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In this paper, we study the effect of spatial average and time delay on the dynamics of a diffusive predator–prey model under the Neumann boundary condition. Compared to the model without spatial average, the delay‐induced Hopf bifurcation at the first critical value of delay is nonhomogeneous due to the joint effects of spatial average and delay, and spatiotemporal patterns arise. Also, we show that the spatially homogeneous and nonhomogeneous periodic patterns may switch for different bifurcation parameter values. Moreover, a double Hopf bifurcation occurs at the intersection point of the homogeneous and nonhomogeneous Hopf bifurcation curves, and spatially nonhomogeneous quasi‐periodic patterns can be observed via numerical simulations near the double Hopf bifurcation point. The normal form algorithm for the spatially nonhomogeneous/homogeneous Hopf bifurcation is derived for a general reaction‐diffusion system with spatial average and delay. [ABSTRACT FROM AUTHOR]

In this paper, we are concerned with a reaction-diffusion model incorporating delay and nonlocal effects. The normal form of double Hopf bifurcation is derived. The diffusive model of pollen tube tip growth is discussed and numerical simulations show that spatially homogeneous and inhomogeneous periodic solutions can be both stable or connected by a heteroclinic orbit under certain conditions. In addition, the diffusive Lotka-Volterra model with delay and nonlocality is considered and spatially inhomogeneous quasi-periodic solution is obtained. • Derive the algorithm of normal form of double Hopf bifurcation for reaction-diffusion system with delay and spatial average. • Investigate the dynamics near the double Hopf bifurcation point for two biological models. • Find bistability of homogeneous and inhomogeneous periodic solutions and pattern transitions for pollen tube tip growth model. • Find stable inhomogeneous quasi-periodic solutions and pattern transitions of periodic and quasi-periodic solutions for Lotka-Volterra model. [ABSTRACT FROM AUTHOR]

This paper presents a method for measuring the reciprocity parameter based on the free field. It is able to achieve accurate measurement of the reverberation constant in a narrow band. The method uses the same transmitting and receiving system, and keeps the same set of parameters to measure the open circuit voltage output under different frequencies in a free field. The open circuit output voltage is measured through average technology in the reverberation control region, then the reverberation radius is calculated and the reciprocity constant obtained. This method uses a single frequency signal and the spatial averaging technique. It is simple, convenient and not suitable for complex measuring instruments. The validity of the method is verified by comparing the measured results with the reverberation time measurement.

Transformation models are used to infer geotechnical properties from indirect measurements. A site-specific transformation model can be calibrated with direct and indirect measurements from a site. When such a model is used, then spatial variability, measurement errors and statistical uncertainty propagate into the uncertainty of the spatial average, which is the variable of interest in most geotechnical analyses. This research shows how all components enter the total uncertainty of a transformation model for undrained shear strength from cone resistance. A method is proposed to estimate the uncertainty in the spatial average undrained shear strength, particularly focusing on the role of averaging of all spatially variable error components. The main finding is that if a considerable share of the measurement and transformation errors is random or spatially variable, the uncertainty estimates can be considerably lower compared to methods proposed earlier, and hence, characteristic values can be considerably higher. [ABSTRACT FROM AUTHOR]

Global climate observations from ground stations require an evaluation of the effectiveness of a station network, which is often an assessment of the geometric distribution of n points on a sphere. The representativeness of the Angell–Korshover 63-station network (AK-network) is assessed in this paper. It is shown that AK-network can effectively sample the January global average temperature data of the NCEP/NCAR Reanalysis from 1948 to 2015 when estimating inter-decadal variations, but it has large uncertainties for estimating linear trends. This paper describes a method for the assessment, and also includes an iterative numerical algorithm used to search for the locations of 63 uniformly distributed stations, named U63. The results of AK-63 and U63 are compared. The Appendix explains a problem of searching for the optimal distribution of n points on a unit sphere in three-dimensional space under the condition of the maximum sum of the mutual distances among the points. The core R code for finding U63 is included. The R code can generate various interesting configurations for different n , among which one is particularly surprising: The configuration of 20 points is not a dodecahedron although the configurations for n = 4 , 6 , 8 , and 12 are tetrahedron, octahedron, cube, and icosahedron, respectively. [ABSTRACT FROM AUTHOR]

