Autoregressive filtering versus linear detrending in estimation of fluxes by the eddy covariance method

The application of autoregressive running mean filtering (RMF) and linear detrending (LDT) in the estimation of turbulent fluxes by the eddy covariance method is analysed. The systematic, as well as the random, errors of the fluxes arising from filtering and/or limited observation time effects are described. To observe negligible systematic errors in fluxes, the RMF has to be applied with moderately long time constants. However, the obtained flux values are subject to increased random errors during periods of non-stationarity and the method leads to systematic overestimation of variances. These shortcomings are not inherent in the LDT approach, which is recommended for use. But the systematic errors of fluxes due to LDT are not negligible under certain experimental conditions and have to be accounted for. The corrections are important because the relatively small errors in short-period fluxes can translate to significant errors in long-period averages. The corrections depend on the turbulence time scales, which should be preferably estimated as ensemble mean variables for a particular site.     

Evaporation from boreal reservoirs: A comparison between eddy covariance observations and estimates relying on limited data

Hydrological models used for reservoir management typically lack an accurate representation of open-water evaporation and must be run in a scarce data context. This study aims to identify an accurate means to estimate reservoir evaporation with simple meteorological inputs during the open-water season, using long-term eddy covariance observations from two boreal hydropower reservoirs with contrasting morphometry as reference. Unlike the temperate water bodies on which the majority of other studies have focused, northern reservoirs are governed by three distinct periods: ice cover in the cold season, warming in the summer and energy release in the fall. The reservoirs of interest are Eastmain-1 (52°N, mean depth of 11 m) and Romaine-2 (51°N, mean depth of 42 m), both located in eastern Canada. Four approaches are analysed herein: a combination approach, a radiation-based approach, a mass-transfer approach, and empirical methods. Of all the approaches, the bulk transfer equation with a constant Dalton number of 1.2 x 10⁻³ gave the most accurate estimation of evaporation at hourly time steps, compared with the eddy covariance observations (RMSE of 0.06 mm h⁻¹ at Eastmain-1 and RMSE of 0.04 mm h⁻¹ at Romaine-2). The daily series also showed good accuracy (RMSE of 1.38 mm day⁻¹ at Eastmain-1 and RMSE of 0.62 mm day⁻¹ at Romaine-2) both in the warming and energy release phases of the open-water season. The bulk transfer equation, on the other hand, was incapable of reproducing condensation episodes that occurred soon after ice breakup. Basic and variance-based sensitivity analyses were conducted, in particular to measure the variation in performance when the bulk transfer equation was applied with meteorological observations collected at a certain distance (~10–30 km) from the reservoir. This exercise illustrated that accurate estimates of open water evaporation require representative measurements of wind speed and water surface temperature.     

The random coin method: Solution of the problem of the simulation of a random function in the plane

This paper presents a new geostatistical method to obtain realizations of stationary random functions in the plane.     

Canonical variate analysis with unequal covariance matrices: Generalizations of the usual solution

Canonical variate analysis is extended for use when the covariance matrices are not equal. Linear combinations of variates are derived by generalizing either a weighted between-groups approach or the likelihood-ratio test and the associated noncentrality matrix. The usual solution and the two generalizations are compared via generated data for a few typical configurations of means in a situation in which the covariance matrices are in fact equal. The MSE of the canonical variate coefficients and group means for the generalizations are approximately three times those for the usual solution, due to corresponding changes in the variances. Two examples are discussed.     

Isotropic reproducing kernels for the inner of a sphere or spherical shell and their use as density covariance functions

Isotropic reproducing kernels for a sphere or spherical shell are derived as weighted product sums o fL ₂ orthonormal base functions. For the sphere these functions are products of the surface spherical harmonics and the Jacobi polynomials of degree (0, 2). Reproducing kernels for a sphere are consistent with the covariance function of the outer anomalous gravity potential of the Earth. These reproducing kernels may be used for gravity field modeling which include density (anomaly) data as observations or which aims at predicting such quantities using optimal estimation methods, that is for solving the inverse gravimetric problem.     

