Vertical Dip-slip Fault Research Articles

Lithospheric plane strain deformation caused by a vertical dip-slip fault

Lithospheric plane strain deformation caused by a vertical dip-slip fault

Dynamic dip-slip fault rupture in a layered geological medium: Broken symmetry of seismic motion

As recorded recently on 22 November 2014 during the Nagano-ken Hokubu (Kamishiro Fault), Japan, earthquake, one important observation in dynamic rupture (fracture) related to a shallow dip-slip earthquake (usually assumed as mode-II shear crack propagation from depth towards a free surface) is the broken symmetry of seismic motion in the vicinity of the fracturing fault plane. Generally, the strong motion (particle motion) on the hanging wall is much larger than that on the footwall, but the physical mechanisms causing this asymmetry have not been fully clarified yet. Here, utilising the techniques of finite difference calculations and dynamic photoelasticity in conjunction with high speed cinematography, we investigate the fracture dynamics of a dip-slip fault plane situated near a free surface and try to explain the mechanics behind the asymmetry, numerically as well as experimentally. In our two-dimensional crack-like rupture models, we prepare a flat fault plane, which dips either vertically or at an angle, in a monolithic (scenario (1)) or layered (scenario (2)) linear elastic medium (representing rocks). In the basic scenario (1), when the primary fault rupture initiated at some depth approaches the free surface, four Rayleigh-type waves may be induced: two of them propagate along the free surface as Rayleigh surface waves into the opposite directions to the far field, and the other two travel back downwards along the fractured fault plane as interface waves into depth. If the fault plane is inclined, in the hanging wall, the interface and Rayleigh waves may interact with each other and a shear wave carrying concentrated energy (corner wave) can be produced to cause stronger disturbances. The corner waves, generated by primary fault rupture, may exist only when the fault plane is inclined, i.e. only when the geometry considered is asymmetric. In the scenario (2), however, we indicate that (anti-)symmetry of mode-II seismic motion can be easily broken even in geometrically symmetric models if the secondary fracture is allowed at an interface between layers. If primary vertical dip-slip fault rupture in an (anti-)symmetric model moves from depth and interacts with a horizontal interface that follows, for example, a tensile fracture criterion, basically only the interface segments where the primary rupture induces dynamic tension (in the relatively subsiding footwall) may be fractured and the segments in compression (in the rising hanging wall) may remain unbroken. In this case, in the hanging wall, the dynamic stresses in the upper layer above the interface become relatively large because the compressive parts of the primary rupture-induced wave (rupture front wave) can propagate from the lower layer across the still bonded interface into the upper layer, with less reflection back to the lower one. On the contrary, in the footwall, much of the energy carried by the rupture front wave in the lower layer is reflected at the broken interface and dynamic disturbances in the upper layer tend to become smaller. Thus, the strong motion on the hanging wall is expected to become larger than that on the footwall not only in geometrically asymmetric cases but also in symmetric ones.

Static deformation due to two-dimensional seismic sources embedded in an isotropic half-space in smooth contact with an orthotropic half-space

Closed form analytical expressions for displacements and stresses at any point of a two-phase medium consisting of a homogeneous, isotropic, perfectly elastic half-space in smooth contact with a homogeneous, orthotropic, perfectly elastic half-space caused by two-dimensional seismic sources located in the isotropic half-space are obtained. The method consists of first finding the integral expressions for two half-spaces in smooth contact from the corresponding expressions for an unbounded medium by applying suitable boundary conditions at the interface and then evaluating the integrals analytically. Here, we discuss the horizontal and vertical displacements for vertical dip-slip fault numerically. Numerical computations indicate that the deformation field due to a source in an isotropic half-space in smooth contact with an anisotropic half-space may differ substantially from the deformation field when both the half-spaces are isotropic. Keywords: Isotropic Half-Space, Orthotropic Half-Space, Static-Deformation, Smooth Contact.

