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.

At the beginning of the adjustment, a few parameters must be defined. 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 system.

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

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:

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

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.

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

Linear systems

In any linear system, where the following function defines observation :

the observation equation is defined as:

Nonlinear systems

Adjustments using non-linear equations have differently constructed model matrices and right-hand vectors. To perform an adjustment of a non-linear system of equations, approximate values for the parameters must be derived first. Then, the functions must be linearized. Let be the observation function for observation :

The unknowns which the adjustment results in are the corrections applied to the approximate parameters (variables with an index of 0 denote that they yield approximate values), resulting in obtaining the values of the adjusted parameters . The partial derivative is taken in regards to the approximate value for each argument (parameter) and is used as the coefficient under its respective parameter in the model matrix:

Thus, the final linear equation, called a correction equation, indexed in the system, can be defined as:

where is a correction applied to each equation, is defined as the difference between the observed and calculated values of :

and the other terms are defined previously.