This section gives a brief overview of different coordinate systems used in this project. It also discusses the reasons and/or advantages of using these particular coordinate systems. The 'natural' coordinate system for the Earth is a geographical system of latitudes and longitudes. To describe 'world-wide' phenomena, such as topography, the data can be sampled at specified latitude and longitude intervals. This data set can be analyzed using, for instance, a spherical harmonic technique. To analyze data over part of a sphere, often with a higher resolution, a local coordinate system is prefered.
One of the main objectives of this project is to produce a uniform grid for the study area, whose surface forms a significant part of a sphere (see Chapter 1). A uniform grid indicates a grid that has a cell size in x and y directions that does not vary significantly over the region. As an example, assume the geographic coordinate system is used to define the grid, with a cell size of 0.1 degrees both in latitude and longitude. At the equator, this corresponds to a cell size of about 11 km, both in x and y direction. However, further north or south, at about 60N or 60S, the cell has changed to about 5.5 km in the x direction, with no change in the y size. Near the North or South Pole, the x size reduces to zero. The resulting grid is non uniform and, moving away from the equator, over-sampled in the x direction. To overcome this, the grid can be defined in a different coordinate system, obtained by projecting the spherical earth onto a plane, cylinder or cone, as is commonly done to produce maps.
a: Map projections:
The shape of the Earth is usually approximated by an ellipsoid or, less accurately, by a sphere. To show this curved surface as a flat map inevitably leads to distortions, the size of which depend upon the extend of the region shown. The technique of representation is called 'map projection', and it involves a transformation of coordinates from a curved surface to a flat one. This can be done in many different ways and therefore, the number of map projections is virtually endless (for an overview, see Richardus and Adler, 1972; Snyder, 1982). Map projections can be divided into three broad groups, with the following main characteristics:
Conformal projection (correct representation of shapes) In a conformal projection, local angles are preserved, i.e. if two lines are locally perpendicular on the surface of the earth (for example a meridian and a parallel), they are also locally perpendicular in the projected plane. This property is useful when local directions of structures are important. Examples are Lambert Conformal, and Mercator projections.
Equal Area Projection In an equal area projection, objects that occupy equal surface areas on the earth will also have equal areas in the projected plane. This property is useful when comparing surface areas over the globe, and when performing Fourier transforms. Examples are Lambert Equal Area and sinusoidal projections (e.g. Sanson-Flamsteed projection).
Others Projections that are neither conformal nor equal area.
There is no single projection that combines all preferred characteristics of map projections. The final choice will be a compromise that minimizes distortion over the region of the map. For the compilation project, the final choice was based on the following three considerations: 1): the use of one single projection for the North Atlantic and Arctic Oceans and surroundings. This is a large geographic region which includes the geographic North Pole; 2): minimal distortion over the largest possible part of the map, in particular over regions that are covered by data; and 3): a projection commonly used in regional compilations, and readily available for computations
The first consideration eliminates all projections that can not properly map the Earth's geographic poles. Since the region of the compilation is larger in extent in the north-south direction than in the east-west direction, an obvious choice is a transverse projection, with a central meridian near the centre of the area at about 50W. The final two candidates are a Transverse Mercator and a Transverse sinusoidal projection.
Transverse Mercator:
The Mercator projection corrects for the convergence of meridians when nearing the poles by enlarging the local scale. The Transverse Mercator projection enlarges the scale away from the specified central meridian, where the map has true scale. In this projection the central meridian is the equivalent of the equator in the Mercator projection. In order to minimize the average scale distortion of a region, a scale factor is often introduced. This has the effect that distances at the central meridian becomes too small and true scale occurs on two lines parallel to the central meridian. For the DNAG compilation of North America, a scale factor of 0.926 was used, which means that the true scale occurs at a distance of 2,343 km on either side of the central meridian (taken at 100W).
The compilation area in a Transverse Mercator projection is shown in Figure 7-10. Using a scale factor of 0.926, the same as for the DNAG compilation, the distance distortions over the area are shown in Figure 7-11. These distortions vary with the distance from the central meridian and they range from about -5% and -8%, along the left and right hand sides of the map, respectively, to about +8% at the central meridian. This means that distances are compressed at the central meridian and stretched at the edges of the map.