Static deformation due to a long buried dip-slip fault in an isotropic half-space welded with an orthotropic half-space

Closed-form analytical expressions for the displacements and the stresses at any point of a two-phase medium consisting of a homogeneous, isotropic, perfectly elastic half-space in welded contact with a homogeneous, orthotropic, perfectly elastic half-space due to a dip-slip fault of finite width located at an arbitrary distance from the interface in the isotropic half-space are obtained. The Airy stress function approach is used to obtain the expressions for the stresses and the displacements. The case of a vertical dip-slip fault is considered in detail. The variations of the displacements with the distance from the fault and with depth have been shown graphically.

Crustal Deformation Associated with Two-Dimensional Thrust Faulting.

The effect of the source-depth and the dip of the fault on the surface displacement field due to a long thrust fault embedded in an elastic half-space is studied. It is found that, for small dip angles, the displacement field is highly asymmetric about the fault strike, the displacement of the footwall is very small, and the horizontal displacement of the hanging wall shows a wavy pattern. The vertical displacement of the hanging wall is an uplift near the fault. For small dip angles, this uplift is maximum at a point almost vertically above the upper edge of the fault. As we go away from the fault strike, the vertical displacement changes sign, i.e., it becomes subsidence. For small dip angles, this subsidence is maximum at a point roughly vertically above the lower edge of the fault. Steeper fault dips decrease the subsidence relative to the uplift. For a vertical dip-slip fault, the horizontal displacement is completely symmetric about the fault strike and the vertical displacement is completely antisymmetric. The variation of the coseismic and postseismic shear stresses and shear strains for a surface-breaking long thrust fault with depth is studied for a viscoelastic half-space. It is found that, while the coseismic and postseismic shear stresses are of opposite signs, the coseismic and postseismic shear strains are of the same sign.

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.

Displacements and stresses in two welded half-spaces caused by two-dimensional sources

Closed-form analytic expressions for the displacements and stresses at any point of either of two homogeneous, isotropic, perfectly elastic half-spaces in welded contact caused by various two-dimensional sources are obtained. The method consists of first finding the integral expressions for two half-spaces in welded contact from the corresponding expressions for an unbounded medium by applying suitable boundary conditions at the interface and then evaluating the integrals analytically. The case of a long dip-slip dislocation source is considered in detail. Graphs showing the variation of the displacements with the distance from the fault and with the distance from the interface (depth) for a vertical dip-slip fault are presented. It is found that, although at a large distance from the fault, the variation with depth is smooth, at short distances the variation with depth is complex, with sharp maxima and minima.

Surface-wave studies of some South Carolina earthquakes

AbstractLong-period surface waves generated by three South Carolina earthquakes are used to infer focal mechanisms and seismic moments of the events. The 3 February 1972 Bowman, South Carolina, earthquake is found to be shallow and to require an almost strike-slip focal mechanism with pressure axes trending northeast-southwest. The 22 November 1974 Charleston, South Carolina, earthquake was smaller and was recorded by only three long-period WWSSN stations. For this earthquake, only an estimate of the seismic moment and focal depth were possible. The focal mechanism of the 2 August 1974 Clarkhill reservoir earthquake indicated motion on a northwest striking almost vertical dip-slip fault. In order to specify preferred motions on the nodal planes, full waveform modeling of the long-period traces at ATL was necessary. The results are not without ambiguity, but represent the best estimates of source parameters for these earthquakes.

