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For over 60 years, an approximation involving spherical geopotentials has underlain the representation of gravity in global numerical models of Earth's atmosphere and oceans. This article explores how departures from sphericity can be allowed by assuming spheroidal geopotentials instead. A route to more accurate model formulation is indicated, and the theoretical basis of the classical spherical‐geopotential approximation is illuminated too. An overview—from the time of Newton to the present—is given of the development of zeroth‐ and first‐order approximations to the geopotential field in terms of two small non‐dimensional parameters. The importance of using geopotential coordinate systems in atmospheric and oceanic models is emphasized. Early suggestions for such systems using non‐spherical coordinates involved qualitatively inappropriate choices of ellipsoids derived from families of confocal ellipses. Specific examples of appropriate ellipsoid choices are considered before presentation of the recently developed Geophysically Realistic, Ellipsoidal, Analytically Tractable (GREAT) system. This is based on a suitably constructed geopotential field approximation, first order accurate, from which—without further approximation—may be analytically derived equations for geopotential surfaces and surfaces orthogonal to them. GREAT coordinates satisfy stated desiderata for geopotential coordinate systems and are applicable both above and below Earth's geoid (assumed to coincide with the WGS 84 [2004, https://gis‐lab.info/docs/nima‐tr8350.2‐wgs84fin.pdf] reference ellipsoid to an excellent approximation). GREAT‐coordinate analysis provides justification for the classical spherical‐geopotential approximation: It is revealed as a mathematical limit, but not a physically realizable one. Attention is drawn to a certain partially spherical limit that is realizable physically. Plain Language Summary Gravity is by far the dominant external force in the equations of motion for Earth's atmosphere and oceans. It is crucially important that it be adequately represented in global atmospheric and oceanic models for climate and weather prediction. In principle, one simply applies Newton's inverse square law of gravitational attraction to represent gravity in the equations of motion. In practice, this is far too complicated to do exactly—due principally to the rotating Earth being closer to spheroidal than spherical in shape, with an inhomogeneous mass distribution—thereby necessitating approximation. The dominant nature of gravity means that forecast accuracy is enhanced if one constructs an orthogonal coordinate system to integrate the equations of motion, whereby gravity only acts in the vertical and not in the horizontal. This is termed a geopotential coordinate system and, to further complicate matters, its construction is intrinsically coupled to a suitable approximation of gravity. We first review the basic concepts to represent gravity in global atmospheric and oceanic models. Next, we discuss the importance and principles of geopotential coordinates for modeling purposes. Various geopotential coordinate systems of varying accuracy are then compared. Finally, we outline some possible developments for representing gravity in future models. Key Points We review the derivation from first principles of geopotential approximations to represent gravity in global atmospheric and oceanic models Various geopotential coordinate systems of varying accuracy are compared, leading to one that satisfies all desiderata for such a system The low‐order, ubiquitous, classical, spherical‐geopotential approximation is shown to be the asymptotic limit of a more accurate one

A geodesic or geodetic line on a sphere is called the orthodrome. Research has shown that the orthodrome can be defined in a large number of ways. This article provides an overview of various definitions of the orthodrome. We recall the definitions of the orthodrome according to the greats of geodesy, such as Bessel and Helmert. We derive the equation of the orthodrome in the geographic coordinate system and in the Cartesian spatial coordinate system. A geodesic on a surface is a curve for which the geodetic curvature is zero at every point. Equivalent expressions of this statement are that at every point of this curve, the principal normal vector is collinear with the normal to the surface, i.e., it is a curve whose binormal at every point is perpendicular to the normal to the surface, and that it is a curve whose osculation plane contains the normal to the surface at every point. In this case, the well-known Clairaut equation of the geodesic in geodesy appears naturally. It is found that this equation can be written in several different forms. Although differential equations for geodesics can be found in the literature, they are solved in this article, first, by taking the sphere as a special case of any surface, and then as a special case of a surface of rotation. At the end of this article, we apply calculus of variations to determine the equation of the orthodrome on the sphere, first in the Bessel way, and then by applying the Euler–Lagrange equation. Overall, this paper elaborates a dozen different approaches to orthodrome definitions.

