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A family of spherical caps of the 2-dimensional unit sphere S2 is called a totally separable packing in short, a TS-packing if any two spherical caps can be separated by a great circle which is disjoint from the interior of each spherical cap in the packing. The separable Tammes problem asks for the largest density of given number of congruent spherical caps forming a TS-packing in S2. We solve this problem up to eight spherical caps and upper bound the density of any TS-packing of congruent spherical caps in terms of their angular radius. Based on this, we show that the centered separable kissing number of unit balls in Euclidean 3-space is 8. Furthermore, we prove bounds for the maximum of the smallest inradius of the cells of the tilings generated by n>1 great circles in S2. Next, we prove dual bounds for TS-coverings of S2 by congruent spherical caps. Here a covering of S2 by spherical caps is called a totally separable covering in short, a TS-covering if there exists a tiling generated by finitely many great circles of S2 such that the cells of the tiling are covered by pairwise distinct spherical caps of the covering. Finally, we extend some of our bounds on TS-coverings to spherical spaces of dimension >2.

The Pacific–North American (PNA) teleconnection pattern has recently been found to be influenced by the South China Sea (SCS) sea surface temperature anomaly (SSTA). This study further demonstrates that, the SCS SSTA can induce a new type of PNA response. The SCS-induced PNA pattern and its evolution are shown to be distinctly different from its conventional counterparts and, particularly, the ENSO-induced pattern. In contrast to the observed conventional patterns, the SCS-induced PNA seems to agree remarkably well with the classical theory of Rossby wave propagation on a sphere by Hoskins and Karoly (1981), with the centers following the great circle route, and it shows a more zonally oriented new pathway of evolution led by a precursory center not seen before. Further study of the dynamical processes underlying the new PNA response reveals a specific air-sea interaction triggered by SCS SSTA, which results in an upper-level Rossby wave source and induces the precursory center. We remark that, although the SCS SSTA is relatively small in amplitude, it has an effect on PNA almost as conspicuous as that from ENSO.

Comparison of in-situ measured antenna radiation patterns (RPs) to modeled ones is vital for validation of both. Inflight measured RPs do not always produce a standard conic or elevation cut (constant θ or ϕ angle, respectively), but rather Great Circle (GC) cuts at the aircraft bank angle of interest. WIPL-D’s post-processing routines, on the other hand, do not produce GC cuts in normal setups. A manipulation of the aircraft orientation in xyz-coordinates is required to accomplish this task. Under standard conditions in WIPL-D, the fuselage is positioned parallel to the x-axis and the wings parallel to the y-axis. A model rotation of 90° with respect to the y-axis allows for the generation of GC cuts, where θ and ϕ swap roles. This makes comparison between in-flight measurements and computed data cumbersome. This paper investigates several options to produce non-standard RPs in WIPL-D and MATLAB (using WIPL-D results) that are equivalent to those of in-flight measurements.

Given a spherical set of points, we consider the detection of cocircular subsets of the data. We distinguish great circles from small circles, and develop algorithms for detecting cocircularities of both types. The suggested approach is an extension of the Hough transform. We address the unique parameter-space quantization issues arising due to the spherical geometry, present quantization schemes, and evaluate the quantization-induced errors. We demonstrate the proposed algorithms by detecting cocircular cities and airports on Earth’s spherical surface. These results facilitate the detection of great and small circles in spherical images.

Determining the appropriate number and position of waypoints on a great circle route (GCR) helps to shorten the sailing distance, reduce the number of course changes, and well-approximate the GCR through a small number of rhumb line (RL) legs. In this study, a genetic algorithm-based method (i.e., the GA method) is proposed to optimize the positions of waypoints on the GCR when the number of waypoints is given. Furthermore, a fuzzy logic-based evaluation method for the number of waypoints (i.e., the FL method) is proposed to judge whether to add a new waypoint or stop the process by using the non-fixed values while considering both the number of waypoints and the remaining benefit of the GCR. According to the example demonstration results, the two methods proposed in this study can well-determine the number and position of waypoints and provide effective support for ocean route planning.

