29.3
ジオイドと楕円体
地球の形状は、楕円を短軸の周りで回転させて作られた、わずかに平らな球体である楕円体として最もよく説明されます。この平らな球体により、極軸は赤道軸より約 21 km 短くなります。対照的に、ジオイドは地球の重力形状を表し、平均海面 (MSL) と一致します。ジオイドは、重力がどの地点でも垂直である不規…
The Earth is best described as an ellipsoid, or an oblate spheroid, formed by rotating an ellipse around its minor axis.
This shape is slightly flattened at the poles, with the polar axis approximately 21 kilometers shorter than the equatorial axis.
In contrast, the geoid represents Earth's shape based on gravitational forces, closely aligning with mean sea level or MSL over oceans when external influences like tides are excluded.
It is an equipotential surface, with gravity acting perpendicular at all points. Unlike the smooth ellipsoid, the geoid is irregular due to mass distribution variations, causing global MSL undulations of up to 100 meters.
Historically, surveyors referenced MSL or the geoid using spirit levels, focusing on local measurements.
However, modern GPS provides a global perspective, measuring elevation relative to the ellipsoid. Based on the World Geodetic System 1984 ellipsoid, this data must be converted using a geoid model to approximate MSL for practical use.
GPS observations calculate Cartesian coordinates in an Earth-Centered, Earth-Fixed system, which are transformed into geographic coordinates such as latitude, longitude, and height for precise mapping and surveying.
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Q1: What is the difference between an ellipsoid and a geoid?
An ellipsoid is a smooth, mathematically defined shape created by rotating an ellipse around its minor axis, with the polar axis approximately 21 kilometers shorter than the equatorial axis. The geoid represents Earth's actual gravitational shape and aligns with mean sea level. Unlike the smooth ellipsoid, the geoid is irregular due to mass distribution variations, causing global mean sea level undulations of up to 100 meters.
Q2: Why is the geoid considered an equipotential surface?
The geoid is an equipotential surface because gravity acts perpendicular at all points on it. This means the gravitational potential is constant everywhere on the geoid surface. The geoid closely aligns with mean sea level over oceans when external influences like tides are excluded, making it the reference surface for vertical measurements in traditional surveying.
Q3: How does GPS measure elevation differently from traditional surveying methods?
Traditional surveying used spirit levels to reference mean sea level or the geoid for local measurements. Modern GPS provides a global perspective by measuring elevation relative to an ellipsoid, such as the World Geodetic System 1984. GPS observations calculate Cartesian coordinates in an Earth-Centered, Earth-Fixed system, which are transformed into geographic coordinates like latitude, longitude, and height for precise mapping and field application of global positioning system.
Q4: What is the Earth-Centered, Earth-Fixed coordinate system used for?
The Earth-Centered, Earth-Fixed (ECEF) coordinate system is a Cartesian system with its origin at Earth's center of mass. The x-axis extends to the equator at the Greenwich meridian, the y-axis lies perpendicular on the equatorial plane, and the z-axis aligns with the polar axis. GPS calculates Cartesian coordinates in this system, which are then converted into latitude, longitude, and ellipsoidal height for practical surveying and mapping applications.
Q5: Why must GPS ellipsoidal heights be converted using a geoid model?
GPS measures elevation relative to the ellipsoid, but practical applications require heights relative to mean sea level, which the geoid represents. A geoid model converts GPS ellipsoidal heights to approximate mean sea level values. This conversion is essential because the geoid and ellipsoid differ by up to 100 meters globally due to Earth's irregular mass distribution.
Q6: How does Earth's mass distribution affect the geoid's shape?
Variations in Earth's mass distribution cause the geoid to be irregular rather than smooth. These mass variations create gravitational anomalies that result in global mean sea level undulations of up to 100 meters. The geoid remains an equipotential surface where gravity acts perpendicular at all points, but its shape reflects the actual gravitational field influenced by subsurface density variations.
Q7: What role does the World Geodetic System 1984 play in modern GPS surveying?
The World Geodetic System 1984 (WGS 84) is the ellipsoidal model that GPS uses as its reference surface for measuring heights. Aligned with the Geodetic Reference System 1980, WGS 84 incorporates satellite data to refine measurements of Earth's size, shape, and gravitational forces. GPS observations based on WGS 84 are converted into geographic coordinates and state plane coordinates for accurate surveying and mapping.