• Sarva Jit Singh

Articles written in Journal of Earth System Science

• Rayleigh wave group velocity in a spherically symmetric gravitating earth model

Expressions for kinetic energy, elastic potential energy and gravitational potential energy for the spheroidal oscillations of a spherically symmetric, self-gravitating, elastic earth model have been obtained. Some inconsistencies in the expressions given by earlier authors have been pointed out. The principle of equipartition of energy and the Rayleigh principle have been used to derive a formula for Rayleigh wave group velocity in terms of energy integrals. This formula can be used to compute the group velocity without the numerical differentiation implied in its definition

• Quasi-static deformation of a layered half-space by a long strike-slip fault

Theoretical expressions for the surface displacement and shear stress caused by a long strike-slip dislocation in an elastic layer overlying an elastic half-space are derived and the correspondence principle is used to obtain the quasi-static response when the half-space is Maxwell-viscoelastic. Variation of the surface displacement and shear stress with horizontal distance is studied for various times and vertical extents of the fault. It is seen that the quasi-static response differs significantly from the corresponding elastic response.

• Toroidal oscillations of a transradially isotropic elastic sphere

In order to consider the effect of anisotropy on the periods of the oscillations of the Earth, the problem of toroidal oscillations of a transradially isotropic elastic sphere is considered. At each point, the medium is assumed to be transversely isotropic about the radius through the point. The roots of the frequency equation are obtained for different values of the anisotropy parameter α. It is found that, for large order oscillations, the percentage change in the frequency of the toroidal oscillations on account of the anisotropy is nearly equal to ¦α-1¦ × 100.

• Static deformation of an orthotropic multilayered elastic half-space by two-dimensional surface loads

The transfer matrix approach is used to solve the problem of static deformation of an orthotropic multilayered elastic half-space by two-dimensional surface loads. The general problem is decoupled into two independent problems. The antiplane strain problem and the plane strain problem are considered in detail. Integral expressions for displacements and stresses at any point of the medium due to a normal line load and a shear line load, acting parallel to a symmetry axis, are obtained. In the case of a uniform half-space, closed form analytic expressions for displacements and stresses are derived. The procedure developed is quite easy and convenient for numerical computations.

• Static deformation of two welded half-spaces due to dip-slip faulting

Closed-form expressions for the displacements and stresses at any point of either of two elastic half-spaces in welded contact caused by a dip-slip line source obtained earlier are integrated analytically to derive the elastic residual field due to a long dip-slip fault of finite width. The results are valid for an arbitrary dip of the fault. The variation of the displacement field with the distance from the fault as well as with the distance from the interface is studied numerically. It is found that the displacement field is heavily dependent on the dip angle. Contour maps showing the residual elastic field in the two half-spaces caused by a vertical dip-slip fault located in one of the half-spaces are also obtained.

• A note on the assumptions made while computing the postseismic lithospheric deformation

The postseismic lithospheric deformation is usually explained as viscoelastic relaxation of the coseismic stresses. In general, for computing the postseismic deformation, the shear modulus (μ) is relaxed, keeping either the bulk modulus (k) or the La’me parameter (A) fixed. It is shown that the two assumptions yield significantly different results. The assumptionk = const. implies that the medium behaves like an elastic body for dilatational changes which can be justified on physical grounds, but such a justification cannot be given in the case of the assumption λ = const.

• A note on the relationship between dip-slip fault parameters and plate parameters in the two-dimensional modelling of preseismic deformation

Preseismic lithospheric deformation at a subduction zone can be modelled as dip-slip dislocation on an inclined fault or as flexure of a thin plate. Both these models predict a region of positive topography known as forebulge or outer rise. By matching the location and the magnitude of the forebulge, we derive useful relations between the dip-slip fault parameters and the plate parameters. In particular, we determine the width of a long dip-slip fault of given dip corresponding to a semi-infinite plate of given thickness. The displacement profiles of the two models are also compared.

• Reflection ofP andSV waves at the free surface of a monoclinic elastic half-space

The propagation of plane waves in an anisotropic elastic medium possessing monoclinic symmetry is discussed. The expressions for the phase velocity ofqP andqSV waves propagating in the plane of elastic symmetry are obtained in terms of the direction cosines of the propagation vector. It is shown that, in general,qP waves are not longitudinal andqSV waves are not transverse. Pure longitudinal and pure transverse waves can propagate only in certain specific directions. Closed-form expressions for the reflection coefficients ofqP andqSV waves incident at the free surface of a homogeneous monoclinic elastic half-space are obtained. These expressions are used for studying numerically the variation of the reflection coefficients with the angle of incidence. The present analysis corrects some fundamental errors appearing in recent papers on the subject.