The last decades have seen an unprecedented increase in the availability of data sets that are inherently global and temporally evolving, from remotely sensed networks to climate model ensembles. This paper provides an overview of statistical modeling techniques for space–time processes, where space is the sphere representing our planet. In particular, we make a distintion between (a) second order-based approaches and (b) practical approaches to modeling temporally evolving global processes. The former approaches are based on the specification of a class of space–time covariance functions, with space being the two-dimensional sphere. The latter are based on explicit description of the dynamics of the space–time process, that is, by specifying its evolution as a function of its past history with added spatially dependent noise. We focus primarily on approach (a), for which the literature has been sparse. We provide new models of space–time covariance functions for random fields defined on spheres cross time. Practical approaches (b) are also discussed, with special emphasis on models built directly on the sphere, without projecting spherical coordinates onto the plane. We present a case study focused on the analysis of air pollution from the 2015 wildfires in Equatorial Asia, an event that was classified as the year’s worst environmental disaster. The paper finishes with a list of the main theoretical and applied research problems in the area, where we expect the statistical community to engage over the next decade.

Heading is a crucial navigation parameter for high-precision flight guidance. Since the heading changes rapidly while unmanned aerial vehicles (UAVs) track great ellipse routes in polar regions, it is necessary to implement special guidance algorithms. This article presents a high-precision polar flight guidance algorithm for fixed-wing UAVs along great ellipse routes based on heading prediction. Specifically, a globally applicable definition of polar grid frame was proposed. On this basis, a novel flight guidance algorithm based on heading prediction was developed. Therein, the calculation method for grid azimuth on great ellipse routes based on the WGS-84 ellipse model was derived in detail, realizing accurate heading estimation and prediction. Subsequently, the predicted grid heading was utilized to tackle the difficulty of heading changes, enabling the UAV to predict and adjust its heading in advance. Moreover, an adaptive predicted lead-time adjustment strategy based on fuzzy decision-making was introduced to improve the prediction accuracy under challenging situations, and an enhanced particle swarm optimization algorithm was employed to determine the hyperparameters in fuzzy rules. To verify the effectiveness of the proposed algorithm, extensive simulations were operated using the Monte Carlo method, and the proposed algorithm demonstrated 3–4 times higher guidance accuracy compared to conventional algorithms.

Geocoding aims to assign unambiguous locations (i.e., geographic coordinates) to place names (i.e., toponyms) referenced within documents (e.g., within spreadsheet tables or textual paragraphs). This task comes with multiple challenges, such as dealing with referent ambiguity (multiple places with a same name) or reference database completeness. In this work, we propose a geocoding approach based on modeling pairs of toponyms, which returns latitude-longitude coordinates. One of the input toponyms will be geocoded, and the second one is used as context to reduce ambiguities. The proposed approach is based on a deep neural network that uses Long Short-Term Memory (LSTM) units to produce representations from sequences of character n-grams. To train our model, we use toponym co-occurrences collected from different contexts, namely textual (i.e., co-occurrences of toponyms in Wikipedia articles) and geographical (i.e., inclusion and proximity of places based on Geonames data). Experiments based on multiple geographical areas of interest—France, United States, Great-Britain, Nigeria, Argentina and Japan—were conducted. Results show that models trained with co-occurrence data obtained a higher geocoding accuracy, and that proximity relations in combination with co-occurrences can help to obtain a slightly higher accuracy in geographical areas with fewer places in the data sources.

