Velocity Gradient Tensor Field Research Articles

A Geodetic Strain Rate Model for the East African Rift System

Here we describe the new Sub-Saharan Africa Geodetic Strain Rate Model v.1.0 (SSA-GSRM v.1.0), which provides fundamental constraints on long-term tectonic deformation in the region and an improved seismic hazards assessment in Sub-Saharan Africa. Sub-Saharan Africa encompasses the East African Rift System, the active divergent plate boundary between the Nubian and Somalian plates, where strain is largely accommodated along the boundaries of three subplates. We develop an improved geodetic strain rate field for sub-Saharan Africa that incorporates 1) an expanded geodetic velocity field, 2) redefined regions of deforming zones guided by seismicity distribution, and 3) updated constraints on block rotations. SSA-GSRM v.1.0 spans longitudes 22° to 55.5° and latitudes −52° to 20° with 0.25° (longitude) by 0.2° (latitude) spacing. For plates/sub-plates, we assign rigid block rotations as constraints on the strain rate calculation that is determined by fitting bicubic Bessel splines to a new geodetic velocity solution for an interpolated velocity gradient tensor field. We derive strain rates, velocities, and vorticity rates from the velocity gradient tensor field. A comparison with the Global Geodetic Strain Rate model v2.1 reveals regions of previously unresolved spatial heterogeneities in geodetic strain rate distribution, which indicates zones of elevated seismic risk.

A geodetic plate motion and Global Strain Rate Model

We present a new global model of plate motions and strain rates in plate boundary zones constrained by horizontal geodetic velocities. This Global Strain Rate Model (GSRM v.2.1) is a vast improvement over its predecessor both in terms of amount of data input as in an increase in spatial model resolution by factor of ∼2.5 in areas with dense data coverage. We determined 6739 velocities from time series of (mostly) continuous GPS measurements; i.e., by far the largest global velocity solution to date. We transformed 15,772 velocities from 233 (mostly) published studies onto our core solution to obtain 22,511 velocities in the same reference frame. Care is taken to not use velocities from stations (or time periods) that are affected by transient phenomena; i.e., this data set consists of velocities best representing the interseismic plate velocity. About 14% of the Earth is allowed to deform in 145,086 deforming grid cells (0.25° longitude by 0.2° latitude in dimension). The remainder of the Earth's surface is modeled as rigid spherical caps representing 50 tectonic plates. For 36 plates we present new GPS-derived angular velocities. For all the plates that can be compared with the most recent geologic plate motion model, we find that the difference in angular velocity is significant. The rigid-body rotations are used as boundary conditions in the strain rate calculations. The strain rate field is modeled using the Haines and Holt method, which uses splines to obtain an self-consistent interpolated velocity gradient tensor field, from which strain rates, vorticity rates, and expected velocities are derived. We also present expected faulting orientations in areas with significant vorticity, and update the no-net rotation reference frame associated with our global velocity gradient field. Finally, we present a global map of recurrence times for Mw=7.5 characteristic earthquakes.

A Long-term Forecast of Shallow Seismicity Based on the Global Strain Rate Map

The Global Strain Rate Map (GSRM) of Kreemer et al. (2003) was the main result of Project II-8 of the International Lithosphere Program. The GSRM is a numerical velocity gradient tensor field model for the entire Earth's surface that describes the spatial variations of horizontal strain rate tensor components, rotation rates, and velocities. The model consists of 25 rigid spherical plates and ∼25,000 0.6° by 0.5° deformable grid areas within the diffuse plate boundary zones ( e.g. , western North America, central Asia, Alpine-Himalaya belt). The model provides an estimate of the horizontal strain rates in diffuse plate boundary zones as well as the motions of the spherical caps. This is one of the first successful models of its kind that includes the kinematics of plate boundary zones in the description of global plate kinematics. The vast majority of the data used to obtain the GSRM comes from horizontal velocity measurements obtained using Global Positioning System (GPS) measurements. The latest model version of May 2004 ( i.e. , GSRM version 1.2) includes 5,170 velocities for 4,214 sites worldwide (Holt et al. 2005). Most geodetic velocities are measured within plate boundary zones. The observed velocities are obtained from 86 different (mostly published) studies. The model includes additional constraints on the style (not magnitude) of the strain rate tensor inferred from moment tensors of shallow earthquakes. In addition, geologic strain rates in central Asia inferred from Quaternary faulting data are fit simultaneously with the geodetic velocities to improve the model there. See Kreemer et al. (2000, 2003) for more details. It was always a goal of the GSRM project to support long-term forecasts of seismicity based on tectonic deformation. Two recent developments make this especially timely. First, the Collaboratory for the Study of Earthquake Predictability (CSEP; Jordan et al. 2007) is accepting global …

