I think that the paper by (hereinafter referred to only as SH) is confusing for the reader in sev... more I think that the paper by (hereinafter referred to only as SH) is confusing for the reader in several aspects and presents several misleading conclusions. In this Comment, I shall attempt to point out some of the controversial results of the paper and offer, according to my opinion, their corrected version.
Physics of the Earth and Planetary Interiors, 2006
The deep part of the Tonga subduction zone consists of two differently oriented slab segments: th... more The deep part of the Tonga subduction zone consists of two differently oriented slab segments: the northern segment within latitudes 17-19 • S, and the southern segment within latitudes 19.5-27 • S. The orientation of the slab is (strike/dip): 110 • /57 • in its northern part and 210 • /46 • in its southern part. Both segments are seismically active at depths from 500 to 700 km. The mechanisms of deep-focus earthquakes reported in the Harvard moment tensor catalogue contain compensated linear vector dipole (CLVD) components that behave differently in both segments. The mean value of the CLVD is 3% for the northern segment but −10% for the southern segment. The mean absolute value of the CLVD is 12% for the northern segment and 16% for the southern segment. The complex behaviour of the CLVD is explained by spatially dependent seismic anisotropy in the slab. The inversion for anisotropy from the non-double-couple components of moment tensors points to orthorhombic anisotropy in the both slab segments. The anisotropy seems to have a uniform strength of 5-7% for P-waves and 9-12% for S-waves, and is oriented according to the orientation of each segment and the stress acting in it. The spatial variation of velocities is roughly similar in both segments. The retrieved anisotropy might have several possible origins. It can be: (1) intrinsic, caused by preferentially aligned anisotropic minerals such as wadsleyite, ringwoodite, ilmenite or others, (2) effective, caused, for example, by intra-slab layering, or (3) partly apparent, produced by systematic errors in the moment tensors due to neglecting 3D slab geometry and the slab/mantle velocity contrast when calculating the Green functions in the moment tensor inversion.
Stress inversions from focal mechanisms require knowledge of which nodal plane is the fault. If s... more Stress inversions from focal mechanisms require knowledge of which nodal plane is the fault. If such information is missing, and faults and auxiliary nodal planes are interchanged, the stress inversions can produce inaccurate results. It is shown that the linear inversion method developed by Michael is reasonably accurate when retrieving the principal stress directions even when the selection of fault planes in focal mechanisms is incorrect. However, the shape ratio is more sensitive to the proper choice of the fault and substituting the faults by auxiliary nodal planes introduces significant errors. This difficulty is removed by modifying Michael's method and inverting jointly for stress and for fault orientations. The fault orientations are determined by applying the fault instability constraint and the stress is calculated in iterations. As a by-product, overall friction on faults is determined. Numerical tests show that the new iterative stress inversion is fast and accurate and performs much better than the standard linear inversion. The method is exemplified on real data from central Crete and from the West-Bohemia swarm area of the Czech Republic. The joint iterative inversion identified correctly 36 of 38 faults in the central Crete data. In the West Bohemia data, the faults identified by the inversion were close to the principal fault planes delineated by foci clustering. The overall friction on faults was estimated to be 0.75 and 0.85 for the central Crete and West Bohemia data, respectively.
The moment tensors are unique and describe the body force equivalents of rupture processes in any... more The moment tensors are unique and describe the body force equivalents of rupture processes in any medium including faulting at a material interface, defined as the contact of two media with non-zero velocity or density contrasts. From a practical point of view, however, the moment tensor inversion of sources near or at a material interface is more involved than if the medium is smooth in the source area. First, the moment tensors of sources characterized by the same displacement discontinuity display jumps when the source crosses the interface. Consequently, the moment tensors become sensitive to the source location. If the source lies near the interface, the location into an incorrect half-space can introduce errors in the moment tensor. Second, if the source lies at the material interface, some of the spatial derivatives of the Green's function are, in general, discontinuous and the radiated wave field must be calculated using a generalized representation theorem. Third, the moment tensors are functions of averaged elastic parameters known from effective medium theory. The theory implies that shear faulting at a material interface in isotropic media is represented by the standard double-couple moment tensor. The scalar seismic moment is calculated as a product of the displacement discontinuity across the fault, the fault size and the effective rigidity at the fault. The effective rigidity is the harmonic mean of rigidities at the individual sides of the fault.
