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Memorandum N 5

This memorandum is a direct sequel of the Memorandum N 5. Its main and single goal is the calculation of the geoid excess above the reference-ellipsoid. We need the value of the excess of quasigeoid above the reference-ellipsoid, but quasigeoid (averaged over small fluctuations geoid) probably weakly differs from geoid. This difference is of the order of tens centimetres and is considerably smaller than the error, inherent to the surface gravimetric survey. The last come to 3-4 metres. The excess can reach up to several tens of metres, so it is necessary to take it into account, since we want to attain the accuracy of several metres (near 5 metres).
Further, we have to distinguish the force of the gravitational attraction from the gravity force being the sum of the gravitational attraction and centrifugal force, which takes place in the rotating reference system motionless with respect to the Earth. On the vertical line of the gravity force the irregularity of the Earth rotation affects, i.e. virtually the astronomical coordinates of the place can vary, though this change is vanishingly small. In spite of smallness of such effects, we formally take irregularity into account, i.e. the changing of the angular velocity of the Earth rotation, in all calculations with only one exception. This is the calculation of the geoid excess above the reference-ellipsoid. Within the framework of this calculation we use a standard value of angular velocity of the Earth rotation, which we put equal to w0=7.292115105 rad/c11). The correction due to the excess is too small to take into account the irregularity of rotation.
It is also necessary to notice that geopotential is a potential of the gravitational attraction force, but the geoid as an equipotential surface of zero potential is in each point perpendicular to the gravity force.

Null point. For each geodetic system the point is given, in which the geoid sticks together with the reference-ellipsoid. It actually sets a level of the zero potential of the gravitational field of the Earth. For the Krassovski reference-ellipsoid these are the Pulkovo heights. For small geodetic system such as of some island it is possible to set this point arbitrary at the sea level as the reference-ellipsoid passes through the sea surface near to this island (islands). Really stipulated null-points are known less than ten. I think, that I know them all. If we find some necessary information missing, I know a first class professional in this field with whom I can discuss this topic.
The system of coordinates, in which we calculate the geopotential, is called linking in our terminology. It does not coincide with the IERS Reference Pole, therefore the coefficients and are not equal to 0.

Potential. The potential of the gravitational attraction by the Earth is calculated according the formula
where r, q , j are the spherical coordinates of a point. The coefficients and are given by the TXT file (model JGM-3). Other constants:
GM Earth5=3.9860044151014 m3/s2
ae=6378136.3 m.
The normalized Legendre polinomials with astronomical specificity consisting in that the latitude is counted from the equator instead of the ordinary polar angle counted from the North Pole are calculated according the recursion relations
where .

Excess of the geoid above the reference-ellipsoid. It is necessary to add the potential of the centrifugal force to the gravitational potential V to receive the potential of the gravity force using the formula
where r is a projection of a vector of a point r onto the plane of the instantaneous equator. To caslculate r it is necessary to take the basis vector N 3 of the trihedron R2(d)R3(E)[1:2:3]CRS (the Capitaine procedure) or
R1(eADe)R3(Dy)R1(eA)R3(zA)R2(qA)R3(zA)[1:2:3]CRS (the Lieske classical procedure). Therefore
At first it is necessary to calculate U0, i.e. the value of the potential in a null-point on the surface of the reference-ellipsoid where the geoid sticks together with the reference-ellipsoid.
Now let us take a point on the surface of the Earth, which geographical (in other words ellipsoidal) coordinates are known (h, B, L). We calculate the geocentric vector r0 of the point on the surface of the ellipsoid with the coordinates (B, L) and its gravity potential U(r0). The excess H of the geoid above the reference-ellipsoid makes
where the value of the free fall acceleration is calculated according the formula:

1. Burca, M., 1995, Report of the I.A.G. Special Commission SC3, Fundamental Constants, XXI, I.A.G. General Assembly.