Finding the point on Bezier Curves with the normal vector passing an external point

Two algorithms for finding the point on non-rational/rational Bezier curves of which the normal vector passes through a given external point are presented. The algorithms are based on Bezier curves generation algorithms of de Casteljau's algorithm for non-rational Bezier curve or Farin's recursion for rational Bezier curve, respectively. Orthogonal projections from the external point are used to guide the directional search used in the proposed iterative algorithms. Using Lyapunov's method, it is shown that each algorithm is able to converge to a local minimum for each case of non-rational/rational Bezier curves. It is also shown that on convergence the distance between the point on curves to the external point reaches a local minimum for both approaches. Illustrative examples are included to demonstrate the effectiveness of the proposed approaches.

Hammerstein model identification algorithm using Bezier-Bernstein approximation

A new identification algorithm is introduced for the Hammerstein model consisting of a nonlinear static function followed by a linear dynamical model. The nonlinear static function is characterised by using the Bezier-Bernstein approximation. The identification method is based on a hybrid scheme including the applications of the inverse of de Casteljau's algorithm, the least squares algorithm and the Gauss-Newton algorithm subject to constraints. The related work and the extension of the proposed algorithm to multi-input multi-output systems are discussed. Numerical examples including systems with some hard nonlinearities are used to illustrate the efficacy of the proposed approach through comparisons with other approaches.

Generalized neurofuzzy network modeling algorithms using Bezier-Bernstein polynomial functions and additive decomposition

This paper introduces a new neurofuzzy model construction algorithm for nonlinear dynamic systems based upon basis functions that are Bezier-Bernstein polynomial functions. This paper is generalized in that it copes with n-dimensional inputs by utilising an additive decomposition construction to overcome the curse of dimensionality associated with high n. This new construction algorithm also introduces univariate Bezier-Bernstein polynomial functions for the completeness of the generalized procedure. Like the B-spline expansion based neurofuzzy systems, Bezier-Bernstein polynomial function based neurofuzzy networks hold desirable properties such as nonnegativity of the basis functions, unity of support, and interpretability of basis function as fuzzy membership functions, moreover with the additional advantages of structural parsimony and Delaunay input space partition, essentially overcoming the curse of dimensionality associated with conventional fuzzy and RBF networks. This new modeling network is based on additive decomposition approach together with two separate basis function formation approaches for both univariate and bivariate Bezier-Bernstein polynomial functions used in model construction. The overall network weights are then learnt using conventional least squares methods. Numerical examples are included to demonstrate the effectiveness of this new data based modeling approach.

Direct observations of the evolution of polar cap ionization patches

Patches of ionization are common in the polar ionosphere where their motion and associated density gradients give variable disturbances to High Frequency (HF) radio communications, over-the-horizon radar location errors, and disruption and errors to satellite navigation and communication. Their formation and evolution are poorly understood, particularly under disturbed space weather conditions. We report direct observations of the full evolution of patches during a geomagnetic storm, including formation, polar cap entry, transpolar evolution, polar cap exit, and sunward return flow. Our observations show that modulation of nightside reconnection in the substorm cycle of the magnetosphere helps form the gaps between patches where steady convection would give a “tongue” of ionization (TOI).

Automated identification and tracking of polar-cap plasma patches at solar minimum

A method of automatically identifying and tracking polar-cap plasma patches, utilising data inversion and feature-tracking methods, is presented. A well-established and widely used 4-D ionospheric imaging algorithm, the Multi-Instrument Data Assimilation System (MIDAS), inverts slant total electron content (TEC) data from ground-based Global Navigation Satellite System (GNSS) receivers to produce images of the free electron distribution in the polar-cap ionosphere. These are integrated to form vertical TEC maps. A flexible feature-tracking algorithm, TRACK, previously used extensively in meteorological storm-tracking studies is used to identify and track maxima in the resulting 2-D data fields. Various criteria are used to discriminate between genuine patches and "false-positive" maxima such as the continuously moving day-side maximum, which results from the Earth's rotation rather than plasma motion. Results for a 12-month period at solar minimum, when extensive validation data are available, are presented. The method identifies 71 separate structures consistent with patch motion during this time. The limitations of solar minimum and the consequent small number of patches make climatological inferences difficult, but the feasibility of the method for patches larger than approximately 500 km in scale is demonstrated and a larger study incorporating other parts of the solar cycle is warranted. Possible further optimisation of discrimination criteria, particularly regarding the definition of a patch in terms of its plasma concentration enhancement over the surrounding background, may improve results.

