Cartography

Introduction toModern Cartography

Analogue cartography printed cartographic data on paper maps, by necessity storing and displaying the data in a single map projection. Modern cartography stores cartographic data in projections that are different than projections for visualization, not requiring a single “catch-all” projection.

Modern cartography also acquires cartographic data that are higher resolution, using a single three-dimensional reference frame, replacing legacy horizontal and vertical datums that were error-prone.

Very Long BaselineInterferometry (VLBI)

In modern cartography, the Greenwich Observatory is no longer at longitude zero. Instead, the earth’s graticule (longitude and latitude grid) is recalculated and repositioned periodically, using least squares fitting, relative to known stationary points around the world, none of which are at the Greenwich Observatory.

The known stationary points around the world are VLBI stations. VLBI is very long baseline interferometry, which accurately determines the positions of the stations relative to each other, to within a few millimeters over thousands of kilometers.

VLBI stations receive radio signals that are dense (have many oscillations) from reliable distant quasars in the sky. The time difference of two stations receiving the same signal reveals the geometry of those stations relative to the direction of the quasar. That geometry is used to calculate the baseline distance between the stations (Figure 1).

Figure 1: VLBI processing. (JPL/AAS)

The baseline distance between two VLBI stations, denoted B in Figure 1, is a straight line, passing through the earth (below the surface of the earth).

Each VLBI station includes an extremely accurate atomic clock. For VLBI station configuration information see VLBI 2010 in the References section at end of this article.

Figure 2:  VLBI station at the Goddard Geophysical and Astronomical Observatory. (NASA/JPL)

“VLBI derived baselines have already contributed scale information to the development of the DoD World Geodetic System in 1972.”
—  Geodesy For The Layman (1983)

International TerrestrialReference Frame (ITRF)

Multiple VLBI stations around the world are used to provide geometric positioning of the stations relative to each other. The resulting grid is now called the International Terrestrial Reference Frame (ITRF).

Coverage (densification) of remote areas far from VLBI stations may be estimated relative to the stations using highly accurate PSI (Persistent Scatterer Interferometry). PSI uses satellite interferometric synthetic aperture radar (InSAR) referenced to persistent scatterers (PS) such as points on buildings, road intersections, rock outcroppings, etc.

Persistent scatters (PS) are objects that do not normally change over time, unlike for example leaves, water, etc.

“With respect to spatial resolution, the cost of maintaining a dense network of permanent GPS stations or of often repeated GPS surveys, turns our interest to different sources of densification data, such as laser scanning and SAR interferometry”
—
Dermanis, Kotsakis, “Estimating Crustal Deformation Parameters from Geodetic Data”, in Sanso and Gil, eds., Geodetic Deformation Monitoring, p. 7.
“PS are detected and their average velocity is then estimated by an automatic procedure, allowing to process large amounts of data relative to large areas… Advanced Analysis…suitable for those small areas where a full exploitation of the information content of the satellite data is required…requires skilled technical staff.”
—
Meisina, Zucca, Fossati, Ceriani, Allievi, “PS InSAR Integrated with Geotechnical GIS”, in Sanso and Gil, eds., Geodetic Deformation Monitoring, p. 67.
“There is a need to increase the processing capability of current PSI algorithms… The objective is to achieve a massive data processing capability to systematically monitor very wide areas.”
—
Crosetto et al., ISPRS Journal, 2015. (Open Access, see References below)
Figure 3:  InSAR requires multiple sensors or multiple passes to triangulate heights. South is up in this map. (JAXA)

ITRF is recalculated periodically to account for movement of the earth’s crust. The ITRF is stored as geocentric X,Y,Z coordinates. An ellipsoid, referred to as GRS80, is derived from ITRF, and is the most accurate reference ellipsoid. Other reference ellipsoids, including WGS84, are based on GRS80.

Datums

A reference ellipsoid (e.g., GRS80) provides a position relative to the ellipsoid as longitude, latitude and normal height (labeled E, N and Z in Figure 3 respectively).

A less accurate ellipsoid called WGS84 has been developed for recreational uses. Unlike the more accurate ellipsoids, which are purely geometric, WGS84 incorporates a gravity model, relative to sea level. Sea level (referred to as the geoid) is directly observable but reduces accuracy. Some applications require a gravity model. To make sure a gravity model is properly applied, some data sets only come with a gravity model already applied, including instructions on how to remove the gravity model.

For environmental studies purposes, equal-area cartographic data may be mapped from a reference ellipsoid to a sphere, using authalic latitudes (changing latitudes while keeping longitudes constant), then mapped from the sphere to a map projection, all transformations equal-area (see below). A sphere radius of 6371007.181 meters is often used for this, referred to as the MODIS Sphere.

Map Projections

The Earth is not flat, but must be displayed on flat surfaces. It is not possible to map (transform) the surface of the Earth onto a flat surface without distortion that changes directions, and/or distortion that changes area sizes relative to each other.

A projection (transformation) that is conformal maintains directions (and therefore shapes), from the transformation preimage (e.g., surface of the Earth), to the transformation image (map projection). This causes distortion of areas relative to each other.

An equal-area (equiareal) projection, on the other hand, maintains area ratios relative to each other. For example, an area that is twice as big as another area in the preimage will still be twice as big as the other area in the image. This causes distortion of angles (directions/shapes).

Projections other than conformal projections are important now that data sampling of wide areas is higher resolution.

Angular map projections define the projection image in terms of preimage angles (longitude and latitude). Gridded map projections define the image in image plane coordinates.

Figure 4:  Sinusoidal equal-area gridded map projection using MODIS Sphere. (USGS/MRT)

References

1.  Petrachenko, Niell, Corey, Behrend, Schuh, Wresnik, “VLBI2010: Next generation VLBI system for geodesy and astrometry”, 2009. pdf o

2.  Defense Mapping Agency, “Geodesy For The Layman”, 1983. pdf o

3.  Crosetto, Monserrat, Cuevas-Gonzalez, Devanthery, Crippa, “Persistent Scatterer Interferometry: A review”, ISPRS Journal, 2015. Open Access pdf o

4.  Shimon Wdowinski, “Collapsed Surfside building showed signs of subsidence in 1990s”.  FIU map & video, 2021.