquake (M: 7.8); September 18, 2011 Mangan (Sikkim) earthquake (M: 6.9); February 14, 2006 Sikkim earthquake (M: 5.3), and the like. The present study mainly focuses on few major shocks and their source mechanism to explain properly the process of tectonics. A numerically stable computational scheme, using method of eigenfunction expansion has been used in the study to compute surface response or theoretical seismogram in a layered vertically stratified media overlying a half-space. Simple dislocation source model has been considered. The transverse (SH) and vertical (P-SV) components of displacement field have been computed exactly as summed modes by propagator matrix approach using Runga–Kutta method of order 4. The present result has been compared with the observed seismograms. The overflow error appearing in the numerical computation has been prevented by approximating layer matrices suitably or using generalized R/T (Reflection and Transmission) coefficients. The numerical result has been represented here graphically. The study has an advantage to get an idea of real earth structure or seismic source model by an inverse iterative technique.

Appendices

Appendix 1

The free elastodynamics equation for an isotropic multilayered half-space can be expressed in terms of three-dimensional displacement vector, $\mathop u\limits^{ \to } (\mathop x\limits^{{}} ,y,z;t)$ as

$$ \mathop \nabla \limits^{ \to } [(\lambda^{j} + 2\mu^{j} )\mathop \nabla \limits^{ \to } .\mathop u\limits^{ \to } ] - \mathop \nabla \limits^{ \to } \times [\mu^{j} \mathop \nabla \limits^{ \to } \times \mathop u\limits^{ \to } ] + 2[(\mathop \nabla \limits^{ \to } \mu^{j} \mathop \nabla \limits^{ \to } )\mathop u\limits^{ \to } + \mathop \nabla \limits^{ \to } \mu^{j} \times (\mathop \nabla \limits^{ \to } \times \mathop u\limits^{ \to } )] = \rho^{j} \frac{{\partial^{2} \mathop u\limits^{ \to } }}{{\partial t^{2} }}\quad {\text{for}}\;j = 1,2,3, \ldots ,{\text{N}},{\text{N}} + 1 $$

(6)

where the depth $z^{(j)}$ of the lower jth layer boundary satisfies the relation

$$ 0 = z^{(0)} < z^{(1)} < \cdots < z^{(N)} < z^{(N + 1)} = + \infty $$

(7)

and $\mathop \nabla \limits^{ \to } \equiv (\mathop i\limits^{ \wedge } \frac{\partial }{\partial x} + \mathop j\limits^{ \wedge } \frac{\partial }{\partial y} + \mathop k\limits^{ \wedge } \frac{\partial }{\partial z})$, $\lambda^{j} ,\mu^{j}$ and $\rho^{j}$ are, respectively, the elastic moduli and density of the jth layer.

The dynamic displacement–stress vectors ($U_{p}^{j} (z,h,k)$ and $D_{p}^{j} (z,h,k)$, p = PSV or SH) in the jth layer of a layered half-space media can be expressed in terms of down and up going P and S waves by using modified R/T coefficients [12] as

$$ \left( \begin{aligned} U_{p}^{j} (z,h,k) \hfill \\ D_{p}^{j} (z,h,k) \hfill \\ \end{aligned} \right) = \left( {\begin{array}{*{20}c} {E_{11}^{j} } & {E_{12}^{j} } \\ {E_{21}^{j} } & {E_{22}^{j} } \\ \end{array} } \right)\left( {\begin{array}{*{20}c} {\Lambda _{d}^{j} (z)} & 0 \\ 0 & {\Lambda _{u}^{j} (z)} \\ \end{array} } \right)\left( \begin{aligned} C_{d}^{j} (h) \hfill \\ C_{u}^{j} (h) \hfill \\ \end{aligned} \right) $$

(8)

where $C_{d}^{j} (h)$ and $C_{u}^{j} (h)$ are, respectively, the down and up going coefficients and $E^{j}$ as layer matrix in the jth layer. A time-dependent stress discontinuity $\Delta (t)$, at a depth “h” below the surface has been considered in the source layer S as

$$ \left( \begin{aligned} U_{p}^{S + } (h) \hfill \\ D_{p}^{S + } (h) \hfill \\ \end{aligned} \right) = \left( \begin{aligned} U_{p}^{S - } (h) \hfill \\ D_{p}^{S - } (h) \hfill \\ \end{aligned} \right) + \left( {\begin{array}{*{20}l} 0 \hfill \\ {\Delta (t)} \hfill \\ \end{array} } \right),\left( {p = PSV\,or\,SH} \right) $$

(9)

where S+ and S− are, respectively, the sub-layers below and above the source.

Now the surface of the layered media is stress-free and applying the stress-free boundary conditions at the surface z⁽⁰⁾ = 0, the secular equation in the layered media can be expressed as

$$ (I - R_{u}^{(0)} \overline{R}_{d}^{(1)} ) = 0 $$

(10)

where I is either a 2 × 2 identity matrix (Rayleigh wave) or 1 (Love wave). For a given frequency ($\omega$), only a set of finite number of wave numbers $\left( {k_{n} ,\,n = 0,1,2, \ldots ,M\left( \omega \right)} \right)$, are the roots of the secular function $(I - R_{u}^{(0)} \overline{R}_{d}^{(1)} )$ and the eigen displacements $W(\omega ,k_{n} ,z)$ can be evaluated at the roots of the secular function.

