Reference ellipsoid

Now we will establish a relationship between the potential U(p) at any point p of the volume V and its values on the spherical surface, surrounding all masses. Fig. 1.11. The reason why we consider this problem is very simple it plays the fundamental role in Stokes s theorem, which allows one to determine the elevation of the geoid with respect to the reference ellipsoid. 

A similar formula describes the dependence of the distance from the origin to any point of the outer surface of the spheroid. In both expressions terms proportional to the third and higher order of flattening are discarded. This reference ellipsoid and its field are defined by four constants. The best-known and widely used values are... 

We assume that with the help of leveling we solved our first problem and found the separation between the geoid and the points of the physical surface of the earth. Our next step is to determine the position of the geoid with respect to the reference ellipsoid. The solution of this fundamental problem was given by Stokes. To begin,... 

In other words, we obtain the reference ellipsoid from the condition that its normal potential U has the same value as the resultant potential of the geoid. At the same time, it is assumed that the geoid and the ellipsoid enclose equal masses and have the same angular velocity. 

Here N is the distance between points p and q measured along the perpendicular from the point q to the geoid. Fig. 2.9a. The linear behavior of the normal potential implies that the field y is constant between the geoid and the reference ellipsoid. The change of sign in Equation (2.260) is related to the fact that the field has a direction, which is opposite to the direction of differentiation. As follows from the first equation of the set (2.258 and 2.260) we have... 

In the previous section we described the Stokes method, which allows us to find the distance between the reference ellipsoid and the physical surface of the earth. The ellipsoid, given by its semi-major axis a, flattening a, and elements of orientation inside of the earth can be considered as the first approximation to a figure of the earth. In order to perform the transition to the real earth we have to know the distance along the normal from each point of the spheroid to the physical surface of the earth. Earlier we demonstrated that this problem includes two steps, namely,... 

Fig. 2.10. Telluroid, quasi-geoid, and reference ellipsoid (after Grushinsky).

Here Wq and Uq are the total and normal potentials on the surface of the geoid and on the surface of the reference ellipsoid, respectively. By definition, y — —dUjdz is the magnitude of the normal gravitational field. Thus, Equation (2.292) becomes... 

Here g is the gravitational field on the physical surface of the earth, y the normal field on the surface S. At the same time, dT/dv and dy/dv have the same values along line V at both surfaces. This is the boundary condition for the disturbing potential and therefore we have to find the harmonic function regular at infinity and satisfying Equation (2.301) on the surface S. In this case, the physical surface of the earth is represented by S formed by normal heights, plotted from the reference ellipsoid. In other words, by leveling the position of the surface S becomes known. 

Besides, these measurements allow us to determine an elevation of the geoid, quite ocean surface, over the reference ellipsoid. The idea of this method is very... 

Fig. 1. (a, b) Determination of geoid elevation over the reference ellipsoid. 

Fig. 4.5. Ellipsoid representing the value of B for A = 1 sphere of radius 1 reference ellipsoid (e.g. A = electric field, B = current)...

The tensors compatible with different symmetries have the forms (4.10), (4.11), (4.13) and (4.14). The reference ellipsoids are described in Section 4.2.5. The eigenvalues of s are all positive. For an alternating electric field, the tensor s is a function of the frequency. 

Note that the deflection of the vertical (i is equal to the inclination —SNfSD of the geoid relative to the reference ellipsoid The linkage between the reference surface of 6h (= surface parallel to the reference ellipsoid) and that of SH (= equipotential surface) is based on two earth fixed physical points P subject to measurements in both systems. 

The reference ellipsoids are the horizontal surfaces to which the geodetic latitude and longitude are referred. 

These equations also define the signs of the deflection components. In North America, however, the sign of is sometimes reversed. We emphasize here that the geodetic coordinates have to refer to the geocentric reference ellipsoid/mean earth ellipsoid. Both geodetic and astronomical coordinates must refer to the same point, either on the geoid or on the surface of the earth. In Section II.B, we mentioned that the astronomical determination of point... 

---