Motion of the dayside polar cap boundary during substorm cycles: II. Generation of poleward-moving events and polar cap patches by pulses in the magnetopause reconnection rate

Using data from the EISCAT (European Incoherent Scatter) VHF and CUTLASS (Co-operative UK Twin- Located Auroral Sounding System) HF radars, we study the formation of ionospheric polar cap patches and their relationship to the magnetopause reconnection pulses identified in the companion paper by Lockwood et al. (2005). It is shown that the poleward-moving, high-concentration plasma patches observed in the ionosphere by EISCAT on 23 November 1999, as reported by Davies et al. (2002), were often associated with corresponding reconnection rate pulses. However, not all such pulses generated a patch and only within a limited MLT range (11:00–12:00 MLT) did a patch result from a reconnection pulse. Three proposed mechanisms for the production of patches, and of the concentration minima that separate them, are analysed and evaluated: (1) concentration enhancement within the patches by cusp/cleft precipitation; (2) plasma depletion in the minima between the patches by fast plasma flows; and (3) intermittent injection of photoionisation-enhanced plasma into the polar cap. We devise a test to distinguish between the effects of these mechanisms. Some of the events repeat too frequently to apply the test. Others have sufficiently long repeat periods and mechanism (3) is shown to be the only explanation of three of the longer-lived patches seen on this day. However, effect (2) also appears to contribute to some events. We conclude that plasma concentration gradients on the edges of the larger patches arise mainly from local time variations in the subauroral plasma, via the mechanism proposed by Lockwood et al. (2000).

Ectomycorrhizal symbiosis can enhance plant nutrition through improved access to discrete organic nutrient patches of high resource quality

It is known that roots can respond to patches of fertility; however, root proliferation is often too slow to exploit resources fully, and organic nutrient patches may be broken down and leached, immobilized or chemically fixed before they are invaded by the root system. The ability of fungal hyphae to exploit resource patches is far greater than that of roots due to their innate physiological and morphological plasticity, which allows comprehensive exploration and rapid colonization of resource patches in soils. The fungal symbionts of ectomycorrhizal plants excrete significant quantities of enzymes such as chitinases, phosphatases and proteases. These might allow the organic residue to be tapped directly for nutrients such as N and P. Pot experiments conducted with nutrient-stressed ectomycorrhizal and control willow plants showed that when high quality organic nutrient patches were added, they were colonized rapidly by the ectomycorrhizal mycelium. These established willows (0.5 m tall) were colonized by Hebeloma syrjense P. Karst. for 1 year prior to nutrient patch addition. Within days after patch addition, colour changes in the leaves of the mycorrhizal plants (reflecting improved nutrition) were apparent, and after I month the concentration of N and P in the foliage of mycorrhizal plants was significantly greater than that in non-mycorrhizal plants subject to the same nutrient addition. It seems likely that the mycorrhizal plants were able to compete effectively with the wider soil microbiota and tap directly into the high quality organic resource patch via their extra-radical mycelium. We hypothesize that ectomycorrhizal plants may reclaim some of the N and P invested in seed production by direct recycling from failed seeds in the soil. The rapid exploitation of similar discrete, transient, high-quality nutrient patches may have led to underestimations when determining the nutritional benefits of ectomycorrhizal colonization.

Earth's ion upflow associated with polar cap patches: global and in-situ observations

We report simultaneous global monitoring of a patch of ionization and in situ observation of ion upflow at the center of the polar cap region during a geomagnetic storm. Our observations indicate strong fluxes of upwelling O+ ions originating from frictional heating produced by rapid antisunward flow of the plasma patch. The statistical results from the crossings of the central polar cap region by Defense Meteorological Satellite Program F16–F18 from 2010 to 2013 confirm that the field-aligned flow can turn upward when rapid antisunward flows appear, with consequent significant frictional heating of the ions, which overcomes the gravity effect. We suggest that such rapidly moving patches can provide an important source of upwelling ions in a region where downward flows are usually expected. These observations give new insight into the processes of ionosphere-magnetosphere coupling.

Causes of the apparent scale independence of fractal indices associated with forest fragmentation in Bolivia

In a recent investigation, Landsat TM and ETM+ data were used to simulate different resolutions of remotely-sensed images (from 30 to 1100 m) and to analyze the effect of resolution on a range of landscape metrics associated with spatial patterns of forest fragmentation in Chapare, Bolivia since the mid-1980s. Whereas most metrics were found to be highly dependent on pixel size, several fractal metrics (DLFD, MPFD, and AWMPFD) were apparently independent of image resolution, in contradiction with a sizeable body of literature indicating that fractal dimensions of natural objects depend strongly on image characteristics. The present re-analysis of the Chapare images, using two alternative algorithms routinely used for the evaluation of fractal dimensions, shows that the values of the box-counting and information fractal dimensions are systematically larger, sometimes by as much as 85%, than the "fractal" indices DLFD, MPFD, and AWMFD for the same images. In addition, the geometrical fractal features of the forest and non-forest patches in the Chapare region strongly depend on the resolution of images used in the analysis. The largest dependency on resolution occurs for the box-counting fractal dimension in the case of the non-forest patches in 1993, where the difference between the 30 and I 100 m-resolution images corresponds to 24% of the full theoretical range (1.0 to 2.0) of the mass fractal dimension. The observation that the indices DLFD, MPFD, and AWMPFD, unlike the classical fractal dimensions, appear relatively unaffected by resolution in the case of the Chapare images seems due essentially to the fact that these indices are based on a heuristic, "non-geometric" approach to fractals. Because of their lack of a foundation in fractal geometry, nothing guarantees that these indices will be resolution-independent in general. (C) 2006 International Society for Photogrammetry and Remote Sensing, Inc. (ISPRS). Published by Elsevier B.V. All rights reserved.