Lambert Azimuthal Equal-Area Research Articles

Slope, aspect, and hillshade algorithms for non‐square digital elevation models

AbstractCommon global digital elevation models (DEMs) at 1 to 3 arc‐second resolution use geographic coordinates for grid definition, and while cells are square (equal dx and dy spacing) near the equator, they become increasingly rectangular at higher latitudes. Some GIS software cannot handle rectangular grid cells, but minor modifications to slope, aspect, and hillshade algorithms can handle them. The algorithms can also handle grids with other projections, including Equi‐7, Lambert conformal conic, Lambert azimuthal equal area, and polar stereographic, converting aspect to true north instead of the potentially rotated grid. Computed slope varies with grid spacing with a logarithmic relationship and negative slope. Slope maps and distributions from rectangular DEMs, or DEMs where the actual grid spacing varies from the nominal values because points are far from the undistorted region, have characteristics equal to those of a Universal Transverse Mercator grid with spacing equal to the average dx and dy spacing of the DEM.

Map production and data analysis with local parameters

In this work we perform a geographical analysis of the Algarve region by using local parameters in the three map projections we consider: Transverse Mercator (TMzn), Lambert Azimuthal Equal-Area (LAEA) and Lambert Conformal Conic (LCC). We compare the Algarve maps resulting by each one of these representations when the origin of coordinates is moved from the center of mainland Portugal (that is in force) to the center of the Algarve. For the LCC map projection, the analysis takes also into account the change of the upper parallel from the North of Portugal to a latitude inside the Algarve border. The area and perimeter of the Algarve is computed and the location of its geographic center is determined in every map representation. Borders lengths that separates the Algarve from Alentejo and from Spain, and the extents of its South and West coasts, are also computed. We conclude by presenting the map, the values of area and perimeter and the location of the geographic center that should be considered for the Algarve, at different scales and for the available data.

Corrigendum

Diversity and DistributionsVolume 23, Issue 3 p. 341-341 CorrigendumOpen Access Corrigendum This article corrects the following: Space invaders; biological invasions in marine conservation planning Sylvaine Giakoumi, François Guilhaumon, Salit Kark, Antonio Terlizzi, Joachim Claudet, Serena Felline, Carlo Cerrano, Marta Coll, Roberto Danovaro, Simonetta Fraschetti, Drosos Koutsoubas, Jean-Batiste Ledoux, Tessa Mazor, Bastien Mérigot, Fiorenza Micheli, Stelios Katsanevakis, Lucy Hawkes, Volume 22Issue 12Diversity and Distributions pages: 1220-1231 First Published online: October 5, 2016 First published: 09 February 2017 https://doi.org/10.1111/ddi.12526AboutSectionsPDF ToolsRequest permissionExport citationAdd to favoritesTrack citation ShareShare Give accessShare full text accessShare full-text accessPlease review our Terms and Conditions of Use and check box below to share full-text version of article.I have read and accept the Wiley Online Library Terms and Conditions of UseShareable LinkUse the link below to share a full-text version of this article with your friends and colleagues. Learn more.Copy URL DOI: 10.1111/ddi.12491 Space invaders; biological invasions in marine conservation planning Volume 22, Issue 12, Pages 1220–1231, Article first published online: 5 October 2016 In Giakoumi et al. (2016), a writing error was made in the preparation of Fig. 4 and the ‘Methods’ section of the manuscript such that a reference has been omitted. This omission is corrected below, with an additional reference to Albouy et al. (2015) for data provenance. The fish distribution data and life-history traits data are from Albouy et al. (2015) and not from Guilhaumon et al. (2015) as previously written (a link to Albouy et al. (2015) exists in the ‘data availability’ section of the manuscript). Correction for the ‘Methods’ section. Methods Conservation plans: applying the ‘protect’, ‘avoid’ or ‘ignore’ approaches in two Mediterranean case studies Case study 2: endemic fish species Data on the distribution of 80 endemic fish species were obtained from Albouy et al. (2015). Correction for the legend of Fig. 4. Figure 4 Fish species case study (data from Albouy et al., 2015). Difference in planning unit (12,828 cells, 10 × 10 km) selection frequency, from Marxan outputs, when following the different approaches: (a) ‘ignore’ versus ‘protect’, (b) ‘ignore’ versus ‘avoid’ and (c) ‘avoid’ versus ‘protect’. Planning units in red are those with a higher selection frequency in the ‘ignore’ scenario, in orange those with higher selection in the ‘protect’ scenario and in blue those with higher selection in the ‘avoid’ scenario. Planning units are black if they had maximum selection frequency (1000) in all three scenarios. Scatter plots show the selection frequency for the planning units under the different scenarios. For the maps, we used ETRS89 Lambert Azimuthal Equal-Area projection. Reference Albouy, C., Ben Rais Lasram, F., Velez, L., Guilhaumon, F., Meynard, C.N., Boyer, S., Benestan, L., Mouquet, N., Douzery, E., Aznar, R., Troussellier, M., Somot, S., Leprieur, F., Le Loc'h, F. & Mouillot, D. (2015) FishMed: traits, phylogeny, current and projected species distribution of Mediterranean fishes, and environmental data. Ecology, 96, 2312– 2313. Volume23, Issue3March 2017Pages 341-341 ReferencesRelatedInformation