We investigate Gauss maps associated to great circle fibrations of the euclidean unit 3-sphere S3. We show that the associated Gauss map to such a fibration is harmonic, respectively minimal, if and only if the unit vector field generating the great circle foliation is harmonic, respectively minimal. These results can be viewed as analogues of the classical theorem of Ruh and Vilms about the harmonicity of the Gauss map of a minimal submanifold in the euclidean space. Moreover, we prove that a harmonic or minimal unit vector field on S3, whose integral curves are great circles, is a Hopf vector field.

We investigated movements of a western population of Veeries (Catharus fuscescens) breeding in the Okanagan region of British Columbia, Canada, in 2013–2014 using light-level geolocators. We tracked 9 individuals and incorporated a state-space Kalman filter model approach to estimate movement parameters. During migration, Veeries traversed the Rocky Mountains, Great Plains, Gulf of Mexico, and Caribbean Sea with stopovers generally closer to the shorter orthodromic (great circle) route than a loxodromic (straight line) route between breeding and first wintering grounds, particularly on fall migration. Birds initially settled in the south-central portion of the Amazon basin in Brazil at sites that were 666 ± 299 km apart, suggesting low migratory connectivity. Intra-tropical movements were observed for 8 of 9 (88.9%) birds, with second wintering sites an average of 1,447 ± 472 km to the northwest (initial bearing x̄ = 316 ± 16°). Veeries typically followed a pattern of loop migration at the Gulf of Mexico, with more birds using the Yucatan Peninsula to stop and reorient toward destinations on spring migration (n = 7) vs. fall migration (n = 2). Western Veeries follow a presumed ancestral (eastern) migration route, but this route is also the shortest (great circle) route between breeding and wintering grounds, even though this route was only ~100 km shorter than the straight line route. Eight Veeries (88.9%) underwent a post-breeding, pre-migratory movement up to 628 km (x̄ = 263 ± 152 km) away from breeding territories, possibly to molt. We encourage researchers utilizing light-level geolocators to apply similar state-space modeling approaches to reduce the influence of observers and erroneous location estimates on analysis and interpretation of geolocator data.

An approach formulated by vector algebra is proposed to deal with great circle sailing problems. Using the technique of the fixed coordinates system and relative longitude concept, derivations of formulae for this approach are simpler than those of the conventional methods. Due to fixing the initial great circle course, the great circle track (GCT) is determined. Since the course is fixed (known as “COFI” in this paper), the proposed approach, which we have named the “COFI method”, can directly calculate the waypoints along the GCT. It is considered that the COFI method is a more understandable and straightforward method to solve waypoint problems than older approaches in the literature. Based on the COFI method, a program has been developed for the navigator. In addition, the spherical triangle method with respect to the equator crossing point (STM-E) is developed by supplemental theorem. Several examples are demonstrated to validate the proposed COFI method and STM-E.

In this article, we propose stationary covariance functions for processes that evolve temporally over a sphere, as well as cross-covariance functions for multivariate random fields defined over a sphere. For such processes, the great circle distance is the natural metric that should be used to describe spatial dependence. Given the mathematical difficulties for the construction of covariance functions for processes defined over spheres cross time, approximations of the state of nature have been proposed in the literature by using the Euclidean (based on map projections) and the chordal distances. We present several methods of construction based on the great circle distance and provide closed-form expressions for both spatio-temporal and multivariate cases. A simulation study assesses the discrepancy between the great circle distance, chordal distance, and Euclidean distance based on a map projection both in terms of estimation and prediction in a space-time and a bivariate spatial setting, where the space is in this case the Earth. We revisit the analysis of Total Ozone Mapping Spectrometer (TOMS) data and investigate differences in terms of estimation and prediction between the aforementioned distance-based approaches. Both simulation and real data highlight sensible differences in terms of estimation of the spatial scale parameter. As far as prediction is concerned, the differences can be appreciated only when the interpoint distances are large, as demonstrated by an illustrative example. Supplementary materials for this article are available online.

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.