Assumption Of Local Homogeneity Research Articles

Tackling local ecological homogeneity: Finding intraspecific trait variability in local populations of Mediterranean plants.

Local homogeneity, in ecology, is the often undisclosed assumption that variability within populations is negligible or mostly distributed evenly. In large areas, this can lead to the aggregation of different populations without regard for their unique needs and characteristics, such as drought sensitivity and functional trait distributions. Here, we discuss whether this assumption can be justified, and we hypothesize that discerning the source of variation between plasticity and adaptation could be a feasible approach to formulate an informed decision. We test this hypothesis on plants, resorting to a common garden experiment to determine the source of variation of several plant functional traits at a local scale (~60 km) of three wild species: Quercus ilex, Pistacia lentiscus, and Cistus salviifolius. Individuals of each species were sourced from three key sites chosen along a local aridity gradient. Our approach led to the rejection of the local homogeneity assumption for Q. ilex and C. salviifolius at this scale due to the adaptive divergence observed among neighboring populations. This case study provides evidence that addressing local homogeneity can highlight diverging populations in a relatively simple way. We conclude that gathering empirical evidence on intraspecific variability is a feasible approach that can provide researchers with solid bases to decide whether to adopt the local homogeneity assumption or not.

Coarse graining for granular materials: micro-polar balances

In this work, we extend a previously discussed averaging method for granular materials (Artoni and Richard in Phys Rev E 91:032202, 2015), by considering average balance equations related to particle rotations (the micro-inertia, the angular momentum and the rotational energy balances). The method is based on a fluctuation decomposition taking into account velocity gradients and on a local homogeneity assumption. After having developed the coarse-grained balances, we study the simplifications related to some classes of simple flows and particularly assess the relation between the micro-inertia balance and the average orientation of particles in a simple shear flow.

Mean wind vector estimation using the velocity–azimuth display (VAD) method: an explicit algebraic solution

Abstract. This paper deals with the analysis of the sampling setup for Doppler profilers aiming at the determination of vertical profiles of the wind. An explicit solution for the retrieval of mean wind vectors under the assumption of local homogeneity is presented for the case of a symmetric velocity–azimuth display sampling, and a stability analysis is performed. Furthermore, the explicit solution allows a detailed investigation of the propagation of radial wind measurement errors on the retrieved wind vector.

Near wall dissipation revisited

The characteristics of the dissipation of the turbulent kinetic energy in wall-bounded flows are revisited through direct numerical simulations of turbulent channel flows realized in large computational domains up to the Reynolds number Reτ=hu¯τν=1100 where u¯τ is the friction velocity, ν and h stands for the kinematic viscosity and channel half width. It is shown that the local homogeneity assumption is acceptable in the whole layer, while the local isotropy is valid only in the far meso-layer and near the centerline. The mean dissipation in inner scale is Reynolds number dependent in the low-buffer and viscous sublayers. This is due to the local shear layers induced by the outer-layer irrotational eddies. A conceptual model is subsequently proposed to link the near wall dissipation to the large-scale structures. The dissipation statistics conditioned by fixed amplitudes of the velocity field are also analyzed in order to clarify whether the zero-crossings of the fluctuating velocity components contribute most to the energy dissipation or not. Incidentally, the occurrence of the Eulerian stagnation points is questioned. The largest contribution to the dissipation in the viscous sublayer comes from the level-crossings of the wall normal velocity v component in the spanwise direction, that peaks to 30% when v=0. The spanwise direction is the most active in terms of the level-crossing frequencies that are inversely proportional to the corresponding Taylor scales.

