Relative positioning is the determination of the location of one point with respect to another, either by measuring directly between the two points, or by measuring indirectly from the two points to extraterrestrial objects. Relative positioning can be done in one, two or three dimensions. All procedures involve a direct and inverse problem.
As shown in Figure 1, the direct problem in 3D positioning reads as follows:
Given the (xi, yi, zi)CT and the astronomical coordinates (Fi, Li) or equivalently the deflection of the vertical components (xi, hi) of the observation point Pi, and given either the astronomical azimuth Aij, the vertical angle vij or the zenith distance zij, and the spatial distance Sij to an observed point Pj, find the CT coordinates of Pj. [It should be pointed out that it is assumed that the astronomical quantities (F, L, A) have already been corrected for effect of polar motion so that they refer to the CIO, and that the vertical/zenith angles have been corrected for refraction]
The observed topocentric position vector can be formulated as
The
is transformed into the CT system by the previously derived rotations in
Transformation
of Terrestrial Positions.
is given.
From the previous discussion, the coordinates in the CT system (xj, yj, zj)CT can then be transformed into the geodetic system (xj, yj, zj)G and finally, if desired, into the curvilinear geodetic coordinates (f, l, h).
If xi and hi are known instead of Fi and Li, the solution can again be obtained by a transformation from the LA to the CT system, but this time by means of the LG and G systems. This procedure can only be used if the misalignment of the G and CT systems are known. When the system are parallel the solution becomes:
as given above.
where DA=Aij - aij, and xi and hi are components of the surface deflection of the vertical.
where (f, l)i are from the same system used to define x and h.
There are three kinds of observables used in relative horizontal positioning (1) astronomical azimuth A, horizontal angle w or direction d, and spatial distance Dr. If done correctly the horizontal position computed must match those computed using either three- or two-dimensional systems.
When computed in three-dimensions, the observations are not corrected, other than for instrumental effects and refraction because the computations are carried out in the same space as the measurements. However if the computations are performed in two-dimensional space, the observations must be corrected before the computations on the reference ellipsoid are performed. Conversely, if performing the inverse computations, the corrections must be inversely applied to the computed values to obtain their equivalent values on the surface.
In the reduction of observations on the ellipsoid, there are two groups of effects to be considered: (1) geometrical effects, and (2) the effects of the Earth's gravity field. The geometrical effects occur because of the peculiarities of the geometry of a biaxial ellipsoid. The gravity field must be considered because geodetic instruments used in the measurements are aligned to the gravity field while computations are carried out in the geometrical space of the ellipsoid. A peculiarity of this problem is that the positions of the points are what we are solving for and yet they are used in the corrections. Thus to be technically correct the solution should be iterative. However, since the corrections are generally very small, iteration is seldom, if ever, performed.
Relationship between astronomical azimuth A and geodetic azimuth a where A refers to the LA system and a refers to the LG system. The difference between the two azimuths is given by the Laplace Equation which is called the Laplace correction, Da = a - A is also known as the correction for the deflection of the vertical.
Daij = -hi tan fi - (xi sin aij - hi cos aij)cot Zij = + C1 + C2 (1)
If the deflection is equal to zero, the correction is also equal to zero. Ideally
for horizontal lines where Zij = 90°, the C2 = 0. The geodetic azimuth
obtained using Eq. (1) is often called the Laplace azimuth. What is really
needed for computations on the ellipsoid is an azimuth on the ellipsoid. To
understand this further, consider the let a'ij be the angle between the
geodetic meridian plane and the plane given by the ellipsoidal normal at Pi,
and the projection P'j and Pj onto the ellipsoid. The difference between aij'
and aij is called the skew-normal correction. From the figure it can be seen that this correction arises because the two normals Pi
and Pj are skewed and not coplanar. The correction is computed as
(2a)
or more simply as
(2b)
where fm = ½(fi + fj), Mm = ½(Mi + Mj), and hj is the height of the target above the ellipsoid of point Pj. Since the equation is also dependent on the height of the target at Pj, this correction is sometimes called the target height correction. Substituting in GRS 80 values into Equation (2b) results in the more familiar expression