Mobile crowdsensing can facilitate environmental surveys by leveraging sensor-equipped mobile devices that carry out measurements covering a wide area in a short time without bearing the costs of traditional field work. In this article, we examine statistical methods to perform an accurate estimate of the mean value of an environmental parameter in a region based on such measurements. The main focus is on estimates produced by considering the mobile device readings at a random instant in time. We compare stratified sampling with different stratification weights to sampling without stratification as well as an appropriately modified version of systematic sampling. Our main result is that stratification with weights proportional to stratum areas can produce significantly smaller bias for a moderate number of strata and gets arbitrarily close to the true area average as the number of mobiles increases. The performance of the methods is evaluated for an application scenario where we estimate the mean area temperature in a linear region that exhibits the so-called Urban Heat Island effect, with mobile users moving in the region according to the Random Waypoint Model. [ABSTRACT FROM AUTHOR]

The purpose of this study is to examine in more detail under what conditions would spatial averaging over some prescribed region be sufficient to reproduce the response statistics arising from a spatially variable field. The spatially variable undrained shear strength is first simulated by a random field, and the actual response of a spatially variable clay in three problems (soil column, retaining wall, shallow foundation) is simulated using the random finite element method. This actual response is then compared to the spatial average response (the response of a homogeneous clay whose undrained shear strength is equal to certain spatial average). It is observed that the actual response can be well approximated by the spatial average response only for situations where the critical slip curve is constrained. This constraint is the most significant for the retaining wall and the least significant for the soil column. [ABSTRACT FROM AUTHOR]

Optimal designs of sampling spatial locations in estimating spatial averages of random fields are considered. The random field is assumed to have correlated values according to a covariance function. The quality of estimation is measured by the mean squared error. Simple nonparametric linear estimators along with sampling designs having a limiting density are considered. For a large class of locally isotropic random fields, we argue for the asymptotic optimality of simple linear estimators. The convergent rates of the mean squared error and optimal limiting densities of sampling designs are determined in every dimension. An example of simulation is given. [ABSTRACT FROM AUTHOR]

Highlights: [•] We develop new variations of methods for simulating Gaussian random vector fields. [•] Randomized Spectral Method can efficiently simulate low-order ensemble statistics. [•] Fourier-Wavelet Method performs better for spatial sampling of single realization. [•] Second-order Lagrangian statistics can be simulated well by either method. [•] Stratified sampling of angles in plane wave decomposition greatly helps convergence. [ABSTRACT FROM AUTHOR]

A Fourier series method (FSM) of simulating spatial averages of stationary Gaussian random fields is presented. The FSM is able to simulate spatial averages over nonequally spaced rectangular cells, and by adopting Gauss quadrature, it can be further applied to nonrectangular cells. It is also capable of simulating line averages over any prescribed line segment in two dimensions. The former (spatial averaging over cells) is essential for finite-element analysis, while the latter (line averaging over line segments) is essential for slope-stability analysis using limit equilibrium. To resolve the issue of unrealistic periodical correlation pertaining to the FSM, a rule of thumb is provided to extend the simulation space. For cases with nonrectangular cells, the required number of Gauss points to achieve a prescribed accuracy is calibrated for both one-dimensional (1D) and two-dimensional (2D) cases and for both the single-exponential (SExp) and squared-exponential (QExp) autocorrelation models. The consistency of the FSM is verified through 1D and 2D examples for both SExp and QExp models. [ABSTRACT FROM AUTHOR]