An open source software for fast grid-based data-mining in spatial epidemiology (FGBASE)

BackgroundExamining whether disease cases are clustered in space is an important part of epidemiological research. Another important part of spatial epidemiology is testing whether patients suffering from a disease are more, or less, exposed to environmental factors of interest than adequately defined controls. Both approaches involve determining the number of cases and controls (or population at risk) in specific zones. For cluster searches, this often must be done for millions of different zones. Doing this by calculating distances can lead to very lengthy computations. In this work we discuss the computational advantages of geographical grid-based methods, and introduce an open source software (FGBASE) which we have created for this purpose.MethodsGeographical grids based on the Lambert Azimuthal Equal Area projection are well suited for spatial epidemiology because they preserve area: each cell of the grid has the same area. We describe how data is projected onto such a grid, as well as grid-based algorithms for spatial epidemiological data-mining. The software program (FGBASE), that we have developed, implements these grid-based methods.ResultsThe grid based algorithms perform extremely fast. This is particularly the case for cluster searches. When applied to a cohort of French Type 1 Diabetes (T1D) patients, as an example, the grid based algorithms detected potential clusters in a few seconds on a modern laptop. This compares very favorably to an equivalent cluster search using distance calculations instead of a grid, which took over 4 hours on the same computer. In the case study we discovered 4 potential clusters of T1D cases near the cities of Le Havre, Dunkerque, Toulouse and Nantes. One example of environmental analysis with our software was to study whether a significant association could be found between distance to vineyards with heavy pesticide. None was found. In both examples, the software facilitates the rapid testing of hypotheses.ConclusionsGrid-based algorithms for mining spatial epidemiological data provide advantages in terms of computational complexity thus improving the speed of computations. We believe that these methods and this software tool (FGBASE) will lower the computational barriers to entry for those performing epidemiological research.Electronic supplementary materialThe online version of this article (doi:10.1186/1476-072X-13-46) contains supplementary material, which is available to authorized users.

An Area Preserving Projection from the Regular Octahedron to the Sphere

In this paper, we propose an area preserving bijective map from the regular octahedron to the unit sphere $${\mathbb{S}^2}$$ , both centered at the origin. The construction scheme consists of two steps. First, each face F i of the octahedron is mapped to a curved planar triangle $${\mathcal{T}_i}$$ of the same area. Afterwards, each $${\mathcal{T}_i}$$ is mapped onto the sphere using the inverse Lambert azimuthal equal area projection with respect to a certain point of $${\mathbb{S}^2}$$ . The proposed map is then used to construct uniform and refinable grids on a sphere, starting from any triangular uniform and refinable grid on the triangular faces of the octahedron.

Gaussian Curvature, Mirrors, and Maps

We present a method to optically measure the Gaussian curvature K of a surface and show how it can be used to establish a link between surfaces with constant K and area preserving maps between a sphere and a planar region. As an example, we show how Lambert's azimuthal equal area projection is naturally related to a sphere.