Calculation of acoustic axes in triclinic elastic anisotropy is considerably more complicated tha... more Calculation of acoustic axes in triclinic elastic anisotropy is considerably more complicated than for anisotropy of higher symmetry. While one polynomial equation of the 6th order is solved in monoclinic anisotropy, we have to solve two coupled polynomial equations of the 6th order in two variables in triclinic anisotropy. Furthermore, some solutions of the equations are spurious and must be discarded. In this way we obtain 16 isolated acoustic axes, which can run in real or complex directions. The real/complex acoustic axes describe the propagation of homogeneous/ inhomogeneous plane waves and are associated with a linear/elliptical polarization of waves in their vicinity. The most frequent number of real acoustic axes is 8 for strong triclinic anisotropy and 4 to 6 for weak triclinic anisotropy. Examples of anisotropy with no or 16 real acoustic axes are presented.
The form of parabolic lines and caustics in homogeneous generally anisotropic solids can be very ... more The form of parabolic lines and caustics in homogeneous generally anisotropic solids can be very complicated, but simplifies considerably in homogeneous weakly anisotropic solids. Assuming sufficiently weak anisotropy, no parabolic lines appear on the S1 slowness sheet. Consequently, the corresponding wave sheet displays no caustics or triplications. Parabolic lines and caustics can appear on the S2 slowness and wave sheets, respectively, but only in directions close to conical or wedge singularities. Each conical and wedge singularity generates parabolic lines, caustics and anticaustics in its vicinity. The parabolic lines cannot touch or pass through a conical singularity, but they touch each wedge singularity. The size of the caustics and anticaustics decreases with decreasing strength of anisotropy. For infinitesimally weak anisotropy, the caustics and anticaustics contract into a single point. No parabolic lines, caustics, anticaustics and triplications can appear in transversely isotropic solids, provided the transverse isotropy is sufficiently weak.
Determination of the ray vector (the unit vector specifying the direction of the group velocity v... more Determination of the ray vector (the unit vector specifying the direction of the group velocity vector) corresponding to a given wave normal (the unit vector parallel to the phase velocity vector or slowness vector) in an arbitrary anisotropic medium can be performed using the exact formula following from the ray tracing equations. The determination of the wave normal from the ray vector is, generally, a more complicated task, which is usually solved iteratively. We present a first-order perturbation formula for the approximate determination of the ray vector from a given wave normal and vice versa. The formula is applicable to qP as well as qS waves in directions, in which the waves can be dealt with separately (i.e. outside singular directions of qS waves). Performance of the approximate formulae is illustrated on models of transversely isotropic and orthorhombic symmetry. We show that the formula for the determination of the ray vector from the wave normal yields rather accurate results even for strong anisotropy. The formula for the determination of the wave normal from the ray vector works reasonably well in directions, in which the considered waves have convex slowness surfaces. Otherwise, it can yield, especially for stronger anisotropy, rather distorted results.
Anisotropy is frequently present in geological structures, but usually neglected when source para... more Anisotropy is frequently present in geological structures, but usually neglected when source parameters are determined through waveform inversion. Due to the coupling of propagation and source effects in the seismic waveforms, such neglect of anisotropy will lead to an error in the retrieved source. The distortion of the mechanism of a double-couple point source located in an anisotropic medium is investigated when inverting waveforms using isotropic Green's functions. The anisotropic medium is considered to be transversely isotropic with six levels of anisotropy ranging from a fairly weak to rather strong anisotropy, up to about 24% in P waves and 11% in S waves. Inversions are based on either only direct P waves or both direct P and S waves. Two different algorithms are employed: the direct parametrization (DIRPAR, a nonlinear algorithm) and the indirect parametrization (INPAR, a hybrid scheme including linear and nonlinear steps) of the source. The orientation of the doublecouple mechanism appears to be robustly retrieved. The inclination of the resulting nodal planes is very small, within 10j and 20j from the original solution, even for the highest degree of anisotropy. However, the neglect of anisotropy results in the presence of spurious isotropic and compensated linear-vector dipole (CLVD) components in the moment tensor (MT). This questions the reliability of non-double-couple components reported for numerous earthquakes. D
Bohemia/Vogtland region display S-wave splitting. The split S waves are usually well defined, bei... more Bohemia/Vogtland region display S-wave splitting. The split S waves are usually well defined, being separated in time and polarized in roughly perpendicular directions in the horizontal projection. In most cases, the polarization of the fast S wave is aligned NW-SE (referred to as "normal splitting"), which is close to the direction of the maximum horizontal compression in the region. However, for some ray directions, the polarization of the fast S wave is aligned NE-SW (referred to as "reverse splitting"). The pattern of normal/reverse splitting on a focal sphere is station-dependent, indicating the presence of inhomogeneities in anisotropy. For some stations, the normal/reverse splitting pattern is asymmetric with respect to the vertical axis, indicating the symmetry axes of anisotropy are probably inclined. The presence of inclined anisotropy is confirmed by observations of directionally dependent delay times between split S waves. A complex and stationdependent anisotropy pattern is probably the result of a complicated anisotropic crust characterized by diverse geological structures. The spatial variation of anisotropy probably reflects the presence of a variety of different types of anisotropic rocks in the region.