An MR-Based Viscosity-Type Regularization Method for Electrical Property Tomography

Here, a method based on viscosity-type regularization is proposed for magnetic resonance electrical property tomography (MREPT) to mitigate persistent artifacts when it is used to reconstruct a map of electrical properties based on data from a magnetic resonance imaging scanner. The challenges for solving the corresponding partial differential equation (PDE) are discussed in detail. The existing artifacts in the numerical results are pointed out and classified. The methods in the literature for MREPT are mainly based on an assumption of local homogeneity, which makes the approach simple but leads to artifacts in the transition region where electrical properties vary rapidly. Recent work has focused on eliminating the assumption of local homogeneity, and one of the solutions is convection–reaction MREPT that is based on a first-order PDE. Numerical solutions of the PDE have persistent artifacts in certain regions and global spurious oscillations. Here, a method based on viscosity-type regularization is proposed to effectively mitigate the aforementioned problems. Finite difference method is used for discretizing the governing PDE. Numerical experiments are presented to analyze the problem in detail. Electrical properties of different phantoms are successfully retrieved. The efficiency, accuracy, and noise tolerance of the proposed method are illustrated with numerical results.

Spatial regimes in regional European growth: an iterated spatially weighted regression approach

This paper proposes a new technique based on an iterated spatially weighted regression procedure to verify the presence of economic growth heterogeneities in EU regions in the period 1981–2009. The approach extends a procedure originally proposed in the field of image analysis based on the assumption of local homogeneity of the signal. The presence of the heterogeneity is a criterion to divide the sample of observations (i.e. regions) into smaller homogeneous groups. Our results highlight the presence in the EU regions of five different clubs with different growth paths within each subgroup. Spatial dependence is also considered in the definition of the economic convergence model.

Dynamically Consistent Parameterization of Mesoscale Eddies—Part II: Eddy Fluxes and Diffusivity from Transient Impulses

This work continues development of the framework for dynamically consistent parameterization of mesoscale eddy effects in non-eddy-resolving ocean circulation models. Here, we refine and extend the previous results obtained for the double gyres and aim to account for the eddy backscatter mechanism that maintains eastward jet extension of the western boundary currents. We start by overcoming the local homogeneity assumption and by taking into account full large-scale circulation. We achieve this by considering linearized-dynamic responses to finite-time transient impulses. Feedback from these impulses on the large-scale circulation are referred to as footprints. We find that the local homogeneity assumption yields only quantitative errors in most of the gyres but breaks down in the eastward jet region, which is characterized by the most significant eddy effects. The approach taken provides new insights into the eddy/mean interactions and framework for parameterization of unresolved eddy effects. Footprints provide us with maps of potential vorticity anomalies expected to be induced by transient eddy forcing. This information is used to calculate the equivalent eddy potential vorticity fluxes and their divergences that partition the double-gyre circulation into distinct geographical regions with specific eddy effects. In particular, this allows approximation of the real eddy effects that maintain the eastward jet extension of the western boundary currents and its adjacent recirculation zones. Next, from footprints and their equivalent eddy fluxes and from underlying large-scale flow gradients, we calculate spatially inhomogeneous and anisotropic eddy diffusivity tensor. Its properties suggest that imposing parameterized source terms, that is, equivalent eddy flux divergences, is a better parameterization strategy than implementation of the eddy diffusion.

Current helicity and magnetic field anisotropy in solar active regions

The electric current helicity density $\displaystyle \chi=\langle\epsilon_{ijk}b_i\frac{\partial b_k}{\partial x_j}\rangle$ contains six terms, where $b_i$ are components of the magnetic field. Due to the observational limitations, only four of the above six terms can be inferred from solar photospheric vector magnetograms. By comparing the results for simulation we distinguished the statistical difference of above six terms for isotropic and anisotropic cases. We estimated the relative degree of anisotropy for three typical active regions and found that it is of order 0.8 which means the assumption of local isotropy for the observable current helicity density terms is generally not satisfied for solar active regions. Upon studies of the statistical properties of the anisotropy of magnetic field of solar active regions with latitudes and with evolution in the solar cycle, we conclude that the consistency of that assumption of local homogeneity and isotropy requires further analysis in the light of our findings.