Uniform spherical grids via equal area projection from the cube to the sphere

We construct an area preserving map from a cube to the unit sphere S2, both centered at the origin. More precisely, each face Fi of the cube is first projected to a curved square Si of the same area, and then each Si is projected onto the sphere by inverse Lambert azimuthal equal area projection, with respect to the points situated at the intersection of the coordinate axes with S2. This map is then used to construct uniform and refinable grids on a sphere, starting from any grid on a square.

Tug of War at the Border

Rugby, North Dakota claims the distinction of being the geographic centre of North America. The author examines this claim and attempts to compute the true location of the geographic centre of North America. The author concludes that an iterative method based on the Lambert Azimuthal Equal Area (LAEA) projection, which slightly understates distance, and the Azimuthal Equidistant projection, which slightly overstates area, is the best method for computing the geographic centre of a given area. Six different definitions of North America are considered, and a proposed centre is offered for each definition.

Automated upscaling of river networks for macroscale hydrological modeling

We developed a hierarchical dominant river tracing (DRT) algorithm for automated extraction and spatial upscaling of basin flow directions and river networks using fine‐scale hydrography inputs (e.g., flow direction, river networks, and flow accumulation). In contrast with previous upscaling methods, the DRT algorithm utilizes information on global and local drainage patterns from baseline fine‐scale hydrography to determine upscaled flow directions and other critical variables including upscaled basin area, basin shape, and river lengths. The DRT algorithm preserves the original baseline hierarchical drainage structure by tracing each entire flow path from headwater to river mouth at fine scale while prioritizing successively higher order basins and rivers for tracing. We applied the algorithm to produce a series of global hydrography data sets from 1/16° to 2° spatial scales in two geographic projections (WGS84 and Lambert azimuthal equal area). The DRT results were evaluated against other alternative upscaling methods and hydrography data sets for continental U.S. and global domains. These results show favorable DRT upscaling performance in preserving baseline fine‐scale river network information including: (1) improved, automated extraction of flow directions and river networks at any spatial scale without the need for manual correction; (2) consistency of river network, basin shape, basin area, river length, and basin internal drainage structure between upscaled and baseline fine‐scale hydrography; and (3) performance largely independent of spatial scale, geographic region, and projection. The results of this study include an initial set of DRT upscaled global hydrography maps derived from HYDRO1K baseline fine‐scale hydrography inputs; these digital data are available online for public access at ftp://ftp.ntsg.umt.edu/pub/data/DRT/.

Developing clear-sky, cloud and cloud shadow mask for producing clear-sky composites at 250-meter spatial resolution for the seven MODIS land bands over Canada and North America

A new technology was developed at the Canada Centre for Remote Sensing (CCRS) for generating Canada-wide and North America continental scale clear-sky composites at 250 m spatial resolution for all seven MODIS land spectral bands (B1–B7). The MODIS Level 1B (MOD02) swath level data are used as input to circumvent the problems with image distortion in the mid latitude and polar regions inherent to the global sinusoidal (SIN) projection utilized for the standard MODIS data products. The MODIS 500 m land bands B3 to B7 are first downscaled to 250 m resolution using an adaptive regression and normalization scheme for compatibility with the 250 m bands B1 and B2. A new method has been developed to produce the mask of clear-sky, cloud and cloud shadow at 250 m resolution. It shows substantial advantages in comparison with the MODIS 250 m standard cloud masks. The testing of new cloud mask showed that it is in reasonable agreement with the MODIS 1-km standard product once it is aggregated to 1-km scale, while the cloud shadow detection looks more reliable with the new methodology. Nevertheless, more quantitative analyses of the presented scene identification technique are required to understand its performance over the range of input scenes in various seasons. The new clear-sky compositing scheme employs a scene-dependent decision matrix. It is demonstrated that this new scheme provides better results than any others based on a single compositing criterion, such as maximum NDVI or minimum visible reflectance. To account for surface bi-directional properties, two clear-sky composites for the same time period are produced by separating backward scattering and forward scattering geometries, which separate pixels with the sun-satellite relative azimuth angles within 90°–270° and outside of this range. Comparison with Landsat imagery and with MODIS standard composite products demonstrated the advantage of the new technique for screening cloud and cloud shadow, and generating high spatial resolution MODIS clear-sky composites. The new data products are mapped in the Lambert Conformal Conic (LCC) projection for Canada and the Lambert Azimuthal Equal-Area (LAEA) projection for North America. Presently this activity is limited to MODIS/TERRA due to known problems with band-to-band registration and noisy SWIR channels on MODIS/AQUA.