We perform a detailed synthetic study on the resolution of non-double-couple (non-DC) components ... more We perform a detailed synthetic study on the resolution of non-double-couple (non-DC) components in the seismic moment tensors from short-period data observed at regional networks designed typically for monitoring aftershock sequences of large earthquakes. In addition, we test two different inversion approaches-a linear full moment tensor inversion and a nonlinear moment tensor inversion constrained to a shear-tensile source model. The inversions are applied to synthetic first-motion P-and S-wave amplitudes, which mimic seismic observations of aftershocks of the 1999 M w = 7.4 Izmit earthquake in northwestern Turkey adopting a shear-tensile source model. To analyse the resolution capability for the obtained non-DC components inverted, we contaminate synthetic amplitudes with random noise and incorporate realistic uncertainties in the velocity model as well as in the hypocentre locations. We find that the constrained moment tensor inversion yields significantly smaller errors in the non-DC components than the full moment tensor inversion. In particular, the errors in the compensated linear vector dipole (CLVD) component are reduced if the constrained inversion is applied. Furthermore, we show that including the S-wave amplitudes in addition to P-wave amplitudes into the inversion helps to obtain reliable non-DC components. For the studied station configurations, the resolution remains limited due to the lack of stations with epicentral distances less than 15 km. Assuming realistic noise in waveform data and uncertainties in the velocity model, the errors in the non-DC components are as high as ±15 per cent for the isotropic and CLVD components, respectively, thus being non-negligible in most applications. However, the orientation of P-and T-axes is well determined even when errors in the modelling procedure are high.
Bulletin of the Seismological Society of America, 2002
We analyze the problem of a heterogeneous formulation of the equation of motion and propose a new... more We analyze the problem of a heterogeneous formulation of the equation of motion and propose a new 3D fourth-order staggered-grid finite-difference (FD) scheme for modeling seismic motion and seismic-wave propagation.
I think that the paper by (hereinafter referred to only as SH) is confusing for the reader in sev... more I think that the paper by (hereinafter referred to only as SH) is confusing for the reader in several aspects and presents several misleading conclusions. In this Comment, I shall attempt to point out some of the controversial results of the paper and offer, according to my opinion, their corrected version.
Physics of the Earth and Planetary Interiors, 2006
The deep part of the Tonga subduction zone consists of two differently oriented slab segments: th... more The deep part of the Tonga subduction zone consists of two differently oriented slab segments: the northern segment within latitudes 17-19 • S, and the southern segment within latitudes 19.5-27 • S. The orientation of the slab is (strike/dip): 110 • /57 • in its northern part and 210 • /46 • in its southern part. Both segments are seismically active at depths from 500 to 700 km. The mechanisms of deep-focus earthquakes reported in the Harvard moment tensor catalogue contain compensated linear vector dipole (CLVD) components that behave differently in both segments. The mean value of the CLVD is 3% for the northern segment but −10% for the southern segment. The mean absolute value of the CLVD is 12% for the northern segment and 16% for the southern segment. The complex behaviour of the CLVD is explained by spatially dependent seismic anisotropy in the slab. The inversion for anisotropy from the non-double-couple components of moment tensors points to orthorhombic anisotropy in the both slab segments. The anisotropy seems to have a uniform strength of 5-7% for P-waves and 9-12% for S-waves, and is oriented according to the orientation of each segment and the stress acting in it. The spatial variation of velocities is roughly similar in both segments. The retrieved anisotropy might have several possible origins. It can be: (1) intrinsic, caused by preferentially aligned anisotropic minerals such as wadsleyite, ringwoodite, ilmenite or others, (2) effective, caused, for example, by intra-slab layering, or (3) partly apparent, produced by systematic errors in the moment tensors due to neglecting 3D slab geometry and the slab/mantle velocity contrast when calculating the Green functions in the moment tensor inversion.