Magnetic resonance-based reconstruction method of conductivity and permittivity distributions at the Larmor frequency

Magnetic resonance electric properties tomography (MREPT) is a recent medical imaging modality for visualizing the electrical tissue properties of the human body using radio-frequency magnetic fields. It uses the fact that in magnetic resonance imaging (MRI) systems the eddy currents induced by the radio-frequency magnetic fields reflect the conductivity (σ) and permittivity distributions inside the tissues through Maxwell’s equations. The corresponding inverse problem consists of reconstructing the admittivity distribution at the Larmor frequency ( MHz for a 3 Tesla MRI machine) from the positive circularly polarized component of the magnetic field . Previous methods are usually based on an assumption of local homogeneity which simplifies the governing equation. However, previous methods that include the assumption of homogeneity are prone to artifacts in the region where γ varies. Hence, recent work has sought a reconstruction method that does not assume local-homogeneity. This paper presents a new MREPT reconstruction method which does not require any local homogeneity assumption on γ. We find that γ is a solution of a semi-elliptic partial differential equation with its coefficients depending only on the measured data , which enable us to compute a blurred version of γ. To improve the resolution of the reconstructed image, we developed a new optimization algorithm that minimizes the mismatch between the data and the model data as a highly nonlinear function of γ. Numerical simulations are presented to illustrate the potential of the proposed reconstruction method.

Direct vibro-elastography FEM inversion in Cartesian and cylindrical coordinate systems without the local homogeneity assumption

To produce images of tissue elasticity, the vibro-elastography technique involves applying a steady-state multi-frequency vibration to tissue, estimating displacements from ultrasound echo data, and using the estimated displacements in an inverse elasticity problem with the shear modulus spatial distribution as the unknown. In order to fully solve the inverse problem, all three displacement components are required. However, using ultrasound, the axial component of the displacement is measured much more accurately than the other directions. Therefore, simplifying assumptions must be used in this case. Usually, the equations of motion are transformed into a Helmholtz equation by assuming tissue incompressibility and local homogeneity. The local homogeneity assumption causes significant imaging artifacts in areas of varying elasticity. In this paper, we remove the local homogeneity assumption. In particular we introduce a new finite element based direct inversion technique in which only the coupling terms in the equation of motion are ignored, so it can be used with only one component of the displacement. Both Cartesian and cylindrical coordinate systems are considered. The use of multi-frequency excitation also allows us to obtain multiple measurements and reduce artifacts in areas where the displacement of one frequency is close to zero. The proposed method was tested in simulations and experiments against a conventional approach in which the local homogeneity is used. The results show significant improvements in elasticity imaging with the new method compared to previous methods that assumes local homogeneity. For example in simulations, the contrast to noise ratio (CNR) for the region with spherical inclusion increases from an average value of 1.5–17 after using the proposed method instead of the local inversion with homogeneity assumption, and similarly in the prostate phantom experiment, the CNR improved from an average value of 1.6 to about 20.

Curl-Based Finite Element Reconstruction of the Shear Modulus Without Assuming Local Homogeneity: Time Harmonic Case

In elasticity imaging, the shear modulus is obtained from measured tissue displacement data by solving an inverse problem based on the wave equation describing the tissue motion. In most inversion approaches, the wave equation is simplified using local homogeneity and incompressibility assumptions. This causes a loss of accuracy and therefore imaging artifacts in the resulting elasticity images. In this paper we present a new curl-based finite element method inversion technique that does not rely upon these simplifying assumptions. As done in previous research, we use the curl operator to eliminate the dilatational term in the wave equation, but we do not make the assumption of local homogeneity. We evaluate our approach using simulation data from a virtual tissue phantom assuming time harmonic motion and linear, isotropic, elastic behavior of the tissue. We show that our reconstruction results are superior to those obtained using previous curl-based methods with homogeneity assumption. We also show that with our approach, in the 2-D case, multi-frequency measurements provide better results than single-frequency measurements. Experimental results from magnetic resonance elastography of a CIRS elastography phantom confirm our simulation results and further demonstrate, in a quantitative and repeatable manner, that our method is accurate and robust.