Implementation and Evaluation of Concurrent Gradient Search Method for Reprojection of MODIS Level 1B Imagery

This paper presents details regarding implementation of a novel algorithm for reprojection of Moderate Resolution Imaging Spectroradiometer (MODIS) Level 1B imagery. The method is based on a simultaneous 2-D search in latitude and longitude geolocation fields by using their local gradients. Due to the segmented structure of MODIS imagery caused by the instrument whiskbroom electrooptical design, the gradient search is realized in the following two steps: intersegment and intrasegment search. This approach resolves the discontinuity of the latitude/longitude geolocation fields caused by overlap between consecutively scanned MODIS multidetector image segments. The structure of the algorithm allows equal efficiency with nearest neighbor and bilinear interpolation. A special procedure that combines analytical and numerical schemes is designed for reprojecting imagery near the polar region, where the standard gradient search may become unstable. The performance of the method was validated by comparison of reprojected MODIS/Terra and MODIS/Aqua images with georectified Landsat-7 Enhanced Thematic Mapper Plus imagery over Canada. It was found that the proposed method preserves the absolute geolocation accuracy of MODIS pixels determined by the MODIS geolocation team. The method was implemented to reproject MODIS Level 1B imagery over Canada, North America, and Arctic circumpolar zone in the following four popular geographic projections: Plate Care (cylindrical equidistant), Lambert Conic Conformal, Universal Transverse Mercator, and Lambert Azimuthal Equal-Area. It was also found to be efficient for reprojection of Advanced Very High Resolution Radiometer and Medium Resolution Imaging Spectrometer satellite images and general-type meteorological fields, such as the North American Regional Reanalysis data sets.

Mineral Resources Map of East Asia (1:3,000,000)

Mineral Resources Map of East Asia (1:3,000,000) M. Kamitani, K. Okumura, Y. Teraoka, S. Miyano and Y. WatanabeTsukuba: Geological Survey of Japan, AIST, 2007 ¥2415 This 1:3,000,000-scale map presents a compilation of the mineral resources of East Asia in two folded sheets, with a brief explanatory text. The area covered includes more than two-thirds of Mongolia, south of Ulaanbaatar, the southeast corner of the Russian Far East, and more than two-thirds of China, east of 95–100° E longitude (east of a line near Yumen in the north extending to near Jinghong in the south); the southern limit is the boundary with Southeast Asia; and to the east the complete Korean peninsula, the Japanese archipelago, and other territories are included. The projection is Lambert azimuthal equal area. Given the large region, the geology varies tremendously in age as well as tectonic setting. The base map is taken from Teraoka and Okumura (2003), also a Geological Survey of Japan map, and consists of 17 age divisions in clearly differentiated colors, plus nine age divisions for intrusive rocks. There is further geological information on sedimentary type as well as composition of extrusive and intrusive rocks, plus metamorphic grade. The mineral deposits are divided into 10 types, where known, with information on principal commodity or commodities (color and symbol, including more than 30 elements and 10 minerals), plus deposit size (small, medium, and large, with divisions listed for each commodity). Large deposits are further identified with a number linked to tables in the accompanying explanatory text, whereas information on medium and small deposits is available on a public website of AIST. (http://www.gsj.jp/Map/Jp/new-maps.htm) Approximately 3200 mineral occurrences are listed (including significant industrial minerals). Metallogenic provinces (which commonly stop at political borders) are not suggested, letting the reader consider the general geological setting and age of various deposit types. The apparent simplicity of the map belies the huge amount of mineral deposit and resource data and information that are represented here, and the assessment that went into the final product. Many of the sources were in the original language, with references listed; some information is from unpublished sources. Various limitations exist for complete, high-quality, and publicly available data in many areas of the map. Thus, despite all of these difficulties, this map compilation is a noteworthy achievement, and the broad experience of the authors was essential to produce this reliable overview. The mineral explorer, particularly if new to the region, will find this map a quick and effective introduction to East Asia, a starting point for more in-depth examination of specific areas of interest. As is the case with any published compilation on mineral resources in a region where some of the areas have intensive exploration, some information presented here may soon become, or already is, incomplete. Nevertheless, the present knowledge base, introduced here in overview fashion, will contribute to building an ever-increasing understanding of the mineral resource potential of East Asia.