A new methodology for multiscale myocardial deformation and strain analysis based on tagging MRI.

Myocardial deformation and strain can be investigated using suitably encoded cine MRI that admits disambiguation of material motion. Practical limitations currently restrict the analysis to in-plane motion in cross-sections of the heart (2D + time), but the proposed method readily generalizes to 3D + time. We propose a new, promising methodology, which departs from a multiscale algorithm that exploits local scale selection so as to obtain a robust estimate for the velocity gradient tensor field. Time evolution of the deformation tensor is governed by a first-order ordinary differential equation, which is completely determined by this velocity gradient tensor field. We solve this matrix-ODE analytically and present results obtained from healthy volunteers as well as from patient data. The proposed method requires only off-the-shelf algorithms and is readily applicable to planar or volumetric tagging MRI sampled on arbitrary coordinate grids.

Contribution of gravitational potential energy differences to the global stress field

SUMMARY Modelling the lithospheric stress field has proved to be an efficient means of determining the role of lithospheric versus sublithospheric buoyancies and also of constraining the driving forces behind plate tectonics. Both these sources of buoyancies are important in generating the lithospheric stress field. However, these sources and the contribution that they make are dependent on a number of variables, such as the role of lateral strength variation in the lithosphere, the reference level for computing the gravitational potential energy per unit area (GPE) of the lithosphere, and even the definition of deviatoric stress. For the mantle contribution, much depends on the mantle convection model, including the role of lateral and radial viscosity variations, the spatial distribution of density buoyancies, and the resolution of the convection model. GPE differences are influenced by both lithosphere density buoyancies and by radial basal tractions that produce dynamic topography. The global lithospheric stress field can thus be divided into (1) stresses associated with GPE differences (including the contribution from radial basal tractions) and (2) stresses associated with the contribution of horizontal basal tractions. In this paper, we investigate only the contribution of GPE differences, both with and without the inferred contribution of radial basal tractions. We use the Crust 2.0 model to compute GPE values and show that these GPE differences are not sufficient alone to match all the directions and relative magnitudes of principal strain rate axes, as inferred from the comparison of our depth integrated deviatoric stress tensor field with the velocity gradient tensor field within the Earth's plate boundary zones. We argue that GPE differences calibrate the absolute magnitudes of depth integrated deviatoric stresses within the lithosphere; shortcomings of this contribution in matching the stress indicators within the plate boundary zones can be corrected by considering the contribution from horizontal tractions associated with density buoyancy driven mantle convection. Deviatoric stress magnitudes arising from GPE differences are in the range of 1–4 TN m−1, a part of which is contributed by dynamic topography. The EGM96 geoid data set is also used as a rough proxy for GPE values in the lithosphere. However, GPE differences from the geoid fail to yield depth integrated deviatoric stresses that can provide a good match to the deformation indicators. GPE values inferred from the geoid have significant shortcomings when used on a global scale due to the role of dynamically support of topography. Another important factor in estimating the depth integrated deviatoric stresses is the use of the correct level of reference in calculating GPE. We also elucidate the importance of understanding the reference pressure for calculating deviatoric stress and show that overestimates of deviatoric stress may result from either simplified 2-D approximations of the thin sheet equations or the assumption that the mean stress is equal to the vertical stress.

