Great Circle Distance Research Articles

From Schoenberg Coefficients to Schoenberg Functions

In his seminal paper, Schoenberg (1942) characterized the class of continuous functions f : [−1,1] → R such that f(cos�) is positive definite over the product space S d ×S d , with S d being the unit sphere of R d+1 and � being the great circle distance. In this paper, we consider the product space S d ×G, for G a locally compact group, and define the class �(S d ,G) of continuous functions f : [−1,1] × G → C such that f(cos�,u −1 · v) is positive definite on S d × S d × G × G. This offers a natural extension of Schoenberg’s Theorem. Schoenberg’s second theorem corresponding to the Hilbert sphere S ∞ is also extended to this context. The case G = R is of special importance for probability theory and stochastic processes, because it characterizes completely the class of space-time covariance functions where the space is the sphere, being an approximation of Planet Earth. MSC: Primary 43A35, Secondary 33C55

Isotropic covariance functions on spheres: Some properties and modeling considerations

Introducing flexible covariance functions is critical for interpolating spatial data since the properties of interpolated surfaces depend on the covariance function used for Kriging. An extensive literature is devoted to covariance functions on Euclidean spaces, where the Matérn covariance family is a valid and flexible parametric family capable of controlling the smoothness of corresponding stochastic processes. Many applications in environmental statistics involve data located on spheres, where less is known about properties of covariance functions, and where the Matérn is not generally a valid model with great circle distance metric. In this paper, we advance the understanding of covariance functions on spheres by defining the notion of and proving a characterization theorem for m times mean square differentiable processes on d -dimensional spheres. Stochastic processes on spheres are commonly constructed by restricting processes on Euclidean spaces to spheres of lower dimension. We prove that the resulting sphere-restricted process retains its differentiability properties, which has the important implication that the Matérn family retains its full range of smoothness when applied to spheres so long as Euclidean distance is used. The restriction operation has been questioned for using Euclidean instead of great circle distance. To address this question, we construct several new covariance functions and compare them to the Matérn with Euclidean distance on the task of interpolating smooth and non-smooth datasets. The Matérn with Euclidean distance is not outperformed by the new covariance functions or the existing covariance functions, so we recommend using the Matérn with Euclidean distance due to the ease with which it can be computed.

Distance to faults as a proxy for radon gas concentration in dwellings

This research was done to demonstrate the usefulness of the local structural geology characteristics to predict indoor radon concentrations. The presence of geologic faults near dwellings increases the vulnerability of the dwellings to elevated indoor radon by providing favorable pathways from the source uranium-rich bedrock units to the surface. Kruskal–Wallis one-way analyses of variance by ranks were used to determine the distance where faults have statistically significant influence on indoor radon concentrations. The great-circle distance between the 640 spatially referenced basement radon concentration measurements and the nearest fault was calculated using the Haversine formula and the spherical law of cosines. It was shown that dwellings located less than 150 m from a major fault had a higher radon potential. The 150 m threshold was determined using Kruskal–Wallis ANOVA on: (1) all the basement radon measurements dataset and; (2) the basement radon measurements located on uranium-rich bedrock units only. The results indicated that 22.8% of the dwellings located less than 150 m from a fault exceeded the Canadian radon guideline of 200 Bq/m3 when using all the basement radon measurements dataset. This percentage fell to 15.2% for the dwellings located between 150 m and 700 m from a fault. When using only the basement radon measurements located on uranium-rich bedrock units, these percentages were 30.7% (0–150 m) and 17.5% (150 m–700 m). The assessment and management of risk can be improved where structural geology characteristics base maps are available by using this proxy indicator.

On Tracking and Disaggregating Center Points of Population

In this article we explore methods for tracking and disaggregating five alternately defined mean and median center points of population, measures that can help in interpreting the forces underlying shifting settlement patterns. We argue that the point that minimizes the sum of squared great circle distances is more conceptually appealing than the center point located via the method currently employed by the U.S. Census Bureau. We also suggest that the point of minimum aggregate distance—as deployed in many other geographic applications—provides an interesting alternative to the median center historically used in population analysis, which is the crossing point of the medial lines of latitude and longitude. We then propose methods to disaggregate any of the alternatively defined center points into multiple points useful for tracking and comparing the relative influences of each of the components of population change: births, deaths, domestic (or internal) migration, and immigration. Similarly, we track and examine the shifting locations of the center points of various age, sex, and race or ethnicity groups. In a final section, we suggest that the increasing average and standard distances of individuals from the median and mean centers result from the increasingly bicoastal distribution of the U.S. population. As summary measures of all of the changes in population occurring anywhere across the nation's land area, centers of population provide an interesting conceptual platform for drilling into the variegated geographic patterns and disparate demographic forces that underlie a country's population distribution.