Reconstructing the Younger Dryas ice dammed lake in the Baltic Basin: Bathymetry, area and volume

A digital 3D-reconstruction of the Baltic Ice Lake's (BIL) configuration during the termination of the Younger Dryas cold phase (ca. 11 700 cal. yr BP) was compiled using a combined bathymetric–topographic Digital Terrain Model (DTM), Scandinavian ice sheet limits, Baltic Sea Holocene bottom sediment thickness information, and a paleoshoreline database maintained at the Lund University. The bathymetric–topographic DTM, assembled from publicly available data sets, has a resolution of 500 × 500 m on Lambert Azimuthal Equal Area projection allowing area and volume calculations of the BIL to be made with an unprecedented accuracy. When the damming Scandinavian ice sheet margin eventually retreated north of Mount Billingen, the high point in terrain of Southern central Sweden bordering to lower terrain further to the north, the BIL was catastrophically drained resulting in a 25 m drop of the lake level. With our digital reconstruction, we estimate that approximately 7800 km 3 of water drained during this event and that the ice dammed lake area was reduced by ca. 18%. Building on previous results suggesting drainage over 1 to 2 years, our lake volume calculations imply that the freshwater flux to the contemporaneous sea in the west was between about 0.12 and 0.25 Sv. The BIL reconstruction provides new detailed information on the paleogeography in the area of southern Scandinavia, both before and after the drainage event, with implications for interpretations of geological records concerning the post-glacial environmental development.

Statistical Comparison of Map Projection Distortions Within Irregular Areas

We describe a statistical sampling approach to characterizing and comparing map projection distortion within irregular areas. Statistical measures of distortion, coupled with traditional distortion isoline maps, give a clear picture of map projection distortion for irregularly shaped areas, like the United States or portions of it. We calculate cumulative distribution functions and several descriptive statistics from the distortion measures. In our example, we compare two common projections, the Lambert azimuthal equal area and the Albers conic equal area, over the conterminous United States and over two subregions. In addition to scale and angle distortion, we develop a new measure of shape distortion. Our analyses show that the Lambert projection has lower mean and median shape distortion when compared over the conterminous U.S., whereas the Albers projection has a lower maximum distortion and distortion variance for all three distortion measures. The cumulative distribution functions are substantially ...

An Equal-Area Map Projection For Polyhedral Globes

Numerous polyhedral shapes have been proposed as approximations for globes, and the projection most often used is the Gnomonic, with considerable scale and area distortion. Complicated conformal projections have been designed, but an equal-area projection has been used only once, for the icosahedron. The Lambert Azimuthal Equal-Area projection can be modified to provide an exactly fitting, perfectly equal-area projection for any polyhedral globe that has regular polygons, but is most satisfactory for the dodecahedron with 12 pentagons and for the truncated icosahedron with 20 hexagons and 12 pentagons. On the application to the truncated icosahedron, the angular deformation does not exceed 3.75°, and the scale variation is less than 3.3 percent. These advantages are at the expense of increased interruptions at the polygon edges when the polyhedral globe is unfolded. On a proposé de nombreuses formes polyédriques comme approximation de globes et la projection gnomonique est la plus souvent utilisée, avec de considérables distortions d'échelles et de superficies. On a conçu des projections conformes compliquées, mais on n'a utilisé qu'une seule fois une projection équivalente pour l'icosaèdre. La projection équivalente azimutale de Lambert peut être modifiée pour permettre un ajustement exact, parfaitement équivalent pour tout globe polyédrique dote de polygones réguliers, mais elle est plus que satisfaisante pour le dodécaèdre à douze pentagones et pour l'icosaèdre tronqué à vingt hexagones et douze pentagones. Dans l'application de l'icosaèdre tronqué, la déformation angulaire ne dépasse pas 3.75° et la variation d'échelle est inférieure à 3.3 pour cent. Ces avantages s'obtiennent aux dépens d'une augmentation des interruptions aux limites des polygones, lorsque l'on déplie le globe polyédrique.