Project helps constrain continental dynamics and seismic hazards

The Global Strain Rate Map project II‐8, initiated in 1998 by the International Lithosphere Program (ILP), provides constraints for understanding continental dynamics and for quantifying seismic hazards in general. To date, the Global Strain Rate Map (GSRM) model is a numerical velocity gradient tensor field solution (i.e.,spatial variations of horizontal strain rate tensor components and rotation rates) for the entire Earth surface [Kreemer et al., 2003].The global model consists of 25 rigid spherical plates and ∼25,000 0.6°×0.5° deformable grid areas within the diffuse plate boundary zones which lie between these plates (e.g., western North America, central Asia, the Alpine‐Himalaya belt). This model provides an estimate of the horizontal strain rates, rotation rates, and velocity fields for the diffuse plate boundary zones as well as an estimate of the motions of the spherical caps.

Dual-plane stereo particle image velocimetry (DSPIV) for measuring velocity gradient fields at intermediate and small scales of turbulent flows

A two-frequency dual-plane stereo particle image velocimetry (DSPIV) technique is described for highly resolved measurements of the complete nine-component velocity gradient tensor field ∂u i /∂x j on the quasi-universal intermediate and small scales of turbulent flows. The method is based on two simultaneous, independent stereo particle image velocimetry (PIV) measurements in two differentially spaced light sheet planes, with light sheet characterization measurements demonstrating the required sheet thicknesses, separation, and two-axis parallelism that determine the measurement resolution and accuracy. The present approach uses an asymmetric forward–forward scatter configuration with two different laser frequencies in conjunction with filters to separate the scattered light onto the individual stereo camera pairs, allowing solid metal oxide particles to be used as seed particles to permit measurements in nonreacting as well as exothermic reacting turbulent flows.

An integrated global model of present-day plate motions and plate boundary deformation

In this paper we present a global model (GSRM-1) of both horizontal velocities on the Earth's surface and horizontal strain rates for almost all deforming plate boundary zones. A model strain rate field is obtained jointly with a global velocity field in the process of solving for a global velocity gradient tensor field. In our model we perform a least-squares fit between model velocities and observed geodetic velocities, as well as between model strain rates and observed geological strain rates. Model velocities and strain rates are interpolated over a spherical Earth using bi-cubic Bessel splines. We include 3000 geodetic velocities from 50 different, mostly published, studies. Geological strain rates are obtained for central Asia only and they are inferred from Quaternary fault slip rates. For all areas where no geological information is included a priori constraints are placed on the style and direction (but not magnitude) of the model strain rate field. These constraints are taken from a seismic strain rate field inferred from (normalized) focal mechanisms of shallow earthquakes. We present a global solution of the second invariant of the model strain rate field as well as strain rate solutions for a few selected plate boundary zones. Generally, the strain rate tensor field is consistent with geological and seismological data. With the assumption of plate rigidity for all areas other than the plate boundary zones we also present relative angular velocities for most plate pairs. We find that in general there is a good agreement between the present-day plate motions we obtain and long-term plate motions, but a few significant differences exist. The rotation rates for the Indian, Arabian and Nubian plates relative to Eurasia are 30, 13 and 50 per cent slower than the NUVEL-1A estimate, respectively, and the rotation rate for the Nazca Plate relative to South America is 17 per cent slower. On the other hand, Caribbean–North America motion is 76 per cent faster than the NUVEL-1A estimate. While crustal blocks in the India–Eurasia collision zone move significantly and self-consistently with respect to bounding plates, only a very small motion is predicted between the Nubian and Somalian plates. By integrating plate boundary zone deformation with the traditional modelling of angular velocities of rigid plates we have obtained a model that has already been proven valuable in, for instance, redefining a no-net-rotation model of surface motions and by confirming a global correlation between seismicity rates and tectonic moment rates along subduction zones and within zones of continental deformation.

