Microtremor surveys based on rotational seismology: theoretical analysis with focus on separation of Rayleigh and Love waves in general wavefield of microtremors

SUMMARY The applicability of rotational seismology to the general wavefield of microtremors is theoretically demonstrated based on a random process model of a 2-D wavefield. We show the effectiveness of taking the rotations (i.e. spatial differentiation) of microtremor waveforms in separating the Rayleigh and Love waves in a wavefield where waves are simultaneously arriving from various directions with different intensities. This means that a method based on rotational seismology (a rotational method) is capable of separating Rayleigh and Love waves without adopting a specific array geometry or imposing a specific assumption on the microtremor wavefield. This is an important feature of a rotational method because the spatial autocorrelation (SPAC) method, a conventional approach for determining phase velocities in microtremor array surveys, requires either the use of a circular array or the assumption of an isotropic wavefield (i.e. azimuthal averaging of correlations is required). Derivatives of the SPAC method additionally require the assumption that Rayleigh and Love waves are uncorrelated. We also show that it is possible to apply a rotational method to determine the characteristics of Love waves based on a simple three-point microtremor array that consists of translational (i.e. ordinary) three-component sensors. In later sections, we assume realistic data processing for microtremor arrays with translational sensors to construct a theoretical model to evaluate the effects of approximating spatial differentiation via finite differencing (i.e. array-derived rotation, ADR) and the effects of incoherent noise on analysis results. Using this model, it is shown that in a short-wavelength range compared to the distance for finite differencing (e.g. $\lambda < 3h$, where $\lambda $ and $h$ are the wavelength and distance for finite differencing, respectively), the leakage of unwanted wave components can determine the analysis limit. It is also shown that in a long-wavelength range (e.g. $\lambda > 3h$), the signal intensity gradually decreases, and thus the effects of incoherent noise increase (i.e. the signal-to-noise ratio, SNR decreases) and determine the analysis limit. We derive the relation between the SNR and wavelength. Although the analysis results quantitatively depend on the array geometry used for finite differencing, the qualitative understanding supported by mathematical expressions with a physically clear meaning can serve as a guideline for the treatment of data obtained from ADR.

Slow-burning instabilities of Dufort-Frankel finite differencing

DuFort–Frankel averaging is a tactic to stabilize Richardson’s unstable three-level leapfrog timestepping scheme. By including the next time level in the right-hand-side evaluation, it is implicit, but it can be rearranged to give an explicit updating formula, thus apparently giving the best of both worlds. Textbooks prove unconditional stability for the heat equation, and extensive use on a variety of advection–diffusion equations has produced many useful results. Nonetheless, for some problems the scheme can fail in an interesting and surprising way, leading to instability at very long times. An analysis for a simple problem involving a pair of evolution equations that describe the spread of a rabies epidemic gives insight into how this occurs. An even simpler modified diffusion equation suffers from the same instability. Finally, the rabies problem is revisited and a stable method is found for a restricted range of parameter values, although no prescriptive recipe is known which selects this particular choice. doi:10.1017/S1446181121000043

SLOW-BURNING INSTABILITIES OF DUFORT–FRANKEL FINITE DIFFERENCING

DuFort–Frankel averaging is a tactic to stabilize Richardson’s unstable three-level leapfrog timestepping scheme. By including the next time level in the right-hand-side evaluation, it is implicit, but it can be rearranged to give an explicit updating formula, thus apparently giving the best of both worlds. Textbooks prove unconditional stability for the heat equation, and extensive use on a variety of advection–diffusion equations has produced many useful results. Nonetheless, for some problems the scheme can fail in an interesting and surprising way, leading to instability at very long times. An analysis for a simple problem involving a pair of evolution equations that describe the spread of a rabies epidemic gives insight into how this occurs. An even simpler modified diffusion equation suffers from the same instability. Finally, the rabies problem is revisited and a stable method is found for a restricted range of parameter values, although no prescriptive recipe is known which selects this particular choice.