Minimum-Error Equal-Area Map Projections

An equal-area map projection minimizing scale variation within a desired region may be prepared by combining the principle of least squares with an iteration of three elementary equal-area transformations. Constants are determined on a one-time basis using the Gram-Schmidt orthonormalization procedure with any number of chosen points on the Earth sphere or ellipsoid for which minimal distortion is desired. A standard initial projection such as the Lambert Azimuthal Equal-Area is then transformed using these constants and the iterative transformations to produce the final equal-area projection with irregular lines of constant distortion suiting the chosen region.

New Equal-Area Map Projections For Noncircular Regions

A series of new equal-area map projections has been devised. Called Oblated Equal-Area, its lines of constant distortion follow approximately oval or rectangular paths instead of the circles of the Lambert Azimuthal Equal-Area projection or the straight lines of the Cylindrical Equal-Area projection. The projection series permits design of equal-area maps of oblong regions with less overall distortion of shape and scale than equal-area maps on other projections.

“Magnifying-Glass” Azimuthal Map Projections

For maps focusing on a region of interest, but including surrounding areas to provide a setting, new azimuthal projections have been developed with a “magnifying-glass” effect. On two such projections, inside a circle bounding the region of interest is a standard Azimuthal Equidistant or Lambert Azimuthal Equal-Area projection. Between this circle and an outer bounding circle, azimuths remain true and the radial or area scale, respectively, remains constant, but at a reduced value. On four other projections, the inner portion is a standard azimuthal projection, which may be Stereographic, Gnomonic, or the above, but beyond this portion, the radial scale is gradually reduced to zero. Equivalents with rectangular boundaries are also available.

A New Oblique Equal-Area Projection

OST published equal-area maps of the world show the major land masses with excessive shape distortion in certain areas or with a break in continuity. L VI v Notably, the relationship between Asia and the Americas is usually unsatisfactory, because of a lack of continuity in the Arctic and sub-Arctic regions. This is the case in such frequently used equal-area projections of the world as the Sinusoidal and the normal forms of the Mollweide and Hammer-Aitoff. Oblique forms of these projections, especially with the center at 450 N. and about 100 E., are preferable, because the Arctic Sea is shown without interruption and the land masses usually fall within areas of small shape distortion. The American Geographical Society many years ago started experimenting with the oblique Mollweide so centered, but examples of maps drawn on this projection were not published until World War II.U More recently it occurred to the writer that a somewhat better result would be obtained by utilizing an oblique version of the HammerAitoff. Such a projection was constructed and was used to illustrate H. Duncan Hall's article Zones of the International Frontier in the Geographical Reviewt.2 In both the Mollweide and the Hammer-Aitoff the major axis of the bounding ellipse is twice the length of the minor axis. In the oblique forms of these projections the parallels of latitude close to the North Pole are shown as ovals. It would be more desirable, however, to show these parallels as near circles; this would preserve closely the true relationship between the northern continents. To this end, the writer developed a projection with a bounding ellipse whose major axis is I.75 the minor axis, and constructed it graphically from the Lambert Azimuthal Equal-Area by a procedure similar to that used in constructing the Hammer-Aitoff. Such a map projection is particularly useful for the plotting of world distributions. The projection has been used for several maps in the Society's Atlas of Diseases, on various scales. In response to a number of requests, outline maps on two convenient scales have now been prepared and are available.3