Correlated crust and mantle strain fields in Tibet

Vertical to subvertical planes of shear within the active crustal deformation field in Tibet align with fast directions of shear-wave polarization. This observation suggests that the present-day velocity gradient tensor field within the crust of Tibet correlates with the lithospheric mantle velocity gradient tensor field beneath Tibet. This inference requires the following. (1) The [100] axes of olivine are aligned within established zones of lithospheric mantle fabric or anisotropy, built up through finite strain. (2) Present-day shear in the mantle lithosphere is occurring within these established zones of fabric, parallel to the direction of olivine [100] axis alignment. (3) The zones of mantle lithospheric fabric, or zones of mantle shear, are aligned with zones of crustal shear (faults). The correlation of crustal and mantle strain fields most simply results from the fact that both crust and mantle lithosphere are under the influence of similar velocity boundary conditions. Furthermore, the observations confirm the distributed nature of lithospheric mantle deformation beneath east-central Tibet and suggest that left-lateral shear has been the dominant component of finite shear there.

Deformation kinematics in the western United States determined from Quaternary fault slip rates and recent geodetic data

We estimate the horizontal velocity gradient tensor field from Quaternary fault slip rates, and recent Global Positioning System (GPS) and very long baseline interferometry (VLBI) velocity solutions in the western United States transform plate boundary zone. The total velocity obtained from the Quaternary fault slip rate data across the entire plate boundary is within 1 mm/yr of the NUVEL‐1A predicted Pacific (PA)‐North American (NA) plate motion velocity, but directions are 5°–6° anticlockwise of directions given by NUVEL‐1A. The total velocity obtained from inversion of recent geodetic data is 2°–3° anticlockwise from the NUVEL‐1A NA‐PA velocity, but the difference between the two is not significant at the 95% confidence level. The discrepancy between the total PA‐NA motion obtained from the geological data and NUVEL‐1A indicates that a marginally significant amount of NE‐SW shortening (possibly as much as 5 mm/yr) is missing overall in the geologic data. Shortening may occur in the long‐term in the offshore and coastal areas of California where such shortening is required in the shorter‐term geodetic solution. The seismic moment released in the last 148 years is ∼59% of the total moment release rate expected from long‐term strain rate field (including both seismic and aseismic deformation) derived from the inversion of geological data with NUVEL‐1A far‐field PA‐NA motion constraints. The accumulated strain in the areas containing the southern San Andreas fault‐San Jacinto fault, the San Francisco Bay area, and the area containing the Ventura basin in the Western Transverse Ranges in the last 148 years is the equivalent to that which could be released by an Mw>7.0 earthquake in each 50 × 100 × 15 (km3) crustal volume if strain is to be released seismically in these areas.

Contemporary kinematics of the western United States determined from earthquake moment tensors, very long baseline interferometry, and GPS observations

Using moment tensors from earthquakes between 1850 and 1995, we determine the horizontal velocity gradient tensor field associated with the seismic deformation in the western U.S. plate boundary zone. The velocity vectors obtained from the integration of the seismic strain rates across the entire plate boundary lie within 5° of the NUVEL‐IA Pacific‐North American plate motion direction. The magnitude of the earthquake‐related velocity is 62% of the NUVEL‐1A total Pacific‐North American plate motion. If earthquake occurrence is a random, Poisson‐like process, then the large formal errors associated with the model velocity estimates are an indication that the 144‐year interval is too short to define the long‐term seismic moment release rate to within an uncertainty less than the deficit between the observed seismic moment release rate and the total long‐term moment release rate associated with both the seismic and aseismic deformation. To spatially resolve the deficit of the earthquake moment release in the last 144 years, we have also estimated the total long‐term horizontal velocity gradient tensor field using seismic strain rate tensors and recent geodetic velocities, both with and without the NUVEL‐1A Pacific‐North American rigid plate motion constraint. We find a systematic difference between the Southern California Earthquake Center (SCEC) geodetic velocity map and NUVEL‐1A North American reference frames with a pole at (38.8°N, 124.2°W, 0.33° m.y.−1). This difference is significant at the 99% confidence level and indicates that 4–7 mm/yr of NE‐SW convergence is absorbed in offshore areas of southern California if both the SCEC velocity map and NUVEL‐1A correctly describe the relative motion between the Pacific and North America plates. Deficit strain rates calculated from comparison of observed seismic strain rates over the last 144 years with long‐term model strain rate tensors are largest in areas east of San Francisco Bay, in the Transverse Ranges, and along the southern San Andreas fault and San Jacinto fault. The deficit in each 100 by 50 km grid area within these regions is equivalent to an accumulation of strain in the last 144 years that would be released by an Mw>7 earthquake. Elsewhere the accumulated strain in the last 144 years is small, except in northern Mojave Desert, along the Garlock fault, and in the region southwest of the southern San Andreas‐San Jacinto fault zone. Within each 100 by 50 km grid area in these regions the accumulated strain is equivalent to that released by an Mw=6.5–7.0 earthquake.

