After Transformation¶

The list of observations after transformation contains all transformed satellite vectors, i.e., distance, direction, and zenith distance.

What happens during a transformation?¶

In Gemini Terrain, a method has been chosen that first transforms the vector in longitude (l) and latitude (j) to a local left-handed coordinate system where the X'-axis is tangent to the local meridian, as shown by the formula below.

\[ \begin{array}{c} \text{X'} \\ \begin{bmatrix} dX' \\ dY' \\ dZ' \end{bmatrix} \end{array} = \begin{array}{c} \text{M} \\ \begin{bmatrix} -\sin \phi \cos \lambda & -\sin\phi\cos \lambda & \cos \phi \\ -\sin \lambda & \cos \lambda & 0\\ \cos \phi \cos \lambda & \cos \phi \sin \lambda & \sin \phi \end{bmatrix} \end{array} \cdot \begin{array}{c} \text{X} \\ \begin{bmatrix} dX \\ dY \\ dZ \end{bmatrix} \end{array} \]

Or abbreviated: \(X' = M \cdot X\)

From this local coordinate system, we can calculate distance, direction angle, and zenith distance.

Distance¶

From the figure, we can see that ellipsoidal distance can be expressed as:

\[ D = R_m\arctan \frac {\sqrt{(dX')^2 + (dY')^2}} {(R_m+h+dZ')} \]

In addition, the distance must be subjected to the usual map projection correction.

Note

The map projection correction is applied first during the adjustment calculation.

Direction Angle¶

From the figure, we can see that azimuth a can be expressed as:

\[ \alpha = \arctan\left(\frac{ dY'} {dX'}\right) \]

We are interested in the direction angle and must therefore subtract the meridian convergence. In addition, we must apply the usual map projection correction.

Note

The map projection correction is applied first during the adjustment calculation.

The meridian convergence γ is according to Jordan/Eggert/Kneissl (1959) for a Transverse Mercator projection:

\[ \gamma = d\lambda \sin \phi + \frac{d\lambda^3}{3} \sin \phi \cos^2 \phi (1 + 3\epsilon^2 + 2\epsilon^4)+\frac{d\lambda^5}{15} \sin\phi \cos^4 \phi(2- \tan^2\phi) \]

Where \(d\lambda\) is the longitude difference from the map projection's contact meridian to the vector's starting point. For a conformal conic projection and for the stereographic projection, there are separate formulas.

Zenith Distances¶

We can see from the figure that the zenith distance z can be expressed as follows:

\[ Z = \arctan\frac{\sqrt{(dX')^2 + (dY')^2}}{dZ'} \]

It differs from a conventional zenith distance in two important aspects:

Refraction-free
Unaffected by deflection of the vertical. In other words, it refers to the ellipsoid normal, and not to the geoid normal

We must therefore apply two corrections: one for earth curvature and one for deflection of the vertical that may be known.

Note

Corrections for earth curvature and deflection of the vertical are applied first during the adjustment calculation.

Earth Curvature¶

\[ Z_{corr} = Z_{obs} - \frac{S_{ell}^2}{2R_m} \]

Where:

\(S_{ell}\): Ellipsoidal distance
\(R_m\): Radius of curvature at the line's midpoint, calculated according to Euler's formula

Deflection of the Vertical¶

\[ Z_{orth} = Z_{corr} + \xi \cos \alpha + \eta \sin \alpha \]

Where:

\(Z_{orth}\): The zenith distance that refers to the geoid normal
\(Z_{corr}\): The zenith distance that refers to the ellipsoid normal after correction for earth curvature
\(\alpha\): The vector azimuth
\(\xi\): Easterly deflection of the vertical
\(\eta\): Northerly deflection of the vertical

Standard Deviations and Correlations¶

In differential GPS, the vector consists of a dX-, dY-, and dZ-component, each with a calculated standard deviation. The components are calculated together, so they become correlated. We must take this correlation into account when the vector is to be used in the adjustment. If we multiply the standard deviations into the right place in the correlation matrix, we get a covariance matrix that we call C_SAT.

\[ C_{SAT} = \begin{bmatrix} \sigma_X & & \\ & \sigma_Y & \\ & & \sigma_Z \end{bmatrix} \begin{bmatrix} 1 & Q_{xy} & Q_{xz} \\ Q_{xy} & 1 & Q_{yz} \\ Q_{xz} & Q_{yz} & 1 \end{bmatrix} \begin{bmatrix} \sigma_X & & \\ & \sigma_Y & \\ & & \sigma_Z \end{bmatrix} \]

But we are interested in the covariance of the derived directions, distances, and zenith distances. With support from the variance propagation law, we get:

\[C_{TERR} = BM \cdot C_{SAT} \cdot M^T \cdot B^T\]

Where:

\(C_{TERR}\): The covariance matrix for distances, directions, and zenith distance of the vector
\(B\): Matrix with differential coefficients for azimuth, distance, and zenith distance with respect to local vector components
\(M\): Matrix for transformation in longitude and latitude
\(C_{SAT}\): Covariance matrix for the vector

If data from more than two receivers are logged simultaneously, and the vectors are calculated using so-called "multibase solution", the vectors will also be mutually correlated.

In the example below, three receivers are used. The vectors are calculated using "multibase solution". Often, three vectors can then be calculated without contradictions, in the sense that the triangle gives complete closure. This provides a third vector that is a linear combination of the other two. It therefore contributes no information and is omitted.

If the "multibase solution" gives no complete closure of the triangle, we must include all three vectors, but the third vector will then be strongly correlated with the other two.

Similarly, it can be shown that four receivers give three correlated vectors.

When vectors are mutually uncorrelated, the matrices will contain many 0-elements outside the main diagonals, and elements different from 0 will typically be placed as the figure below attempts to illustrate.

In the figure, we have two groups of correlated vectors, each consisting of three vectors. Each small square illustrates a (3 × 3) matrix.

In classical surveying, each measurement is considered independent, and the weights are calculated as:

\[\frac{m_0^2}{m_i^2}\]

Where:

\(m_0\): The standard error (standard deviation) of the unit weight
\(m_i\): The standard error (standard deviation) of observation number i

The quantities we derive from the GPS vectors become group-wise correlated, so we can no longer operate with one weight per observation, but with a weight matrix for each "group" of correlated vectors. This weight matrix is calculated by multiplying \(m_0^2\) into the inverted covariance matrix \(C_{TERR}\):

\[P_{GPS} = m_0^2C_{TERR}^{-1}\]

The contribution to the normal equations is calculated by replacing the usual weight matrix \(P\) with \(P_{GPS}\), and we get:

\[N_{GPS} = A^T \cdot P_{GPS} \cdot A\]