Stress inversions from focal mechanisms require knowledge of which nodal plane is the fault. If s... more Stress inversions from focal mechanisms require knowledge of which nodal plane is the fault. If such information is missing, and faults and auxiliary nodal planes are interchanged, the stress inversions can produce inaccurate results. It is shown that the linear inversion method developed by Michael is reasonably accurate when retrieving the principal stress directions even when the selection of fault planes in focal mechanisms is incorrect. However, the shape ratio is more sensitive to the proper choice of the fault and substituting the faults by auxiliary nodal planes introduces significant errors. This difficulty is removed by modifying Michael's method and inverting jointly for stress and for fault orientations. The fault orientations are determined by applying the fault instability constraint and the stress is calculated in iterations. As a by-product, overall friction on faults is determined. Numerical tests show that the new iterative stress inversion is fast and accurate and performs much better than the standard linear inversion. The method is exemplified on real data from central Crete and from the West-Bohemia swarm area of the Czech Republic. The joint iterative inversion identified correctly 36 of 38 faults in the central Crete data. In the West Bohemia data, the faults identified by the inversion were close to the principal fault planes delineated by foci clustering. The overall friction on faults was estimated to be 0.75 and 0.85 for the central Crete and West Bohemia data, respectively.
The moment tensors are unique and describe the body force equivalents of rupture processes in any... more The moment tensors are unique and describe the body force equivalents of rupture processes in any medium including faulting at a material interface, defined as the contact of two media with non-zero velocity or density contrasts. From a practical point of view, however, the moment tensor inversion of sources near or at a material interface is more involved than if the medium is smooth in the source area. First, the moment tensors of sources characterized by the same displacement discontinuity display jumps when the source crosses the interface. Consequently, the moment tensors become sensitive to the source location. If the source lies near the interface, the location into an incorrect half-space can introduce errors in the moment tensor. Second, if the source lies at the material interface, some of the spatial derivatives of the Green's function are, in general, discontinuous and the radiated wave field must be calculated using a generalized representation theorem. Third, the moment tensors are functions of averaged elastic parameters known from effective medium theory. The theory implies that shear faulting at a material interface in isotropic media is represented by the standard double-couple moment tensor. The scalar seismic moment is calculated as a product of the displacement discontinuity across the fault, the fault size and the effective rigidity at the fault. The effective rigidity is the harmonic mean of rigidities at the individual sides of the fault.
Calculation of acoustic axes in triclinic elastic anisotropy is considerably more complicated tha... more Calculation of acoustic axes in triclinic elastic anisotropy is considerably more complicated than for anisotropy of higher symmetry. While one polynomial equation of the 6th order is solved in monoclinic anisotropy, we have to solve two coupled polynomial equations of the 6th order in two variables in triclinic anisotropy. Furthermore, some solutions of the equations are spurious and must be discarded. In this way we obtain 16 isolated acoustic axes, which can run in real or complex directions. The real/complex acoustic axes describe the propagation of homogeneous/ inhomogeneous plane waves and are associated with a linear/elliptical polarization of waves in their vicinity. The most frequent number of real acoustic axes is 8 for strong triclinic anisotropy and 4 to 6 for weak triclinic anisotropy. Examples of anisotropy with no or 16 real acoustic axes are presented.
The form of parabolic lines and caustics in homogeneous generally anisotropic solids can be very ... more The form of parabolic lines and caustics in homogeneous generally anisotropic solids can be very complicated, but simplifies considerably in homogeneous weakly anisotropic solids. Assuming sufficiently weak anisotropy, no parabolic lines appear on the S1 slowness sheet. Consequently, the corresponding wave sheet displays no caustics or triplications. Parabolic lines and caustics can appear on the S2 slowness and wave sheets, respectively, but only in directions close to conical or wedge singularities. Each conical and wedge singularity generates parabolic lines, caustics and anticaustics in its vicinity. The parabolic lines cannot touch or pass through a conical singularity, but they touch each wedge singularity. The size of the caustics and anticaustics decreases with decreasing strength of anisotropy. For infinitesimally weak anisotropy, the caustics and anticaustics contract into a single point. No parabolic lines, caustics, anticaustics and triplications can appear in transversely isotropic solids, provided the transverse isotropy is sufficiently weak.
Determination of the ray vector (the unit vector specifying the direction of the group velocity v... more Determination of the ray vector (the unit vector specifying the direction of the group velocity vector) corresponding to a given wave normal (the unit vector parallel to the phase velocity vector or slowness vector) in an arbitrary anisotropic medium can be performed using the exact formula following from the ray tracing equations. The determination of the wave normal from the ray vector is, generally, a more complicated task, which is usually solved iteratively. We present a first-order perturbation formula for the approximate determination of the ray vector from a given wave normal and vice versa. The formula is applicable to qP as well as qS waves in directions, in which the waves can be dealt with separately (i.e. outside singular directions of qS waves). Performance of the approximate formulae is illustrated on models of transversely isotropic and orthorhombic symmetry. We show that the formula for the determination of the ray vector from the wave normal yields rather accurate results even for strong anisotropy. The formula for the determination of the wave normal from the ray vector works reasonably well in directions, in which the considered waves have convex slowness surfaces. Otherwise, it can yield, especially for stronger anisotropy, rather distorted results.