Scalar imaging velocimetry measurements of the velocity gradient tensor field in turbulent flows. II. Experimental results

Scalar imaging velocimetry is here applied to experimental turbulent flow scalar field data to yield the first fully resolved, non-intrusive laboratory measurements of the spatio-temporal structure and dynamics of the full nine-component velocity gradient tensor field ∇u(x,t), as well as the pressure gradient field ∇p(x,t), in a turbulent flow. Results are from turbulent flows at outer scale Reynolds numbers in the range 3,000≤Reδ≤4,200, with Taylor scale Reynolds numbers Reλ≊45. These give a previously inaccessible level of detailed experimental access to the spatial structure in the velocity gradient tensor field at the small scales of turbulent flows, and through the much longer temporal dimension of these four-dimensional data spaces allow access to the inertial range of scales as well. Sample spatio-temporal data planes and probability distributions spanning more than 75 advection time scales (λν/U) are presented for various dynamical fields of interest, including the three components of the velocity field u(x,t), the nine components of the velocity gradient tensor field ∇u(x,t) through the full vector vorticity field ωi(x,t) and tensor strain rate field εij(x,t), the kinetic energy dissipation rate field Φ(x,t)≡2νε:ε(x,t), the enstrophy field 1/2ω⋅ω(x,t), the enstrophy production rate field ω⋅ε⋅ω(x,t), and the pressure gradient field ∇p(x,t). Continuity tests show agreement with the zero divergence requirement that exceeds the highest values reported from single-point, invasive, multi-probe measurements. Distributions of strain rate eigenvalues as well as alignments of the strain rate eigenvectors with both the vorticity and scalar gradient vectors are in agreement with DNS results, as are distributions of the measured helicity density fields u⋅ω(x,t). Results obtained for the true kinetic energy dissipation rate field show good agreement, up to 14th-order, with previous inertial range structure function exponents measured by Anselmet et al. [J. Fluid Mech. 140, 63 (1984)] at much higher Reynolds numbers. In addition, probability distributions scaled on inner variables show good agreement among buoyant and non-buoyant turbulent flow cases, further suggesting that these results are largely indicative of the high Reynolds number state of the inner scales of fully developed turbulent flows.

Scalar imaging velocimetry measurements of the velocity gradient tensor field in turbulent flows. I. Assessment of errors

The concept of flow field velocimetry based on scalar imaging measurements [Phys. Fluids A 4, 2191 (1992)] is here formulated in terms of an integral minimization implementation, where the velocity field u(x,t) is found by minimizing weighted residuals of the conserved scalar transport equation, along with the continuity condition and a smoothness condition. We apply this technique to direct numerical simulation (DNS) data for the limiting case of turbulent mixing of a Sc=1 passive scalar field. The spatial velocity fields u(x,t) thus obtained demonstrate good correlation with the exact DNS fields, as do the statistics of the velocity and the velocity gradient fields. The results from this integral minimization implementation also show significant improvement over those from the direct inversion technique reported earlier. These results are shown to be largely insensitive to noise at levels characteristic of current fully resolved scalar field measurements.