Appendix 2

In a multilayered half-space, the differential equation satisfied by the transformed displacement field (U(ω, k), V(ω, k), W(ω, k)) in the cylindrical coordinate system is singular in nature. From the theory of the partial differential equation, it follows that the spectrum of surface wave dispersion equation consists of a finite number of real discrete eigen spectrum $U_{n} (\omega ,k,z)$ and continuous eigen spectrum, also called improper eigen spectrum, $\phi (\nu ,z)$ due to the branch cut (Γ) for Rayleigh wave and continuous and discrete spectrum $\psi (\nu ,z)$ and $W_{n} (\omega ,k,z)$, respectively, for Love wave [23]. The orthogonal property holds between the discrete eigen displacement $U_{n} (\omega ,k,z)$ and the continuous eigen displacement $\phi (\nu ,z)$ and among themselves, i.e.,

$$ \begin{aligned} & \left\langle {U_{n} (k,\omega ,z),U_{l} (k,\omega ,z)} \right\rangle = \int\limits_{0}^{\infty } {\rho (z)U_{n}^{t} (z)U_{l} (z){\text{d}}z\delta_{nl} } \\ & \left\langle {U_{n} (k,\omega ,z),\phi (\nu ,z)} \right\rangle = \int\limits_{0}^{\infty } {\rho (z)U_{n}^{t} (z)\phi (\nu ,z){\text{d}}z = 0} \\ & \left\langle {\phi (\nu ,z),\phi (\nu^{{\prime }} ,z)} \right\rangle = \int\limits_{0}^{\infty } {\rho (z)\phi^{t} (\nu ,z)\phi^{ * } (\nu^{{\prime }} ,z){\text{d}}z = \delta (\nu - \nu^{{\prime }} )} \\ \end{aligned} $$

(11)

where $\phi^{ * }$ is the complex conjugate of $\phi$.

Similar type of relations between discrete and continuous eigen spectrum, $W_{n} (\omega ,k,z)$ and $\psi (\nu ,z)$ for Love wave can be deduced. It is to be noted that $U_{n} (z)$ has been written for $U_{n} (\omega ,k,z)$ and $\nu$ represents the integration variable in continuous eigen displacement instead of $\omega^{2}$.

The discrete and continuous orthogonal eigen spectrum form a complete system while individually each one is not a complete set. The completeness property implies that the transformed radial and vertical displacement field for the Rayleigh wave (i.e., $U(\omega ,k,z)$) and also for the Love wave (i.e., $W(\omega ,k,z)$) can be expressed in terms of the complete set of eigen spectrum. Thus, considering only the Rayleigh wave displacement

$$ U(\omega ,k,z) = \sum\limits_{n} {c_{n} U_{n} (z) + \int\limits_{\Gamma } {c(\nu )\phi (\nu ,z){\text{d}}\nu } } $$

(12)

Operating both sides of the above equation by the operator L_R which is defined as

$$ L_{R} U = \frac{d}{dz}\tau - kB^{t} \frac{dU}{dz} - k^{2} CU $$

(13)

where $\tau (k,\omega ) = A\frac{dU}{dz} + kBU$ and

$$ A = \left( {\begin{array}{*{20}l} {\mu (z)} \hfill & 0 \hfill \\ 0 \hfill & {\lambda (z) + 2\mu (z)} \hfill \\ \end{array} } \right),B = \left( {\begin{array}{*{20}l} 0 \hfill & { - \mu } \hfill \\ \lambda \hfill & 0 \hfill \\ \end{array} } \right),C = \left( {\begin{array}{*{20}l} {\lambda (z) + 2\mu (z)} \hfill & 0 \hfill \\ 0 \hfill & {\mu (z)} \hfill \\ \end{array} } \right) $$

(14)

and using the results

$$ \begin{aligned} & LU(z) = - \rho \omega^{2} U(z) + \rho F(\omega ,k,z) \\ & LU_{n} (z) = - \rho \omega^{2} U_{n} (z) \\ & L\phi (\nu ,z) = - \rho \nu \phi (\nu ,z) \\ \end{aligned} $$

(15)

the following relations are obtained

$$ \begin{aligned} & c_{n} (\omega^{2} - \omega_{n}^{2} )\left\langle {U_{n} (z),U_{n} (z)} \right\rangle = \left\langle {F(\omega ,k,z),U_{n} (z)} \right\rangle \\ & c(\nu )(\omega^{2} - \nu ) = \left\langle {F(\omega ,k,z),\phi (\nu ,z)} \right\rangle \\ \end{aligned} $$

(16)

where $F(\omega ,k,z)$ is the transformed component of force field.

Hence

$$ U(\omega ,k,z) = \sum\limits_{n} {\frac{{\left\langle {F(\omega ,k,z),U_{n} (z)} \right\rangle U_{n} (z)}}{{(\omega^{2} - \omega_{n}^{2} )\left\langle {U_{n} (z),U_{n} (z)} \right\rangle }}} + \int\limits_{\Gamma } {\frac{{\left\langle {F(\omega ,k,z),\phi (\nu ,z)} \right\rangle }}{{(\omega^{2} - \nu )}}} \phi (\nu ,z){\text{d}}\nu $$

(17)

The corresponding transformed displacement associated with the Love wave which is horizontal having only the cross-radial component is given by

$$ W(\omega ,k,z) = \sum\limits_{n} {\frac{{\left\langle {F(\omega ,k,z),W_{n} (z)} \right\rangle W_{n} (z)}}{{(\omega^{2} - \omega_{n}^{2} )\left\langle {W_{n} (z),W_{n} (z)} \right\rangle }}} + \int\limits_{\Gamma } {\frac{{\left\langle {F(\omega ,k,z),\psi (\nu ,z)} \right\rangle }}{{(\omega^{2} - \nu )}}} \psi (\nu ,z){\text{d}}\nu $$

(18)

The displacement at any point can be obtained on taking the inverse transform.