Anisotropy is frequently present in geological structures, but usually neglected when source para... more Anisotropy is frequently present in geological structures, but usually neglected when source parameters are determined through waveform inversion. Due to the coupling of propagation and source effects in the seismic waveforms, such neglect of anisotropy will lead to an error in the retrieved source. The distortion of the mechanism of a double-couple point source located in an anisotropic medium is investigated when inverting waveforms using isotropic Green's functions. The anisotropic medium is considered to be transversely isotropic with six levels of anisotropy ranging from a fairly weak to rather strong anisotropy, up to about 24% in P waves and 11% in S waves. Inversions are based on either only direct P waves or both direct P and S waves. Two different algorithms are employed: the direct parametrization (DIRPAR, a nonlinear algorithm) and the indirect parametrization (INPAR, a hybrid scheme including linear and nonlinear steps) of the source. The orientation of the doublecouple mechanism appears to be robustly retrieved. The inclination of the resulting nodal planes is very small, within 10j and 20j from the original solution, even for the highest degree of anisotropy. However, the neglect of anisotropy results in the presence of spurious isotropic and compensated linear-vector dipole (CLVD) components in the moment tensor (MT). This questions the reliability of non-double-couple components reported for numerous earthquakes. D
Bohemia/Vogtland region display S-wave splitting. The split S waves are usually well defined, bei... more Bohemia/Vogtland region display S-wave splitting. The split S waves are usually well defined, being separated in time and polarized in roughly perpendicular directions in the horizontal projection. In most cases, the polarization of the fast S wave is aligned NW-SE (referred to as "normal splitting"), which is close to the direction of the maximum horizontal compression in the region. However, for some ray directions, the polarization of the fast S wave is aligned NE-SW (referred to as "reverse splitting"). The pattern of normal/reverse splitting on a focal sphere is station-dependent, indicating the presence of inhomogeneities in anisotropy. For some stations, the normal/reverse splitting pattern is asymmetric with respect to the vertical axis, indicating the symmetry axes of anisotropy are probably inclined. The presence of inclined anisotropy is confirmed by observations of directionally dependent delay times between split S waves. A complex and stationdependent anisotropy pattern is probably the result of a complicated anisotropic crust characterized by diverse geological structures. The spatial variation of anisotropy probably reflects the presence of a variety of different types of anisotropic rocks in the region.
We perform a detailed synthetic study on the resolution of non-double-couple (non-DC) components ... more We perform a detailed synthetic study on the resolution of non-double-couple (non-DC) components in the seismic moment tensors from short-period data observed at regional networks designed typically for monitoring aftershock sequences of large earthquakes. In addition, we test two different inversion approaches-a linear full moment tensor inversion and a nonlinear moment tensor inversion constrained to a shear-tensile source model. The inversions are applied to synthetic first-motion P-and S-wave amplitudes, which mimic seismic observations of aftershocks of the 1999 M w = 7.4 Izmit earthquake in northwestern Turkey adopting a shear-tensile source model. To analyse the resolution capability for the obtained non-DC components inverted, we contaminate synthetic amplitudes with random noise and incorporate realistic uncertainties in the velocity model as well as in the hypocentre locations. We find that the constrained moment tensor inversion yields significantly smaller errors in the non-DC components than the full moment tensor inversion. In particular, the errors in the compensated linear vector dipole (CLVD) component are reduced if the constrained inversion is applied. Furthermore, we show that including the S-wave amplitudes in addition to P-wave amplitudes into the inversion helps to obtain reliable non-DC components. For the studied station configurations, the resolution remains limited due to the lack of stations with epicentral distances less than 15 km. Assuming realistic noise in waveform data and uncertainties in the velocity model, the errors in the non-DC components are as high as ±15 per cent for the isotropic and CLVD components, respectively, thus being non-negligible in most applications. However, the orientation of P-and T-axes is well determined even when errors in the modelling procedure are high.
Bulletin of the Seismological Society of America, 2002
We analyze the problem of a heterogeneous formulation of the equation of motion and propose a new... more We analyze the problem of a heterogeneous formulation of the equation of motion and propose a new 3D fourth-order staggered-grid finite-difference (FD) scheme for modeling seismic motion and seismic-wave propagation.
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Papers by V. Vavrycuk