A New Method for Finding Geographic Centers, with Application to U.S. States

The geographic center of a region is a fundamental geographic concept, and yet there is no commonly accepted method for its determination. This article discusses some of its history as well as its definition and calculation, and a new method for its calculation is suggested. The new method minimizes the sum of squared great circle distances from all points in the region to the center. This entails (1) projecting regional boundary points using an azimuthal equidistant projection, (2) finding the geographic center of the projected two-dimensional region, and (3) then transforming this location back to a latitude and longitude. This new approach is used to find the geographic center of the contiguous United States and to provide a new list of the geographic centers for U.S. states. This list improves on the widely used but inaccurate list published by the United States Geological Survey in 1923.

Covariance models on the surface of a sphere: when does it matter?

There is a growing interest in developing covariance functions for processes on the surface of a sphere because of the wide availability of data on the globe. Utilizing the one‐to‐one mapping between the Euclidean distance and the great circle distance, isotropic and positive definite functions in a Euclidean space can be used as covariance functions on the surface of a sphere. This approach, however, may result in physically unrealistic distortion on the sphere especially for large distances. We consider several classes of parametric covariance functions on the surface of a sphere, defined with either the great circle distance or the Euclidean distance, and investigate their impact upon spatial prediction. We fit several isotropic covariance models to simulated data as well as real data from National Center for Environmental Prediction (NCEP)/National Center for Atmospheric Research (NCAR) reanalysis on the sphere. We demonstrate that covariance functions originally defined with the Euclidean distance may not be adequate for some global data. Copyright © 2015 John Wiley & Sons, Ltd.

Detecting and locating seismic events with using USArray as a large antenna

Abstract. We design an earthquake detection and location algorithm that explores coherence and characteristic behavior of teleseismic waves recorded by a large-scale seismic network. The procedure consists of three steps. First, for every tested source location we construct a time-distance gather by computing great-circle distances to all stations of the network and aligning the signals respectively. Second, we use the constructed gather to compute a Tau-P transform. For waves emitted by teleseismic sources, the amplitude of this transform has a very characteristic behavior with maxima corresponding to different seismic phases. Relative location of these maxima on the time-slowness plane strongly depends on the distance to the earthquake. To explore this dependence, in a third step, we convolve the Tau-P amplitude with a time-slowness filter whose maxima are computed based on prediction of a global travel-time calculator. As a result of this three-step procedure, we obtain a function that characterizes a likelihood of occurrence of a seismic event at a given position in space and time. We test the developed algorithm by applying it to vertical-component records of USArray to locate a set of earthquakes distributed around the Globe with magnitudes between 6.1 and 7.2.

Facility Location Decisions Based on Driving Distances on Spherical Surface

Facility location problems are concerned with the location of one or more facilities in a way that optimizes a certain objective such as minimizing transportation cost, providing equitable service to customers, capturing the largest market share, etc. Many facility location decisions involving distance objective functions on Spherical Surface have been approached using algorithmic, metaheuristic algorithms, branch-and-bound algorithm, approximation algorithms, simulation, heuristic techniques, and decomposition method. These approaches are most based on Euclidean distance or Great circle distance functions. However, if the location points are widely separated, the difference between driving distance, Euclidean distance and Great circle distance may be significant and this may lead to significant variations in the locations of the corresponding optimal source points. This paper presents a framework and algorithm to use driving distances on spherical surface and explores its use as a facility location decision tool and helps companies assess the optimal locations of facilities.

Geo distance-based event detection in social media

In the past two decades, the increase in the digital technology created an interest within the people to capture the daily activities they come across. The popularity of the social media sites enabled sharing of these experiences leading to large number of multimedia content available on the web. Especially, large numbers of photo collections are made into the social media sites, making the task of retrieving photos difficult. The retrieval task of the social photo collections has been made easy by identifying them as events, known as social event detection SED. The proposed work describes the SED task, by utilising the various sources of metadata coupled with photos on an image community website Flickr. The photos are grouped into an event-oriented way by utilising the photo ID, title, tags, description, date, time and geo-location. Especially, three different measures of great circle distance GCD have been used on the latitude and longitude values of the photos for event detection. The proposed methodology grouped the photos into respective events of sports, technical and protest events simultaneously.