Anelastic response of the Earth to a dip slip earthquake

The deformation of the earth's surface immediately following an earthquake is well modeled by a dislocation in an elastic half space. Some time after the event, however, the elastic stresses begin to relax due to flow in the more fluid regions of the upper mantle (the asthenosphere). This relaxation causes further slow displacements observable at the earth's surface. This paper examines the pattern and timing of these postseismic motions for a vertical dip slip fault and a 30° dip thrust fault. A variety of rheological models are studied. These models encompass variations in asthenosphere thickness, mesosphere viscosity, fault depth, and, most important, power law flow. Both Newtonian (n = 1) and non‐Newtonian (n = 3) Maxwell rheologies are considered for the asthenosphere and mesosphere. The results for a vertical dip slip fault show that a peripheral upwarp develops within a distance of a few lithosphere thicknesses from the rupture zone on the downthrown side of the fault (a comparable downwarp develops on the upthrown side). These peripheral warps grow to a maximum height of about 5% of the total fault slip some time after the earthquake event and then collapse at much greater times. The vertical motions for a 30° dip thrust fault show that after the earthquake a general uplift occurs in the region near the rupture zone. Horizontal displacements propagate asymmetrically away from the rupture zone, the largest displacements occurring in the overthrust plate. The principal result of the comparison of n = 1 and n = 3 rheological laws is that the basic pattern of uplifts and horizontal motions is very similar: the major difference is the time dependence of the development of the pattern. In spite of the complex geometrical structure of the subduction zone the displacements ui(1)(t) for Newtonian rheology are nearly the same as ui(3)(t′) for the non‐Newtonian rheology if t′ ∼ t3. Thus although the qualitative pattern of the displacements is not diagnostic of power law flow, the great difference in time dependence allows a clear discrimination of the two flow laws.

Source process and tectonic implications of the Spanish deep-focus earthquake of March 29, 1954

The source process of the deep-focus Spanish earthquake of March 29, 1954 (m_b = 7.1, h = 630 km) has been studied by using seismograms recorded at teleseismic distances. Because of its unusual location, this earthquake is considered to be one of the most important earthquakes that merit detailed studies. Long-period body-wave records reveal that the earthquake is a complicated multiple event whose wave form is quite different from that of usual deep earthquakes. The total duration of P phases at teleseismic distances is as long as 40 s. This long duration may explain the considerable property damage in Granada and Malaga, Spain, which is rather rare for deep earthquakes. Using the azimuthal distribution of the differences between the arrival times of the first, the second and later P phases, the hypocenters of the later events are determined with respect to the first event. The focus of the second event is located on the vertical nodal plane of the first shock suggesting that this vertical plane is the fault plane. This fault plane which strikes in N2°E and dips 89.1°E defines a nearly vertical dip-slip fault, the block to the west moving downwards. The time interval and spatial separation between the first and the second events are 4.3 s and 19 km respectively, giving an apparent rupture velocity of 4.3 km/s which is about 74% of the S-wave velocity at the source. A third event occurred about 8.8 s after the first event and about 35.6 km from it. At least six to ten events can be identified during the whole sequence. The mechanism of some of the later events, however, seems to differ from the first two events. Synthetic seismograms are generated by superposition of a number of point sources and are matched with the observed signals to determine the seismic moment. The seismic moments of the later events are comparable to, or even larger than, that of the first. The total seismic moment is determined to be 7 · 10^(27) dyn cm while the moments of the first and the second shocks are 2.1 · 10^(26) dyn cm and 5.1 · 10^(26) dyn cm, respectively. The earthquake may represent a series of fractures in a detached piece of the lithosphere which sank rapidly into the deep mantle preserving the heterogeneity of material property at shallow depths.

Quasi-static deformation of a viscoelastic half-space by a displacement dislocation

The correspondence principle is used to get the Laplace transformed solution of the problem of the quasi-static deformation of a viscoelastic half-space by a shear displacement dislocation from the corresponding static elastic results. The transformed solution is inverted for the Voigt and the Maxwell viscoelastic models. It is shown that, for a vertical dip-slip fault, the surface displacements for the viscoelastic case are identical with the elastic displacements. In the case of a vertical strike-slip fault, detailed numerical results are obtained for both a point source and a finite rectangular source. It is found that the results for the viscoelastic models differ significantly from the corresponding elastic results.

Quasi-static strains and tilts due to faulting in a viscoelastic half-space

abstract Recently derived quasi-static surface displacements resulting from a finite rectangular vertical strike-slip fault in a viscoelastic half-space are used to derive the surface strains and tilts for both Voigt and Maxwell viscoelastic models. Contour maps are obtained for various strain and tilt components. The variation with time and epicentral distance is studied in some representative cases. Detailed numerical calculations reveal significant differences between the viscoelastic and the elastic results in the case of a vertical strike-slip fault. This contrasts with the results for a vertical dip-slip fault in a uniform half-space, where the surface strains and tilts for the viscoelastic and the elastic models are identical.