Covariance Localization with the Diffusion-Based Correlation Models

Abstract Improving the performance of ensemble filters applied to models with many state variables requires regularization of the covariance estimates by localizing the impact of observations on state variables. A covariance localization technique based on modeling of the sample covariance with polynomial functions of the diffusion operator (DL method) is presented. Performance of the technique is compared with the nonadaptive (NAL) and adaptive (AL) ensemble localization schemes in the framework of numerical experiments with synthetic covariance matrices in a realistically inhomogeneous setting. It is shown that the DL approach is comparable in accuracy with the AL method when the ensemble size is less than 100. With larger ensembles, the accuracy of the DL approach is limited by the local homogeneity assumption underlying the technique. Computationally, the DL method is comparable with the NAL technique if the ratio of the local decorrelation scale to the grid step is not too large.

Conductivity and Permittivity Image Reconstruction at the Larmor Frequency Using MRI

This paper presents a new method of providing both conductivity ($\sigma$) and permittivity ($\epsilon$) images at the MR Larmor frequency in magnetic resonance electrical property tomography (MREPT), a relatively new MR-based electrical tissue property imaging modality. In MREPT, the RF coil of the MR scanner is used to feed a sinusoidal current at the Larmor frequency, $\omega/2\pi$ (approximately 128 MHz for a 3 Tesla MRI machine), to an imaging object within the MR scanner. Inside the object, this injection current induces a time-harmonic magnetic field, $\mathbf{H}=(H_x,H_y,H_z)$, that is influenced by the body's admittivity distribution, $\gamma=\sigma+i\omega\epsilon$, via Maxwell's equations. Currently, the positive rotating field, $H^+=(H_x+iH_y)/2$, is the only measurable quantity which can be obtained from $B_1$ mapping techniques. The inverse problem of MREPT is to reconstruct $\gamma$ from the $H^+$ data. Existing MREPT reconstruction methods use the local homogeneity assumption ($\nabla\gamm...

Elasticity reconstruction: Beyond the assumption of local homogeneity

Elasticity imaging is a novel domain which is currently gaining significant interest in the medical field. Most inversion techniques are based on the homogeneity assumption, i.e. the local spatial derivatives of the complex-shear modulus are ignored. This analysis presents an analytic approach in order to overcome this limitation, i.e. first order spatial derivatives of the real-part of the complex-shear modulus are taken into account. Resulting distributions in a gauged breast lesion phantom agree very well with the theoretical expectations. An in-vivo example of a cholangiocarcinoma demonstrates that the new approach provides maps of the viscoelastic properties which agree much better with expectations from anatomy.

Shear Modulus Decomposition Algorithm in Magnetic Resonance Elastography

Magnetic resonance elastography (MRE) is an imaging modality capable of visualizing the elastic properties of an object using magnetic resonance imaging (MRI) measurements of transverse acoustic strain waves induced in the object by a harmonically oscillating mechanical vibration. Various algorithms have been designed to determine the mechanical properties of the object under the assumptions of linear elasticity, isotropic and local homogeneity. One of the challenging problems in MRE is to reduce the noise effects and to maintain contrast in the reconstructed shear modulus images. In this paper, we propose a new algorithm designed to reduce the degree of noise amplification in the reconstructed shear modulus images without the assumption of local homogeneity. Investigating the relation between the measured displacement data and the stress wave vector, the proposed algorithm uses an iterative reconstruction formula based on a decomposition of the stress wave vector. Numerical simulation experiments and real experiments with agarose gel phantoms and human liver data demonstrate that the proposed algorithm is more robust to noise compared to standard inversion algorithms and stably determines the shear modulus.

Multiscale local change point detection with applications to value-at-risk

This paper offers a new approach to modeling and forecasting of nonstationary time series with applications to volatility modeling for financial data. The approach is based on the assumption of local homogeneity: for every time point, there exists a historical interval of homogeneity, in which the volatility parameter can be well approximated by a constant. The proposed procedure recovers this interval from the data using the local change point (LCP) analysis. Afterward, the estimate of the volatility can be simply obtained by local averaging. The approach carefully addresses the question of choosing the tuning parameters of the procedure using the so-called “propagation” condition. The main result claims a new “oracle” inequality in terms of the modeling bias which measures the quality of the local constant approximation. This result yields the optimal rate of estimation for smooth and piecewise constant volatility functions. Then, the new procedure is applied to some data sets and a comparison with a standard GARCH model is also provided. Finally, we discuss applications of the new method to the Value at Risk problem. The numerical results demonstrate a very reasonable performance of the new method.