A class of Matérn-like covariance functions for smooth processes on a sphere

There have been noticeable advancements in developing parametric covariance models for spatial and spatio-temporal data with various applications to environmental problems. However, literature on covariance models for processes defined on the surface of a sphere with great circle distance as a distance metric is still sparse, due to its mathematical difficulties. It is known that the popular Matérn covariance function, with smoothness parameter greater than 0.5, is not valid for processes on the surface of a sphere with great circle distance. We introduce an approach to produce Matérn-like covariance functions for smooth processes on the surface of a sphere that are valid with great circle distance. The resulting model is isotropic and positive definite on the surface of a sphere with great circle distance, with a natural extension for nonstationarity case. We present extensive numerical comparisons of our model, with a Matérn covariance model using great circle distance as well as chordal distance. We apply our new covariance model class to sea level pressure data, known to be smooth compared to other climate variables, from the CMIP5 climate model outputs.

A Comparison of HY-2 and QuikSCAT Vector Wind Products for Tropical Cyclone Track and Intensity Development Monitoring

China is the country worst affected by tropical cyclones in the Asia-Pacific region. The use of remote sensing data can help improve emergency preparedness for such events. In this letter, we evaluate the performance of HY-2 scatterometer data for monitoring Saola, Haikui, and Usagi typhoons, and consider QuikSCAT data for Florence, Ioke, and Gorden hurricanes. We use the means and standard deviation of the great circle distance to assess the typhoon track, and the t-test, mean error, and scatter index to evaluate the accuracy of typhoon intensity. Our results show that the HY-2 has the ability similar to that of QuikSCAT, suggesting that HY-2 has potential for application in typhoon monitoring.

Derivation of paleolongitude from the geometric parametrization of apparent polar wander path: Implication for absolute plate motion reconstruction

AbstractObtaining ancient longitude position of continents in the past has always been a challenge for plate tectonic reconstructions. Paleomagnetism has been commonly used to reconstruct paleolatitudes and relative rotations but not paleolongitudes. In this work, we present a synthesized method to derive paleolongitude by geometrically parametrizing apparent polar wander path (APWP). Great and small circle modeling are implemented concurrently to the identified APWP tracks to calculate the paleomagnetic Euler parameters (stage rotation pole and rotation angle). From the Euler parameters of the optimal fitting option, the absolute motion history can be restored for the reference geometries. Using our method as well as the results from relative plate motion studies, we reevaluate the dispersion history of East Gondwana since 140 Ma. To further test the validity of our method, we compare the predictions from four other absolute motion models mainly in paleolatitude movement, longitudinal variation, and great circle distance, which suggest the most similarity with the global hybrid reference frame.

New methods for reconstructing geographical effects on dispersal rates and routes from large-scale radiocarbon databases

We introduce a methodology for reconstructing geographical effects on dispersal and diffusion patterns, using georeferenced archaeological radiocarbon databases. Fast Marching methods for modelling front propagation enable geographical scenarios to be explored regarding barriers, corridors, and favoured and unfavoured habitat types. The use of genetic algorithms as optimal search tools also enables the derivation of new geographical scenarios, and is especially useful in high-dimensional parameter spaces that cannot be characterized exhaustively due to computer runtime constraints. Model selection is guided by goodness-of-fit statistics for observed and predicted radiocarbon dates.We also introduce an important additional model output, namely, modelled phylogenies of the dispersing population or diffusing cultural entity, based on branching networks of shortest or 'least cost' paths. These 'dispersal trees' can be used as an additional tool to evaluate dispersal scenarios, based on their degree of congruence with phylogenies of the dispersing population reconstructed independently from other kinds of information.We illustrate our approach with a case study, the spread of the Neolithic transition in Europe, using a database from the literature (Pinhasi, Fort and Amerman 2005). Our methods find support for a geographical model in which dispersal is limited by an altitudinal cut-off and in which there is a climate-related latitudinal gradient in rate of spread. This model leads to a deceleration in front propagation rate with geodesic distance, which is also consistent with models of the propagation of the Neolithic transition under space competition with pre-existing populations of hunter-gatherers. Our genetic algorithms meanwhile searched the parameter space and found support for an alternative model involving fast spread along the northern Mediterranean coast and the Danube/Rhine riverine corridor. Both these models outperformed the geography-free Great Circle distance model, and both also outperformed another, almost geography-free, model that constrains dispersal to land to and near-offshore coastal waters. The adjusted coefficient of determination for modelled and observed radiocarbon dates for first arrival supports the GA-derived model; the shortest path network analysis, however, gives greater support to the model with altitudinal cut-off and latitudinal gradient in dispersal rate, since it produces branching 'dispersal trees' that are more congruent with these archaeological sites' clade memberships (as defined by archaeological material culture).