Elucidating how life history traits vary geographically is important to understanding variation in population dynamics. Because many aspects of ectotherm life history are climate-dependent, geographic variation in climate is expected to have a large impact on population dynamics through effects on annual survival, body size, growth rate, age at first reproduction, size-fecundity relationship, and reproductive frequency. The Eastern Massasauga (Sistrurus catenatus) is a small, imperiled North American rattlesnake with a distribution centered on the Great Lakes region, where lake effects strongly influence local conditions. To address Eastern Massasauga life history data gaps, we compiled data from 47 study sites representing 38 counties across the range. We used multimodel inference and general linear models with geographic coordinates and annual climate normals as explanatory variables to clarify patterns of variation in life history traits. We found strong evidence for geographic variation in six of nine life history variables. Adult female snout-vent length and neonate mass increased with increasing mean annual precipitation. Litter size decreased with increasing mean temperature, and the size-fecundity relationship and growth prior to first hibernation both increased with increasing latitude. The proportion of gravid females also increased with increasing latitude, but this relationship may be the result of geographically varying detection bias. Our results provide insights into ectotherm life history variation and fill critical data gaps, which will inform Eastern Massasauga conservation efforts by improving biological realism for models of population viability and climate change.

We consider foliations of the whole three dimensional hyperbolic space H 3 by oriented geodesics. Let L be the space of all the oriented geodesics of H 3 , which is a four dimensional manifold carrying two canonical pseudo-Riemannian metrics of signature 2 , 2 . We characterize, in terms of these geometries of L , the subsets M in L that determine foliations of H 3 . We describe in a similar way some distinguished types of geodesic foliations of H 3 , regarding to which extent they are in some sense trivial in some directions: On the one hand, foliations whose leaves do not lie in a totally geodesic surface, not even at the infinitesimal level. On the other hand, those for which the forward and backward Gauss maps φ ± : M → H 3 ∞ are local diffeomorphisms. Besides, we prove that for this kind of foliations, φ ± are global diffeomorphisms onto their images. The subject of this article is within the framework of foliations by congruent submanifolds, and follows the spirit of the paper by Gluck and Warner where they understand the infinite dimensional manifold of all the great circle foliations of the three sphere.

We studied the numerical approximation problem of distortion in map projections. Most widely used differential methods calculate area distortion and maximum angular distortion using partial derivatives of forward equations of map projections. However, in certain map projections, partial derivatives are difficult to calculate because of the complicated forms of forward equations, e.g., equations with iterations, integrations, or multi-way branches. As an alternative, the spherical great circle arcs–based metric employs the inverse equations of map projections to transform sample points from the projection plane to the spherical surface, and then calculates a differential-independent distortion metric for the map projections. We introduce a novel forward interpolated version of the previous spherical great circle arcs–based metric, solely dependent on the forward equations of map projections. In our proposed numerical solution, a rational function–based regression is also devised and applied to our metric to obtain an approximate metric of angular distortion. The statistical and graphical results indicate that the errors of the proposed metric are fairly low, and a good numerical estimation with high correlation to the differential-based metric can be achieved.

Ecological and environmental monitoring has become increasingly important, with increasing threats from human disturbances. Monitoring usually involves sampling from several sites of a similar habitat at regular (or irregular) intervals through time. The purpose of monitoring is to determine where and when an impact may have occurred or, once detected, may still be occurring. Sequential statistical methods, including control charts, as developed for industrial applications, offer some promise in this regard. These provide a way of identifying when a system (e.g., in a factory) is going \"out of control,\" so as to trigger an alarm to stop the system and employ appropriate remedial measures. Such techniques clearly would be useful in the context of environmental monitoring. Traditional control charts, however, cannot be used for many ecological applications because they do not handle multivariate data, and individual counts of species abundances do not generally fulfill the necessary statistical assumptions. A distance-based multivariate control chart method is described here, with some examples of its use in monitoring coral reef fish assemblages of the Great Barrier Reef, Australia. The method is flexible, as it can be based on any dissimilarity measure of choice, and useful, as it does not require any specific assumptions regarding distributions of variables. Bootstrapping techniques are used to provide control-chart limits for an appropriate multivariate distance-based criterion through time. The method is designed to identify impacts at individual sites as quickly as possible, thus triggering an \"alarm bell\" in the context of ecological monitoring. It can also be applied at several spatial scales in hierarchical designs.