GMT322 – GEODESY-1

1. GMT322 – GEODESY-1 Lecture7 Dr. Kamil Teke

2. Physical interpretation of the spherical harmonic coefficients • The spherical harmonic expansion has transformed the single volume integral over theEarth’s masses into an infinite series. • The lower degree harmonics physical interpretation: • The zero degree term (n = 0) of the spherical harmonic expansion of the Earth’s gravitational potential field, when it is 1, represents the potential of ahomogeneous or radially layered spherical Earth: . In other words, zero degree term when equals to 1 yields the potential of the real Earth’s mass M concentrated at the center of the mass. Thus, when the mass of the spherical harmonic Earth model (e.g. GOCE Earth model) is intended to be the same with the mass of the real Earth zero degree term, that is, is taken as 1. • First degree terms (n = 1) are taken as zero when the origin of the coordinate system (defines the geocentric latitude () and longitude () used in harmonics) is placed at the center of the mass of the Earth model. • Thus, for the spherical harmonic series expansion of the Earth model (e.g. GRACE Earth model). Here, . • Second degree terms (n = 2) are • when Z axis coincides with the maximum moment of inertia (principal axis of inertia of the Earth). • dynamical form factor (the polar flattening of the Earth body, that is, the difference between mean equatorial moment of inertia and the polar moment of inertia. The largest deviation from a spherical Earth model.) describe the asymetry of the equatorial mass distribution in relation to rotational axis (ellipticity of the equator)

3. Path dependency to geometric leveling Potential differences are the result of leveling combined with gravity measurements. An integral of over a circuit must be zero. However, the measured height difference, that is, the sum of the leveling increments depends on the path of the integration and is, thus, not in general zero for a circuit: Leveling and orthometric height Leveling without gravity measurements, although applied in practice, is not perfectly true from a rigorous point of view, for the use of leveled heights (*) as such leads misclosures.

4. Geopotential numbers and dynamic heights • Let O be a point at sea level, that is, simply speaking on the geoid. Let A be another point connected to O by a leveling line. The potential difference between A and O can be determined by: • The difference between the potential at the geoid and the potentail at the point A is called as geopotential number (C) of Aand is defined so as to be always positive. As a potential difference, the geopotentail number C is independent of the particular leveling line used for relating the point to the sea level and considered as natural measure of height. • The geopotential number C is measured in geopotential units (gpu): 1 gpu = 1 kgal m = 1000 gal m. Geopotential number of a point is: • The dynamic height of a point is defined by: where is normal gravity for an arbitrary latitude, usually 45o . The dynamic height differs from the geopotential number only in scale or the unit. The dynamic height has no geometrical meaning whatsoever, so that the division by an arbitrary somehow obscures the true physical meaning of a potential difference. Therefore, the geopotential numbers are preferable to the dynamic heights for practically establishing a national or continental height system.

5. Dynamic correction: DCAB • It is sometimes convenient to convert the measured height difference into a dynamic height by adding a small correction: Orthometric heights Orthometric height, that is, the length of the plumb line segment between P0 and P • The above formulas refer to the normal density,

6. Orthometric correction : OCAB • The orthometric correction is added to the measured height difference in order to convert it into a difference in orthometric height. • We need the mean values of gravity along the plumb lines, and . • is an arbitrary constant for which we always take normal gravity at 45o latitude.

7. Normal heights • Assume for the moment the gravity field of the Earth to be normal, that is, . On this assumption orthometric heights will be equal normal heights. • Consider a point P on the physical surface of the Earth. It has a certain potential WP and also a certain normal potential UP, but in general ≠. However, there is a certain point on the ellipsoid normal of P such that ; that is the normal potential U at is equal to the actual potential at P. The normal height, H* of P is nothing but the ellipsoidal height of above the ellipsoid, just as the orthometric height of P is the height of P above geoid. Iterative approach Direct expression Normal correction: NCAB • The normal correction is added to the measured height difference in order to convert it into a difference in normal height. • We need the mean values of normal gravity along the ellipsoid normals, and. • is an arbitrary constant for which we always take normal gravity at 45o latitude.

8. GPS Leveling and geoid determination with GPS • Spirit leveling is a very time-consuming opeartion. GPS has introduced a revolution here. The basic equation is: • This equation relates the orthometric height H, the ellipsoidal height h, and the geoid undulation (geoid height) N. If any two of these quantities are measured, then the third quantity can be computed. • If h is measured by GPS, and if there exists a reliable geoid map (or model e.g. Spherical harmonics), then the orthometric height H can be obtained. • Also, the above equation can be used for geoid determination. If h is measured by GPS and H is available from leveling then the geoid heights N can be determined by: N = h – H. The same principle can be applied even on the oceans with the satellite altimetry observations.

9. The relation of Geoid with mean sea level (MSL) -1.8 m < mean SST (ODT) < + 1.2 m (global scale) (LeGrand et al. 2003) http://www.geod.nrcan.gc.ca/hm/msl_e.php

10. THANKS FOR YOUR ATTENTION