THE LINK AMONG TRADE, FDI, AND IMMIGRATION

Using the gravity model, this paper provides a new empirical framework that analyzes the importance of the link among trade, FDI, and immigration. A further significant contribution of this analysis is the appropriate handling of a large number of zeroes in migration statistics. In this way, the unbalanced panel database of the 15 core European member states (EU15) as destination countries is formed. The results of the estimation show that the introduced explanatory variables, such as the common language, destination country’s population, and great circle distance between two countries, represent the most significant deterministic factors that generally explain the share of the immigrant population. It is also confirmed that the sending country’s population, trade, FDI, and sending country’s landlocked position are important determinants positively influencing the share of immigration.

The shortest distance between two points isn't always a great circle: getting around landmasses in the calibration of marine geodisparity

In the assessment of Phanerozoic marine global biodiversity, there has been longstanding interest in quantifying compositional similarities among sampling points as a function of their distances from one another (geodisparity). Previous research has demonstrated that faunal similarity between any two locations tends to decrease significantly as the great circle distance (GCD) between the locations increases, but the rate of decrease begins to stabilize at transoceanic distances. The accuracy of these assessments, and comparisons among different temporal intervals, may suffer, however, because of intervening landmasses that are not accounted for when distance is calibrated simply as GCD. Here, we present a new method for determining the shortest overwater distance (WD) between two marine locations, and we use the method to recalibrate for several Phanerozoic intervals previous measures of global geodisparity in the taxonomic compositions of marine biotas. WD was determined by using a cost-distance approach in ArcGIS, modified to work on a spherical, as opposed to a planar, surface. Results demonstrate two notable effects of using WD. First, mean compositional similarity between locations tends to decrease more continuously as a function of distance with WD than with GCD. Second, pairs of locations with WDs that are at least 50% greater than their GCDs tend to have lower compositional similarity to one another than those with more closely matching WDs and GCDs. These differences are expected as WD better represents the “true” distance between locations; they diminish at GCDs of 5000 km or more when clear, transoceanic paths between locations become more common. Despite these effects, using WD does not alter fundamental temporal trends in global geodisparity through the Phanerozoic observed in previous research, but it is likely to have more significant ramifications for more confined paleobiogeographic investigations.

A comparative study of measured amplitude and phase perturbations of VLF and LF radio signals induced by solar flares

Very Low Frequency (VLF) and Low Frequency (LF) signal perturbations were examined to study ionospheric disturbances induced by solar X-ray flares in order to understand processes involved in propagation of VLF/LF radio signals over short paths and to estimate specific characteristics of each short path. The receiver at the Belgrade station is constantly monitoring the amplitude and phase of a coherent and subionospherically propagating LF signal operated in Sicily NSC at 45.90 kHz, and a VLF signal operated in Isola di Tavolara ICV at 20.27 kHz, with the great circle distances of 953 km and 976 km, respectively. A significant number of similarities between these short paths is a direct result of both transmitters and the receiver?s geographic location. The main difference is in transmitter frequencies. From July 2008 to February 2014 there were about 200 events that were chosen for further examination. All selected examples showed that the amplitude and phase of VLF and LF signals were perturbed by solar X-ray flares occurrence. This six-year period covers both minimum and maximum of solar activity. Simultaneous measurement of amplitude and phase of the VLF/LF signals during a solar flare occurrence was applied to evaluate the electron density profile versus altitude, to carry out the function of time over the middle Europe.

