Table Of Contents

• Example
• Other Observation Types - Coordinates
• Other Observation Types - GPS Baselines
  • GPS Baselines
  • Rotation Matrix
  • Least Squares Solution
    

Example

Assume all angles are of equal precision (s_d), and uncorrelated, C_d = s²_d I

• Consider only three angles (similar to last weeks example):

  a1 a2 a3

  =

  -1   1   0   0 0   -1   1   0 0   0   -1   1

  d1 d2 d3 d4

  Ca = s2d

  2   -1   0 -1   2   -1 0   -1   2

• Now assume all 4 angles with d₁ being used to obtain a₁ and a₂:

  a1 a2 a3 a4

  =

  -1   1   0   0 0   -1   1   0 0   0   -1   1 1   0   0   -1

  d1 d2 d3 d4

  Ca = s2d

  2   -1   0   -1 -1   2   -1   0 0   -1   2   -1 -1   0   -1   2

  
  

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Other Observation Types - Coordinates

These can be coordinates from a previous adjustment with a covariance matrix, or may be "dummy" observations with a large observation weight to fix the datum, orientation and scale of the network. The observation equation is simple.

The observation may be for x, y, or z - but will tend to be for a coordinate, or a group of coordinates where correlation is a concern. Considering a point only let the measurement and covariance matrix be:

x_m y_m z_m and C_m (3 by 3 matrix)

Sticking with the notation in the hand-written notes:

x_m = x' or linearised x_m - x' = Dx

y_m = y' or linearised y_m - y' = Dy

z_m = z' or linearised z_m - z' = Dz

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Other Observation Types - GPS Baselines

GPS Baselines

Generally speaking GPS baselines would not be used in an adjustment on a local coordinate system (as we are considering). The baseline components will be known in an ellipsoid centred cartesian coordinate system (as will their covariance matrix) and the relationship between this frame of reference and the local frame of reference is not always known.

With reference to the figure: the GPS baseline (X) is known in the [X, Y, Z] coordinate system. The local system used for our adjustment is [x, y, z]. If the latitude and longitude of the origin of the local system (point P) are known, then a rotational relationship between [X, Y, Z] and [x, y, z] can be established in the form of a rotation matrix:

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Rotation Matrix

R = f (j, l ) (3 by 3 orthogonal matrix)

R =	-sinl	cosl	0
	-sinjcosl	-sinjsinl	cosj
	cosjcosl	cosjsinl	sinj

R is constructed so that a GPS baseline [D_X D_Y D_Z] can be expressed in the local frame of reference as [D_x D_y D_z]:

Dx	=	-sinl	cosl	0	DX	, say
Dy		-sinjcosl	-sinjsinl	cosj	DY
Dz		cosjcosl	cosjsinl	sinj	DZ

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Least Squares Solution

x = RX

Where x is the local frame of reference representation of the ellipsoid centred cartesian X. Note this equation is useful for propagation of variance. The GPS baseline covariance matrix (C_X) is also in the ellipsoid centred cartesian frame of reference and must be transformed (C_x in the local frame):

C_x = RC_x R^t

Now we have x and C_x the observation equations are linear, and similar to the previous point observations:

Dx = x'2 - x'1 which linearises to Dx - ( x'2 - x'1 ) =	f(Dx)	Dx1 +	f(Dx)	Dx2
	fx'1		fx'2
Dy = y'2 - y'1 which linearises to Dy - ( y'2 - y'1 ) =	f(Dy)	Dy1 +	f(Dy)	Dy2
	fy'1		fy'2
Dz = z'2 - z'1 which linearises to Dz - ( z'2 - z'1 ) =	f(Dz)	Dz1 +	f(Dz)	Dz2
	fz'1		fz'2

because of the linearity all the derivatives evaluate to either +1 or -1.

Notice how all references to the original baseline X and C_x disappear after the rotation to a local frame of reference. Again note that if GPS baselines are to be used, the adjustment would usually be performed on the ellipsoid or in ellipsoid centred cartesian coordinates.

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