Abstract: Spectral multi-scaling postulates a power-law type of scaling of spectral distribution functions of stationary processes of spatial averages, over nested and geometrically similar sub-regions of the spatial parameter space of a given spatio-temporal random field. Presently a new framework is formulated for down-scaling processes of spatial averages, following naturally from the postulate of spectral multi-scaling, and key ingredients required for its implementation are described. Moreover, results from an extensive diagnostic study are presented, seeking statistical evidence supportive of spectral multi-scaling. Such evidence emerges from two sources of data. One is a 13year long historical record of radar observations of rainfall in southeastern UK (Chenies radar), with high spatial (2km) and temporal (5min) resolution. The other is an ensemble of rain rate fields simulated by a spatio-temporal random pulse model fitted to the historical data. The results are consistent between historical and simulated rainfall data, indicating frequency-dependent scaling relationships interpreted as evidence of spectral multi-scaling across a range of spatial scales. [Copyright &y& Elsevier]

Methods of computational prediction can be used or specific measurements have to be performed for the estimation of human exposure to nonionizing electromagnetic fields at a certain place. According to the exposure standards, the predicted or the measured values have to be spatially averaged in an area representing the dimensions of the human body and compared with the adopted maximum permissible levels of exposure. In this paper, methods for spatial averaging nonionizing electromagnetic fields are discussed. Moreover, methods for estimating spatial average uncertainty of the total field strength values and of the summation of exposure quotients when narrowband measurements are performed in a multiple-frequency environment are proposed. Furthermore, since maximum field values have to be averaged according to the most widely accepted exposure standards, the traffic-dependent power level variation issue is also addressed. In particular, for GSM systems, an extrapolation method to the maximum possible field levels using data from real-time measurements when the frequency allocation per operator base station and the spatial distribution of the exposure sources are unknown is also proposed. Finally, an uncertainty estimation procedure associated to the extrapolated levels is presented. [ABSTRACT FROM AUTHOR]

We propose a new approach to derive spatially-averaged momentum equations for an urban canopy model that resolves buildings vertically and not horizontally. First, in order to mathematically describe the actual momentum field as a completely continuous field, the underling concepts of the immersed boundary method are employed, where we assume that (i) the entire simulation space, including that occupied by buildings, is filled with a fluid, and (ii) an external body force field exists that reduces the wind speed to zero at all positions coinciding with the space occupied by the buildings. Then, in order to obtain the required spatially-averaged momentum equations in a self-consistent manner, a spatial-averaging operation is applied to the Navier–Stokes equations that include a term representing the external force field. The applied spatial-averaging operation is equivalent to the conventional spatial filtering operation used in large-eddy simulations. To examine the significance of the subgrid-scale (SGS) stresses of the spatially-averaged momentum equations, a numerical simulation is performed for a flow around a regular array of cubical blocks with a grid resolution that is sufficient to resolve the blocks. By estimating the individual terms in the spatially-averaged momentum equations using the simulation results, we show that the SGS stresses contribute significantly to the spatially-averaged momentum budget, and therefore they should not be neglected in urban canopy modelling. [ABSTRACT FROM AUTHOR]

Recently, several attempts have been made to model the wind velocity in an urban canopy in order to accurately predict the mixing and transport of momentum, heat, and pollutants within and above the canopy on an urban scale. For this purpose, unverified assumptions made by Macdonald (Boundary-Layer Meteorol 97:25–45, 2000) to develop a model for the profile of the mean wind velocity within an urban canopy have been used. In the present study, in order to provide foundations for improving the urban canopy models, the properties of the spatially-averaged mean quantities used to make these assumptions have been investigated by performing large-eddy simulations (LES) of the airflow around square and staggered arrays of cubical blocks with the following plan area densities: λ p = 0.05, 0.11, 0.16, 0.20, 0.25, and 0.33. The LES results confirm that the discrepancy between the spatial average of wind velocity and Macdonald’s five-point average of wind velocity can be large in both types of arrays for large λ p. It is also confirmed that Prandtl’s mixing length varies significantly with height within the canopy, contrary to Macdonald’s assumption for both types of arrays and for both small and large λ p. On the other hand, in accordance with Macdonald’s assumption, the sectional drag coefficient is found to be almost constant with height except in the case of staggered arrays with high λ p. [ABSTRACT FROM AUTHOR]