C"3 = 0.108" cos2fi sin(2aij)hj/1000
This correction is very small, and is usually in a range from 0" to 0.07" for ellipsoid heights between 0 m and 1000 m, but should be considered since it is systematic when performing the most precise geodetic work. By combining Eq. (1) and Eq. (2), we have
aij' = aij + C3 = Aij + C1 + C2 + C3 (3)
In the figure to the right, it can be seen when considering three points that
there are six possible normal sections. Thus, the use of the normal sections
introduces ambiguity in the definition of an ellipsoidal triangle. This
problem is removed through the use of the geodesic curve which is the
curve that is the shortest length of all curves between two points. Letting the
azimuth of the geodesic curve be aE, which is known as the ellipsoidal
azimuth, the normal section to geodesic correction is given as
(4)
where Nm = ½(Ni + Nj). As with the skewed-normal correction, this correction is very small (0" to 0.014") but should be considered for very precise geodetic work since it is systematic. Finally, the complete azimuth correction is
(5)
A horizontal angle w is reduced by applying Eq. (5) to both of its directions. Since C1 is the same for both directions it will obviously disappear from the differences in the corrections, and thus the resultant horizontal angle correction is
wE - w = DC2 + DC3 + DC4 (6)
The spatial distance Dr must also be reduced to its ellipsoid equivalent SE. From simple
geometric relationships shown in the figure to the right, it can be seen that
(7)
where
(8)
Rm = ½[Ri(a) + Rj(a)] (9)
the hi' = hi + hi, hj' = hj + hr, and the radius of the curvature in the azimuth was previously given as
(10)
To compute the chord distance between two points at different elevations which is commonly referred to as the mark-to-mark distance, such as D1 in the figure, the following equation is used.
This distance is equivalent to the baseline vector the is determined by relative positioning using GPS.
In most cases, the magnitude of the corrections are small, and there is a temptation not to apply them. However, it must be realized that since these are systematic errors, the accumulation will become significant and lead to distortions in a network as it increases in size. It is left the geodesist to judge if these corrections are significant for any given project.
The
figure to the right illustrates a slope distance S measured from A
to B. Points A and B represent an EDM instrument and
a reflector, respectively, O is the earth's center, and R
its radius in the direction of the azimuth as defined above. Vertical angles a
and b were measured at A and B,
respectively. Arc AB2, which is closely approximated by its
chord, is the required horizontal distance. A short-line reduction procedure
would result in AB1 being in error by B1B2.
Arc A'B' is the required ellipsoid distance.
From the figure, the following equations can be written to compute required horizontal distance AB2:
AB1 = S cos d
BB1 = S sin d
AB2 = AB1 - B1B2
Finally, ellipsoid length A'B' can be computed from
where hA is the ellipsoid height.
A slope distance of 5000.000 m is measured between two
points A and B whose orthometric heights are 451.200 m and 221.750
m, respectively. The geoidal undulation at point A is -29.7
m, and is -29.5
m at point B. The height of the instrument at the time of the observation
was 1.500 m, and the height of the reflector was 1.250 m. What are the
geodetic and mark-to-mark distances for this observation? (Use a value of
6,386,152.318 m for R"
in the direction AB.)
The geodetic heights at points A
and B are
hA = 451.200 -
29.7 = 421.500 m
hB = 221.750 -
29.5 = 192.250 m
Thus, hA’
= 421.500 + 1.500 = 423.000 m, hB’ = 192.250 +1.250 =
193.500,
Dh’ = 193.500 -
423.000 = -229.500
m, and the ellipsoidal chord distance lij is
The reduced ellipsoidal arc, or geodetic length, for
the line AB is
Finally, the mark-to-mark distance is
Note that the ellipsoid arc and chord lengths are the same to the nearest millimeter. As lines become longer, however, this will not necessarily be the case. Nevertheless for most geodetic measurements, these arc and chord values will generally be nearly the same. Note also that the measured slope distance differs from the mark-to-mark distance by 13 mm.
PROBLEM 2
The slope distance L and vertical angles a
and b
were measured as 14,250.590 m, 4°32'18",
and -4°038'52",
respectively. If the ellipsoid height at A is 438.4 m, what is distance A'B'
reduced to the ellipsoid? (Use the mean radius of 6,371,000 m for R".)
SOLUTION
Solving the equations in
sequence,
AB1
= 14,250.590 cos 4°35'35" = 14,204.826 m
BB1 = 14,250.590 sin 4°35'35"
= 1141.160 m
AB2
= 14,204.826 - 1.272 = 14,203.554 m
Last Updated:October 05, 2006
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