A Two-Way Nesting Unstructured Quadrilateral Grid, Finite-Differencing, Estuarine and Coastal Ocean Model with High-Order Interpolation Schemes

The balance between the need of improving horizontal resolution in simulating local small-scale ocean processes and computational costs makes it desirable to refine model mesh locally. A three-dimensional, two-way nesting unstructured quadrilateral grid, primitive equations, finite-differencing, estuarine and coastal ocean model is developed for multi-scale modeling. Because the model grid is capable of multi-area nesting and multi-level refinement at each subdomain, the model is highly compatible with simulations involved in complex topography and studies of local small-scale ocean processes. The two-way information exchange is achieved by a virtual grid method, and its basic idea is to implement numerical integrations of variables at nesting interfaces with the support of virtual grid variables, which are interpolated or updated from actual grid variables. The model is novel for two interpolation schemes: the high-order spatial interpolation at the middle temporal level (HSIMT) parabolic interpolation scheme and HSIMT advection-equivalent interpolation scheme, and they have high-order accuracy and good consistency with the advection scheme applied to solving the tracer equations. The conservation of both volume and tracer contents is ensured via a flux correction algorithm. The two original interpolation schemes are examined in an ideal salinity advection experiment in the peak preservation skill, stability, and conservation properties. A realistic application to the Deep Waterway Project area in the Changjiang Estuary showed that the nested grid model can reproduce the hydrodynamic processes at the observed sites successfully while it failed to maintain the performance with the structured grid model in simulating the variance of salinity, for which the enforced conservation had primary responsibility. The HSIMT parabolic interpolation scheme was distinguished from other schemes for its outstanding performances in simulating salinity.

Potential magnetic field calculator for solar physics applications using staggered grids

A program has been designed to generate accurately a potential magnetic field on a staggered grid by extrapolating the magnetic field normal to the photospheric surface. The code first calculates a magnetic potential using the Green’s function method and then uses a finite differencing scheme to calculate the magnetic field from the potential. A new finite differencing formula was derived which accounts for grid staggering; it is shown that this formula gives a numerical approximation that is closest to the real potential field. It is also shown that extending the region over which normal photospheric field is specified can improve the accuracy of the potential field produced. The program is a FORTRAN 90 code that can be used to generate potential magnetic field inputs for Lare3d and other MHD solvers that use a staggered grid for magnetic field components. The program can be parallelised to run quickly over multiple computing cores. The code and supporting description are provided in the appendices.

The Influence of Vertical Advection Discretization in the WRF-ARW Model on Capping Inversion Representation in Warm-Season, Thunderstorm-Supporting Environments

Previous studies have suggested that the Advanced Research version of the Weather Research and Forecasting (WRF-ARW) Model is unable, in its default configuration, to adequately resolve the capping inversions that are commonly found in the warm-season, thunderstorm-supporting environments of the central United States. Since capping inversions typically form in environments of synoptic-scale subsidence, this study tests the hypothesis that this degradation results, in part, from implicit numerical damping of shorter-wavelength features associated with the model-default third-order-accurate vertical advection finite-differencing scheme. To aid in testing this hypothesis, two short-range, deterministic, convection-allowing model forecasts, one using the default third-order-accurate vertical advection finite-differencing scheme and another using a fourth-order-accurate differencing scheme (which lacks implicit damping but is numerically dispersive), are conducted for 25 days during the 2017 NOAA Hazardous Weather Testbed Spring Forecasting Experiment. Model-derived vertical profiles at lead times of 11 and 23 h are validated against available rawinsonde observations released in regions located in the Storm Prediction Center’s 0600 UTC day 1 convection outlook’s “general thunderstorm” forecast area. The fourth-order-accurate vertical advection finite-differencing scheme is shown to not result in statistically significant improvements to model-forecast capping inversions or, more generally, the vertical thermodynamic profile in the lower troposphere. Instead, the fourth-order-accurate differencing scheme primarily impacts the representation of longer-wavelength features already reasonably well resolved by the model. The analysis does, however, provide quantitative evidence over a large sample that, on average, the WRF-ARW model forecasts capping inversions that are too weak, with negative buoyancy spread out over too deep of a vertical layer, compared to observations.

Optimization of Finite-Differencing Kernels for Numerical Relativity Applications

A simple optimization strategy for the computation of 3D finite-differencing kernels on many-cores architectures is proposed. The 3D finite-differencing computation is split direction-by-direction and exploits two level of parallelism: in-core vectorization and multi-threads shared-memory parallelization. The main application of this method is to accelerate the high-order stencil computations in numerical relativity codes.