Transverse Sinusoidal:
The sinusoidal transformation (also known as the Sanson-Flamsteed projection, e.g. Snyder, 1987) compresses parallels in an east-west direction. This is an equal area projection, and therefore not conformal. In a transverse form, the projection preserves true scale along small circles parallel to the central meridian. The angular deformation increases towards the corners of the map region. Figure 7-12 shows the compilation area in a Transverse sinusoidal projection. Note how the corners of the area are more deformed when compared to the Transverse Mercator projection ( Figure 7-10 ). Relative to a Transverse Mercator, a Transverse sinusoidal projection covers a larger region with more area coming into the corners of the map (for example, the Bahamas in the southwest corner). The Transverse sinusoidal projection does not preserve distance, as is shown in Figure 7-13, where the relative difference from true scale in the x-direction is mapped. In the far corners of the map, the distances along the horizontal axis deviate by more than 10 % from the true distance. Distances along the vertical are always correct. The non-conformity of the projection is shown in Figure 7-14, which gives the local change in angle of two lines that are perpendicular on the sphere. Again, the largest distortions are found in the corners, where the deviations exceed 20 degrees from perpendicular.
Choice of map projection:
After considering the information given above, it was decided to use a Transverse Mercator projection for the final grid. The angular and distance distortion of the Transverse sinusoidal projection, though limited to the corners, are considered a significant disadvantage when compared to the Transverse Mercator. In addition, the Transverse Mercator projection is used in many regional compilations and readily available in existing computer software.
b: Geomagnetic coordinates and geomagnetic time
Many of the phenomena that we are dealing with in the 'cleaning and editing' of the magnetic data, are directly related to the Earth's magnetic field: its characteristics (both from internal and external sources), and its variability on time scales of minutes (from external sources, such as solar flares) to days (daily variations) to years (secular variation, from internal sources).
The (internal) magnetic field of the Earth can be approximated by an axially centred dipole field, with an angle of about 11.5 degrees with the rotation axis. This means that the magnetic poles do not coincide with the rotation poles. Some characteristics of the internal magnetic field, such as intensity and inclination, are directly related to the distance from the magnetic pole. Also some effects of the external magnetic field disturbances, such as auroral phenomena, are directly related to the internal magnetic field lines. To describe these parameters, it is often easier to work in a magnetic coordinate system, rather than in the geographic system. For an axially centred dipole field, the coordinate transformation needed can be simply described by a rotation. The location of the magnetic pole was taken from the 1980 magnetic models (Langel, 1987): 78.8N; 70.7W. Figure 7-15 compares the geomagnetic coordinates with the geographic coordinates in the compilation area.
Some of the magnetic phenomena that are discussed are not only location dependent, but also vary with time. For example, effects of the solar wind strongly depend on a particular time of the day: locations will be effected not globally at the same time, but only at a certain local time. Therefore, local time instead of Greenwich Time (GMT) is often used to describe and correct some of the error sources of magnetic observations. For effects that depend on the geomagnetic field orientation, such as the auroral effect, which depends on the angle between the solar wind and the magnetic field lines, local time is not adequate to describe these effects. In this case geomagnetic time is introduced, which gives the solar orientation with respect to the geomagnetic meridians (cf. local time which gives it with respect to geographic meridians). In the geomagnetic coordinate system, the change to geomagnetic time is very simple (e.g. Simonow, 1963). The local difference (t' - t) is:
with: E) the geomagnetic clock error (small effect of less than 20 minutes, which can be ignored in most cases); dlon): difference between geographic longitude and geomagnetic longitude; and gl: longitude of geomagnetic pole.
Figure 7-16 shows the difference between local time and geomagnetic time for the compilation area. For a large part of the area, the differences are less than one hour. They increase to more than 2 hours only in regions of less than 2000 km away from the line joining the geographic pole with the geomagnetic one.