• Deformation of a layered half-space due to a very long tensile fault

The problem of the coseismic deformation of an earth model consisting of an elastic layer of uniform thickness overlying an elastic half-space due to a very long tensile fault in the layer is solved analytically. Integral expressions for the surface displacements are obtained for a vertical tensile fault and a horizontal tensile fault. The integrals involved are evaluated approximately by using Sneddon’s method of replacing the integrand by a finite sum of exponential terms. Detailed numerical results showing the variation of the displacements with epicentral distance for various source locations in the layer are presented graphically. The displacement field in the layered half-space is compared with the corresponding field in a uniform half-space to demonstrate the effect of the internal boundary. Relaxed rigidity method is used for computing the postseismic deformation of an earth model consisting of an elastic layer of uniform thickness overlying a viscoelastic half-space.

• Deformation of two welded elastic half-spaces due to a long inclined tensile fault

The calculation of the deformation caused by shear and tensile faults is necessary for the investigation of seismic and volcanic sources. The solution of the two-dimensional problem of a long inclined shear fault in two welded half-spaces is well known. The purpose of this note is to present the corresponding solution for a tensile fault. Closed-form analytical expressions for the Airy stress function for a tensile line source in two welded half-spaces are first obtained. These expressions are then integrated analytically to derive the Airy stress function for a long tensile fault of arbitrary dip and finite width. Closed-form analytical expressions for the displacements and stresses follow immediately from the Airy stress function. These expressions are suitable for computing the displacement and stress fields around a long inclined tensile fault near an internal boundary.

• A note on 2-D lithospheric deformation due to a blind strike-slip fault

Analytical solution for the problem of a surface-breaking long strike-slip fault in an elastic layer overlying an elastic half-space is well known. The purpose of this note is to obtain the corresponding solution for a blind fault. Since the solution is valid for arbitrary values of the fault-depth and the dip angle, the effects of these two important fault parameters can be studied numerically. The variation of the parallel displacement and shear stress with the distance from the fault is studied numerically for different values of the fault-depth and dip angle.

• Plane strain deformation of a multi-layered poroelastic half-space by surface loads

The Biot linearized quasi-static theory of fluid-infiltrated porous materials is used to formulate the problem of the two-dimensional plane strain deformation of a multi-layered poroelastic half-space by surface loads. The Fourier-Laplace transforms of the stresses, displacements, pore pressure and fluid flux in each homogeneous layer of the multi-layered half-space are expressed in terms of six arbitrary constants. Generalized Thomson-Haskell matrix method is used to obtain the deformation field. Simplified explicit expressions for the elements of the 6 × 6 propagator matrix for the poroelastic medium are obtained. As an example of the possible applications of the analytical formulation developed, formal solution is given for normal strip loading, normal line loading and shear line loading.

• Quasi-static deformation due to two-dimensional seismic sources embedded in an elastic half-space in welded contact with a poroelastic half-space

The Biot linearized theory of ﬂuid saturated porous materials is used to study the plane strain deformation of a two-phase medium consisting of a homogeneous, isotropic, poroelastic half-space in welded contact with a homogeneous, isotropic, perfectly elastic half-space caused by a twodimensional source in the elastic half-space. The integral expressions for the displacements and stresses in the two half-spaces in welded contact are obtained from the corresponding expressions for an unbounded elastic medium by applying suitable boundary conditions at the interface. The case of a long dip-slip fault is discussed in detail. The integrals for this source are solved analytically for two limiting cases: (i) undrained conditions in the high frequency limit, and (ii) steady state drained conditions as the frequency approaches zero. It has been veriﬁed that the solution for the drained case (𝜔 → 0) coincides with the known elastic solution. The drained and undrained displacements and stresses are compared graphically. Diffusion of the pore pressure with time is also studied.

• Consolidation of a poroelastic half-space with anisotropic permeability and compressible constituents by axisymmetric surface loading

The fully coupled Biot quasi-static theory of linear poroelasticity is used to study the consolidation of a poroelastic half-space caused by axisymmetric surface loads.The ﬂuid and solid constituents of the poroelastic medium are compressible and its permeability in the vertical direction is different from its permeability in the horizontal direction.An analytical solution of the governing equations is obtained by taking the displacements and the pore pressure as the basic state variables and using a combination of the Laplace and Hankel transforms.The problem of an axisymmetric normal load is discussed in detail.An explicit analytical solution is obtained for normal disc loading.Detailed numerical computations reveal that the anisotropy in permeability as well as the com-pressibilities of the ﬂuid and solid constituents of the poroelastic medium have signiﬁcant effects on the consolidation of the half-space.The anisotropy in permeability may accelerate the consolidation process and may lead to a dilution in the theoretical prediction of the Mandel –Cryer effect. The compressibility of the solid constituents may also accelerate the consolidation process.

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Posted on July 25, 2019

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