THREE-DIMENSIONAL NETWORK ADJUSTMENTS

The Three-Dimensional Geodetic Model

Figure 1 shows a local geodetic coordinate system in relation to the geocentric coordinate system. The n axis points north along the local meridian; the e axis points east in the direction of the prime vertical; and the h axis points along the direction of the normal to the ellipsoid.

Figure 2, relates our measurements to the local geodetic system. The change in dn, de, du are related to slope distance (S), azimuth (A), and altitude angle (v). From the figure it can be seen that

a. dn = S cos v cos A = S sin z cos A
b. de = S cos v sin A = S sin z sin A (1)
c. du = S sin v = S cos z

The inverse of these equations is

(2)

As shown in Figure 1, the local geodetic coordinate system can be related to the geocentric coordinate system by rotating the u axis an arc distance of the F - 90°, and by rotating the e axis the 180° - L. This rotation can be written as

(3)

Expanding the rotation matrices in Equation (3), it becomes:

(5)

Substituting Equation (4) back into Equation (2), we get

(6)

The Observation Equations

Letting S = s, a₁ = A₁₂ and a₁ = v₁₂, the observation equations the in their nonlinear form can be written as

where a₁ is the azimuth of the line from station 1, a₁ is the altitude angle from station 1 to 2, and s is the mark-to-mark distance between station 1 and 2. The coefficient matrix in linearized form for Equations (6) are

where

Often it is preferred to compute the observations in terms of the changes in geodetic coordinates. The required transformation was discussed previously when going from geodetic coordinates to space rectangular coordinates. Differentiating those equations yields

where M is the radius of curvature in the meridian and N is the radius of curvature in the prime vertical.

Again it maybe preferred to express the changes in terms of the local geodetic coordinate system of (e, n, h). The transformation from the geodetic coordinate system to this system is

Substituting Equation (9) into (8) and finally into Equation (7), the final expression for the linearized observation equations becomes

where the derivatives with respect to the unknowns are:

One of the advantages of the three-dimensional adjustment is that there is no need to reduce the observations. In fact, the observations can be taken from their measuring positions if the elevation is defined as h + dh, where dh is the height of the instrument or reflector above the ground when computing the constants matrix, K. At the completion of the adjustment, dh can be utilized to compute the mark-to-mark distances. The use of geodetic or local geodetic coordinates is useful, since it also allows the introduction of differential leveling measurements. However if differential leveling observations are to be used in the adjustment, then geoid separations must be known. If the adjustment is done in geocentric coordinates (x, y, z), it can readily be converted to local geodetic coordinates using the transformation

Using GLOPOV, 3 × 3 covariance submatrices can be computed from the geocentric coordinate adjustment as

Note that HJ^-1 is the rotation matrix in Equation (4). Utilizing the last equations, it is possible to transfer the uncertainties into terms more understandable by the user. If vertical angles are to included into the adjustment one must either correct these angles for refraction or incorporate the correction formulas into the adjustment. However, one must be careful not to include too many parameters such that the adjustment becomes unique or singular. Furthermore note that horizontal distances may be entered into the adjustment. However, it must be remembered that the distance measured from station 1 to 2 will not equal the distance measured from station 2 to 1 if the stations are at different elevations due to the nonparallelism of the directions of the plumb lines.