Geoid

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. 

Here Uq is the value of the potential on the surface. It is proper to notice that potentials of the attraction field and the centrifugal force usually vary on the level surface of the gravitational field. Changing the value of the constant, Uq, we obtain different level surfaces, including one which coincides in the ocean with the free undisturbed surface of the water and, as was pointed out earlier, this is called the geoid. As follows from Equation (2.73) the projection of the field g on any direction / is related to the potential U by... 

As was pointed out earlier, the determination of the elevation of points of the physical surface of the earth with respect to the geoid is the only the first step in studying the shape of the earth. The next step is the calculation of the distance between corresponding points of the ellipsoid of rotation and the geoid, and this subject will be described in the following section. 

The vector line of the field g between the earth s surface and the geoid can be replaced by a straight line which has the same direction as the plumb line, (vertical), at the point b of the earth s surface. 

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,... 

Suppose that the influence of masses above the geoid is removed and we know the gravitational field g p) on the surface of the geoid. The procedure which allows one to solve this is called reduction of the gravitational field, and it is necessary in order to apply Stokes theory. Let us form the difference ... 

The function T p) is called the disturbing potential and it is very small, T U). Now we focus our attention on the space between the geoid and the ellipsoid of rotation and assume the potential of the gravitational field of the geoid, W p), and the normal potential of the spheroid, U q), are equal ... 

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 order to determine the function T we focus on the space, bounded by the geoid and infinity and formulate the boundary value problem with respect to... 

Now, making use of Bruns theorem we have at points of the geoid... 

The latter represents the boundary condition for the function T at the geoid since it is assumed that the anomaly of the gravitational field, Agf is known at each point. 

Before we make use of Equation (2.265), let us transform the boundary condition (2.264) in the following way. With an accuracy of small quantities, which have the same order as the square of the geoid heights, a differentiation along the normal can be replaced by differentiation along the radius vector, and correspondingly the condition (2.264) becomes... 

In essence, in place of the surface of the geoid we use a spherical surface. Now let us make one more approximation, which has the same order as the flattening, and, therefore, causes a very small error in determining the height N. We assume that the normal field is inversely proportional to the square of r. This gives... 

It is proper to notice that the function E is harmonic outside the geoid. Now, multiplying both sides of this equation by rdr and integrating within the range ... 

This is the Stokes formula, which permits us to find the elevation of the geoid at point p. Imagine that the z-axis of the spherical system of coordinates goes through the point p. Then for this point the angle lA plays the role of the azimuth 9 and... 

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

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