Seismic depth imaging with the Gabor transform

Wavefield extrapolation in depth, a vital component of wave-equation depth migration, is accomplished by repeatedly applying a mathematical operator that propagates the wavefield across a single depth step, thus creating a depth marching scheme. The phase-shift method of wavefield extrapolation is fast and stable; however, it can be cumbersome to adapt to lateral velocity variations. We address the extension of phase-shift extrapolation to lateral velocity variations by using a spatial Gabor transform instead of the normal Fourier transform. The Gabor transform, also known as the windowed Fourier transform, is applied to the lateral spatial coordinates as a windowed discrete Fourier transform where the entire set of windows is required to sum to unity. Within each window, a split-step Fourier phase shift is applied. The most novel element of our algorithm is an adaptive partitioning scheme that relates window width to lateral velocity gradient such that the estimated spatial positioning error is bounded below a threshold. The spatial positioning error is estimated by comparing the Gabor method to its mathematical limit, called the locally homogeneous approximation — a frequency-wavenumber-dependent phase shift that changes according to the local velocity at each position. The assumption of local homogeneity means this position-error estimate may not hold strictly for large scattering angles in strongly heterogeneous media. The performance of our algorithm is illustrated with imaging results from prestack depth migration of the Marmousi data set. With respect to a comparable space-frequency domain imaging method, the proposed method improves images while requiring roughly 50% more computing time.

“Locally homogeneous turbulence”: Is it an inconsistent framework?

In his first 1941 paper Kolmogorov assumed that the velocity has increments that are homogeneous and independent of the velocity at a suitable reference point. This assumption of local homogeneity is consistent with the nonlinear dynamics only in an asymptotic sense when the reference point is far away. This inconsistency is illustrated numerically using the Burgers equation. Kolmogorov’s derivation of the four-fifths law for the third-order structure function and its anisotropic generalization are actually valid only for homogeneous turbulence, but a local version due to Duchon and Robert still holds. A Kolmogorov-Landau approach is proposed to handle the effect of fluctuations in the large-scale velocity on small-scale statistical properties; it is only a mild extension of the 1941 theory and does not incorporate intermittency effects.

On Lagrangian time scales and particle dispersion modeling in equilibrium turbulent shear flows

As intermediate quantities available from various existing numerical computations, the fluid Lagrangian time scales are of primary importance in the development of probability density function models for turbulent flows. Similarly, the time scales of the fluid seen by discrete particles in two-phase flows are essential for the development of dispersion models based on stochastic differential equations. Such time scales are obviously depending not only on the particle properties but also on the fluid Lagrangian and Eulerian time scales. A model is proposed here to estimate the directional dependence of the fluid Lagrangian time scales and drift coefficients in one-directional equilibrium turbulent shear flows, based on a local homogeneity assumption in the frame of the generalized Langevin model. Through comparison with available direct and large eddy simulation predictions in channel flows and in a homogeneous shear flow, the model is shown to lead to significant improvements in the streamwise and spanwise directions, where the existing empirical laws for the Lagrangian time scales are far from being satisfactory. We examine the way this model can be used to build a suitable stochastic process for the fluid seen by inertial particles in such basic turbulent shear flows.

Phase‐space marching algorithm in the presence of a planar wave velocity discontinuity—A qualitative study

The methodology of marching phase‐space transformations has been previously considered for slowly varying media. The present work implements the approach for a medium with a planar wave velocity discontinuity. It is shown that propagation in the water column can be systematically separated from the scattering events at the discontinuity. The latter take place in a narrow interaction zone, out of which the previously invoked local homogeneity assumption can be used again. The width of the interaction zone is obtained by the investigation of the recently developed evolution equation of the local spectra. The associated phase‐space propagators and the isolation of phenomena such as reflection, refraction, and head waves in phase space are demonstrated. Computations are performed and compared to reference solutions.