Least Squares Adjustment and Survey Networks

Least Squares and Survey Networks
Observation Equations - Notation
Observation Equations - Linearisation
Observation Equations - Matrix Form
Least Squares Solution
Covariance Matrices

Least Squares And Survey Networks

Mathematical justification for the use of least squares will be given in the maths course, however least squares:

Easy to apply since normal equations are linear
Gives a unique solution
Provides a covariance matrix of parameters, allowing statistical testing
Can be applied to a wide variety of problems
LS estimate is unbiased (on average equal to the true solution)

Observation Equations - Notation

Bx = m + v

B - design matrix

x - unknown parameters (increments to the assumed coordinates)

m - observations

v - residuals

C_m = P^-1_m- Covariance matrix of the observations

C_x- Covariance matrix of the parameters

^ - least squares estimate of the parameter beneath

Generally P₁ is diagonal except for:

GPS baselines
Previously adjusted coordinates with full covariance matrix
Consideration of correlated angles (previous notes)

which makes computation simpler.

Observation Equations - Linearisation

Because virtually all observation equations - equations that express the observations in terms of the unknown coordinates - are non linear, these equations are linearised with the first using a Taylor expansion ignoring second and higher order terms (see previous notes on linearisation and numerical linearisation).

The least squares solution then involves:

Estimating the coordinates of unknown stations
Solving x - a set of increments to the estimated coordinates
Iteration

Observation Equations - Matrix Form

The linearised observation equations are set up in matrix form as follows:

Dx₁ = +

Dy₁

ff₁ ff₁ ff₁ ff₁ ff₁ ff₁ ff₁ ff₁ ff₁ Dz₁

fx₁ fy₁ fz₁ fx₂ fy₂ fz₂ fx_m fy_m fz_m Dx₂ m₁ - c₁ v₁

ff₂ ff₂ ff₂ ff₂ ff₂ ff₂ ff₂ ff₂ ff₂ Dy₂ m₂ - c₂ v₂

fx₁ fy₁ fz₁ fx₂ fy₂ fz₂ fx_m fy_m fz_m Dz₂

ff_n ff_n ff_n ff_n ff_n ff_n ff_n ff_n ff_n m_n - c_n v_n

fx₁ fy₁ fz₁ fx₂ fy₂ fz₂ fx_p fy_p fz_p Dx_p

Dy_p

Dy_p

p - the number of unknown (x, y, z) points

n - the number of observations

m_i - observation i

c_i - value of observation computed with the assumed coordinates for observation i

v_i - residual for observation i

f - function that expresses the observation as a function of the unknowns (coordinates), a different f is required for each different observation type

D - increments to the assumed coordinate values

Least Squares Solution

Again without mathematical justification the least squares solution is based on minimising the weighted sum squares of the residuals:

v^t P_mv = v^t ( C^-1_m ) v = minimum

This gives a prescription:

B^t P_m v = 0

This leads to normal equations:

B^t P_m Bx = B^t P_m m

Nx = l

With N being the normal matrix. The solution is:

x = N^-1 l = (B^t P_m B)^-1 B^t P_m m

Note the use of the ^ notation to denote a least squares estimate.
P₁ allows for observations to be of differing weights and be correlated.

Covariance Matrices

The covariance of the parameter estimates is given by the inverted normal matrix:

C = N^-1

Covariance matrices for the residuals and adjusted measurements are computed as (these matrices are required for statistical analysis of the adjustment results):

C = BN^-1B^t