STATIC DEFORMATIONS DUE TO THE FAULT SPREADING OVER SEVERAL LAYERS IN A MULTI-LAYERED MEDIUM

Surface deformations due to the fault spreading over several layers in a multi-layered medium are investigated. Machine program is developed by which contour map of vertical components and vector map of horizontal components around the focal area can be drawn, designating focal para-meters as the inputs.Some numerical examples of surface deformations are presented for the purpose of comparison with those obtained for a homogeneous semi-infinite model.For a vertical dip-slip fault, for example, it is known that the vertical displacement at the surface is upheaval on one side of the strike of the fault and subsidence on the other side. For a multi-layered model, however, subsidence area appears even in the upheaval area mentioned above. We call this reverse area. Thus the displacement field shows different pattern from that for a semi-infinite model.

Static Deformation of a Multilayered Sphere by Internal Sources

Summary The Thomson-Haskell matrix device is used to solve the problem of the static deformation of a multilayered spherical Earth model by buried sources. The model consists of p-1 concentric spherical shells plus an inner core; each shell as well as the core being homogeneous, isotropic and perfectly elastic. The point source is represented as a discontinuity in the motion-stress vector across the spherical surface passing through the source. Explicit series expressions in terms of layer matrices are obtained for the displacements and stresses at any point in the medium for three sources: a vertical strike-slip fault, a vertical dip-slip fault and a centre of explosion. The singular case corresponding to the Legendre polynomial of the first degree (n= 1) has been discussed in detail.

Static deformation of a multilayered half-space by internal sources

The method of layer matrices is applied to solve the problem of the static deformation of a multilayered elastic half-space by buried sources. Each layer of the multilayered medium is assumed to be homogeneous and isotropic, and the interfaces are assumed to be in welded contact. The point source ia represented as a discontinuity in the z-dependent coefficients of the displacement and stress integrands at the source level. Source functions are obtained for the six elementary displacement dislocations. Explicit expressions for the surface displacements are calculated for a vertical strike-slip and a vertical dip-slip fault.

The Torsional Free Oscillation of the Earth due to a Dislocation

The torsional free oscillation due to a dip-slip fault or a strike-slip fault is studied in an earth model with homogeneous mantle and homogeneous liquid core. Because of the simple earth model assumption, only the fundamental mode of order number two, 0T2, is investigated. It is found that the 0T2 is effectively excited by a vertical strike-slip fault. The surface amplitude of the free oscillation due to a horizontal or a vertical dip-slip fault is about three percent of that due to a vertical strike-slip fault. Radiation patterns, except for a horizontal or a vertical dip-slip fault, are governed by P22 (cos θ) (cos 2ψ sin 2ψ). If a step function time variation fault is assumed, then the maximum surface amplitude of 6×10-2cm is estimated when the area of the fault is 104km2 and the mean offset over the fault-plane is 5m.

Multipolar elastic fields in a layered half space

Abstract Hansen's expansion is used to derive integral expressions for the displacement field due to a localized buried source of the mth order in a layered half space. The dipolar case (m ≦ 2) is worked out in detail for arbitrary source-depth in the layer and in the substratum. A new type of representation of the source is used which gives the final results in a concise form. Explicit expressions for the displacements at the free surface are obtained for a center of explosion, a vertical strike-slip fault and a vertical dip-slip fault. The results for a horizontal thrust are found to be the same as for a vertical dip-slip fault. The relations between the Galerkin vector and the biharmonic eigenvectors are clarified. It is shown that the Galerkin-Boussinesq solution for the elastic half space cannot be extended to structures of higher complexity, except for a few simple sources. On the other hand, the Hansen Solution is valid for a wide class of sources and structures. Both dynamic and static regimes are considered.