HF Radio Angle-of-Arrival Measurements and Ionosonde Positioning

Since 2010 a 2 nd generation NOAA MF/HF radar, also referred to as the VIPIR ionosonde, has been operated at Hualien, Taiwan (23.8973°N, 121.5503°E). The Hualien VIPIR ionosonde is a modern ionospheric radar, fully digitizing complex signal records and using multiple parallel receiver channels for simultaneous signal measurements from multiple spaced receiving antennas. This paper considers radio direction finding based on interferometric phase measurements from a horizontal antenna array in the Hualien VIPIR ionosonde system. We applied the Hermite normal form method to solve the phasemeasurement aliasing and least squares problems and improve the radio angle-of-arrival (AOA) measurements. Backward ray-tracing simulation has been proposed to determine radio transmitter position. This paper presents a numerical, step by step ray-tracing method based on the IGRF superimposed onto a phenomenological ionospheric electron density model, the TaiWan Ionospheric Model (TWIM). The proposed methodology is successfully applied to locate two experimental HF radio transmitters at Longquan and Chungli with distance errors within 5 km and less than 5% of the great circle distances.

Strictly and non-strictly positive definite functions on spheres

Isotropic positive definite functions on spheres play important roles in spatial statistics, where they occur as the correlation functions of homogeneous random fields and star-shaped random particles. In approximation theory, strictly positive definite functions serve as radial basis functions for interpolating scattered data on spherical domains. We review characterizations of positive definite functions on spheres in terms of Gegenbauer expansions and apply them to dimension walks, where monotonicity properties of the Gegenbauer coefficients guarantee positive definiteness in higher dimensions. Subject to a natural support condition, isotropic positive definite functions on the Euclidean space $\mathbb{R}^3$, such as Askey's and Wendland's functions, allow for the direct substitution of the Euclidean distance by the great circle distance on a one-, two- or three-dimensional sphere, as opposed to the traditional approach, where the distances are transformed into each other. Completely monotone functions are positive definite on spheres of any dimension and provide rich parametric classes of such functions, including members of the powered exponential, Mat\'{e}rn, generalized Cauchy and Dagum families. The sine power family permits a continuous parameterization of the roughness of the sample paths of a Gaussian process. A collection of research problems provides challenges for future work in mathematical analysis, probability theory and spatial statistics.

Spatial Decay of Corruption in Africa and The Middle East

This study contributes to a small, but growing stream of literature exploring the contagious nature of corruption by examining the rate at which the correlation between countries' corruption measures decays as the geographical distance between country capital cities increases. Focusing on countries in Africa and the Middle East and using historical data from Transparency International to measure perceived levels of country corruption and the great circle distance (in miles) between country capitals, the results of this analysis indicate that corruption is contagious and can spread beyond its neighbour's borders. The rate at which corrupt practices are adopted decays as the distance between the countries' capitals increases and the first order conditions suggest that corrupt practices can spread and impact practices beyond 2700 miles. The results also suggest that reducing corruption within a nation's borders can be viewed as a positive externality as reductions in corruption levels can spillover to other countries and help to lower corruption within a larger geographical region.

A Regional Assessment of Mismatch between Joint Commission Certified Stroke Centers and Population Requirements (P02.194)

Objective: The objective of this study was to examine the relationship of the location of Joint Commission certified stroke centers and population. Background The certification program does not incorporate a population based needs assessment. We hypothesized that some population centers may not contain certified centers and that inter-competitive behaviors of centers may be lead to clustering of certified centers. Design/Methods: We identified the primary stroke centers (PSCs) in the Upper Midwest that were certified by Joint Commission and found their addresses and determined the lat and long coordinates. We then obtained a list of all the census blocks in the upper Midwest that included the populations and latitude and longitude coordinates. We used the Haversine formula to calculate the great circle distance between each stroke center and each census block. Results: We identified a total of 56 PSC hospitals that are serving a population of 15.4 million in 2010 in upper Midwest. Of the population served, 53% of the population lives within 15 miles of a PSC. Of these, 3%, 6%, and 1% live within 15 miles of 2,3, and 4 PSCs; 21% live within 15 miles of 5 or more PSCs. Seven PSCs have a population of ≤100,000 per PSC within 15 miles, and 49 PSCs have a population of >100,000 per PSC within 15 miles; 18 PSCs have a population of 1 million or greater within 15 miles. Conclusions: We found that almost half of the population in the Midwest does not reside within 15 miles of a PSC while one-fifth of the population resides within 15 miles of 5 or more PSCs. Such mismatches highlight the need for PSC designation to consider population needs to ensure homogeneity of care. Disclosure: Mr. Connelly has received personal compensation for activities with Cooperative Development Services. Dr. Meland has nothing to disclose. Dr. Chaudhry has nothing to disclose. Dr. AbdAllah has nothing to disclose. Dr. Reshetar has nothing to disclose. Dr. Qureshi has nothing to disclose.