ISO2002: an analytical stochastic model of multi-difference GPS carrier-phase data

The differencing technique is useful in global positioning system (GPS) positioning when two or more GPS receivers collect simultaneous observables from common satellites at each epoch, and all carrier-phase observables have the same normal distribution. An analytical probability distribution of the single-, double-, triple- and multi-difference GPS observables is obtained. This analytical model, called ISO2002, has a good matrix structure, in which I indicates the number of receivers, S indicates the number of observed satellites, and O indicates the number of epochs. The variance–covariance matrix can be expressed as the Kronecker product of several small matrices, so its inverse is equal to the Kronecker product of the inverses of these sub-matrices. Moreover, these small matrices are circulant or symmetric diagonal Toeplitz matrices, so their inverses have analytical solutions. The analytical model ISO2002 proposed to compute the inverse variance–covariance matrix is shown to be very effective.     

On criteria for measures of compositional difference

Simple perceptions about the nature of compositions lead through logical necessity to certain forms of analysis of compositional data. In this paper the consequences of essential requirements of scale, perturbation and permutation invariance, together with that of subcompositional dominance, are applied to the problem of characterizing change and measures of difference between two compositions. It will be shown that one strongly advocated scalar measure of difference fails these tests of logical necessity, and that one particular form of scalar measure of difference (the sum of the squares of all possible logratio differences in the components of the two compositions), although not unique, emerges as the simplest and most tractable satisfying the criteria.     

New Equations For Sonic Temperature Variance And Buoyancy Heat Flux With An Omnidirectional Sonic Anemometer

Many new types of sonic anemometer obtain sonic temperature from an average value of temperature measured along three paths, unlike previous sonic anemometers that generally used one path. New equations are derived to calculate temperature variance from sonic temperature variance and sensible heat flux from buoyancy flux considering the influence of a crosswind. These equations can be applied to CSAT3, Solent R2, R3, R3A, HS, and USA-1 sonic anemometers with the corresponding correction factors given in this paper. The equations are verified by data measured by a CSAT3 sonic anemometer in the LITFASS-1998 field study.     

On the condition number of covariance matrices in kriging, estimation, and simulation of random fields

The numerical stability of linear systems arising in kriging, estimation, and simulation of random fields, is studied analytically and numerically. In the state-space formulation of kriging, as developed here, the stability of the kriging system depends on the condition number of the prior, stationary covariance matrix. The same is true for conditional random field generation by the superposition method, which is based on kriging, and the multivariate Gaussian method, which requires factoring a covariance matrix. A large condition number corresponds to an ill-conditioned, numerically unstable system. In the case of stationary covariance matrices and uniform grids, as occurs in kriging of uniformly sampled data, the degree of ill-conditioning generally increases indefinitely with sampling density and, to a limit, with domain size. The precise behavior is, however, highly sensitive to the underlying covariance model. Detailed analytical and numerical results are given for five one-dimensional covariance models: (1) hole-exponential, (2) exponential, (3) linear-exponential, (4) hole-Gaussian, and (5) Gaussian. This list reflects an approximate ranking of the models, from best to worst conditioned. The methods developed in this work can be used to analyze other covariance models. Examples of such representative analyses, conducted in this work, include the spherical and periodic hole-effect (hole-sinusoidal) covariance models. The effect of small-scale variability (nugget) is addressed and extensions to irregular sampling schemes and higher dimensional spaces are discussed.     

集合卡尔曼滤波同化多普勒雷达资料的观测系统模拟试验

本文将集合卡尔曼滤波同化技术应用到对流尺度系统中,实施了基于WRF模式的同化单部多普勒雷达径向风和反射率因子的观测系统模拟试验,验证了其在对流尺度中应用的可行性和有效性,并对同化系统的特性进行了探讨。试验表明:WRF-EnKF雷达资料同化系统能较准确分析模式风暴的流场、热力场、微物理量场的细致特征;几乎所有变量的预报和分析误差经过同化循环后都能显著下降,同化分析基本上能使预报场在各层上都有所改进,对预报场误差较大层次的更正更为显著;约8个同化循环后,EnKF能在雷达反射率、径向风观测与背景场间建立较可靠的相关关系,使模式各变量场能被准确分析更新,背景场误差协方差在水平方向和垂直方向都有着复杂的结构,是高度非均匀、各项异性和流依赖的;集合平均分析场做的确定性预报在短时间内能较好保持真值场风暴的细节结构,但预报误差增长较快。