Parametric adjustment using least squares. General mathematic procedure

Least squares adjustments are the most significant capability of Geolyth. In short, a least squares adjustment is a solution to a series of linear equations, such that the mean square error of each unknown is as small as possible.

Theoretical basis

The observation function is a function defining the means of computation for its observation . It returns the computed value of the quantity.

The observation equation is the equation which creates the connection between the unknowns , the computed quantity and the observed quantity.

For linearly homogenous observation functions, the observation equation is

where is the correction applied to the -th equation after the adjustment is complete, to make it fit the unknowns. Due to it being linear, and according to Euler's Theorem for Homogeneous Functions, this can be rewritten to

Non-linear functions, however, first need to be explicitly linearized as

where variables with an index of 0 denote that they yield approximate values. Then, the observation equation for non-linear systems, now called the correction equation, since the adjustment now results in corrections to the initial approximations of the unknowns, becomes

And finally, after rearranging, this yields

Matrix form

In matrix form, the observation equations become

is the vector that will contain the computed values of the unknowns once the adjustment is complete. In linear systems, the unknowns will be direct values, whereas in non-linear systems, they are corrections that must be applied to the initial approximations for the parameters. is the vector that will contain the corrections that need to be applied to the measured quantities for them to fit the new parameter values. The other terms are defined later.

Adjustment construction

Firstly, the matrix, also known as the model matrix, is defined as having columns, where is the number of unknown parameters, and rows, where is the number of linear equations (in geodetic network adjustments, the number of relevant observations). The model matrix can be described as a "table" of coefficients. Each equation can be described as a function of any number of the unknowns.

The right-hand vector has elements, its contents vary depending on the linearity of the function.

Linear
Non-linear

The weight matrix is an diagonal matrix, containing the values of each equation's weight.

Solving the adjustment

The next step in solving a least squares adjustment is solving the system of normal equations. Construct the matrix and the vector :

Invert the matrix to get the covariance matrix . Then compute the vector , which contains the unknowns of the system:

For linear systems, the adjustment at this point is complete. In nonlinear systems, due to inaccurate initial estimations of the unknowns, the adjustment often cannot be resolved in one iteration. It therefore needs to be reiterated until the elements of the vector have become too insignificant to account for.

Then, is used as the initial estimation for the next iteration. This process repeats until

at which point the adjustment is concluded. This last iteration is the one used in further analysis.

A-posteriori analysis

After completing an adjustment, it is imperative to perform an a-posteriori analysis to check for the overall quality of the system and single out any outliers.

Compute the observation corrections, using the coefficients in and the values in :

This is required for performing the test, used for checking the overall quality of the weighing model used in the adjustment.

It is then checked if belongs to the interval , both values derived from the picked confidence level and the degrees of freedom of the network.

	Measurement accuracy was underestimated
	Measurement accuracy OK
	Measurement accuracy was overestimated, or there are blunders present

Now that correction data has been computed, a check can be performed to see if there were any errors in computing the adjustment itself thus far:

At this point, it's time to get into a-posterior adjustment analysis. The per-unit standard deviation is computed as:

From that, the standard deviations of each parameter can be determined using the previously computed covariance matrix :

However, to compute the standard deviations of the observations, a separate covariance matrix must be built specifically for the observations. After that, the formula used is the same as for parameters.

The standard deviation of the un-adjusted observation is

The standard deviation of the adjusted observation is

The standard deviation of the observation's correction is

When it comes to checking for outliers in the observations, a simple confidence interval check (W-test) can be performed between the correction and its standard deviation . Their ratio should not be higher than the confidence interval coefficient defined at the beginning of the adjustment (, , ).