# About the fabric least-squares adjustment

This topic applies to ArcEditor and ArcInfo only.

A fabric least-squares adjustment is run on a selection of parcels in automatic mode or on all the parcels in an open fabric job in manual mode. The least-squares adjustment engine in the parcel fabric uses dimensions on parcel lines together with control points to determine the statistically most likely coordinate location for every parcel point in the network. Parcel boundary dimensions accurately define the shape of a parcel, and a least-squares adjustment with control points accurately defines the spatial location of a parcel.

Learn more about the least-squares adjustment

A minimum of two active control points are required to run a fabric least-squares adjustment. Control points can be imported into the parcel fabric or manually inserted in an edit session.

## An overview of the parcel fabric least-squares adjustment

In the first step of the fabric least-squares adjustment, transformation parameters between the original coordinates of the control points and the corresponding coordinates of their underlying parcel points are determined. If the transformation residuals are within acceptable limits (the differences between the two coordinate systems), the transformation parameters are applied to all parcel fabric coordinates to transform them into the coordinates of the control system. The recorded bearing and distance of every parcel line is compared with the same bearing and distance of the line shapes computed in the transformed coordinate system (the coordinate system of the control points). This is done by calculating the difference between the bearing and distance computed from the transformed coordinates and the original bearing and distance. Any parcel line with a bearing and distance difference that exceeds the tolerances you specify in the Adjust Coordinates dialog box is reported in the least-squares adjustment report. After the coordinates of the parcel fabric have been transformed into the coordinates of the control system, the adjustment engine averages (computes a mean) the coordinates and determines the most optimal, best fit solution for all points in the network. The adjustment is a weighted least-squares adjustment, where parcels with a higher accuracy level (higher weight) adjust less than those parcels with lower accuracy level (lower weight).

Learn more about accuracy in the parcel fabric

The least-squares adjustment process determines a more accurate location and a more accurate representation of the line geometry for each parcel line. The original parcel line dimensions (attributes) are not altered. The geometric and spatial representation—the parcel line shape—of the dimensions is updated from the newly adjusted coordinates.

## Redundancy

A least-squares adjustment produces the most reliable results when there are redundant measurements in a network. Redundancy implies that there are repeated observations for a single measurement. Repeated observations validate the measurement network. A parcel fabric is a redundant measurement network.

In the graphic below, a single parcel has four lines and four points. Corner point 2 is defined by two lines (measurements).

In the parcel fabric, corner point 2 from the same parcel is now defined by eight lines (measurements).

With the redundant eight lines defining the same point 2, it's now easier to identify a line that defines a coordinate for point 2 that is significantly different from the coordinates defined by the other lines. Thus, the more lines defining the same point coordinate, the more reliable the detection of outliers or inconsistent lines. The least-squares adjustment uses redundancy to identify those lines that do not fit with the best fit solution. Redundancy in the parcel fabric is created through common points and connectivity.

## Adjustment tolerances

Before running a fabric least-squares adjustment , you need to specify adjustment tolerances, which are necessary for evaluating data in your parcel fabric. Tolerances should be based on how much you expect recomputed parcel line shapes to differ from the original recorded bearings and distances once the parcel network is adjusted to the more accurate control network. Running a check fit on your control network will give you a good idea of how large your adjustment tolerances should be. The check fit residuals will indicate the least amount a parcel line has to adjust to transform to the control network.

The following adjustment tolerances are specified before running a least-squares adjustment:

- Bearings
The difference in bearing in seconds. If the difference between the bearing computed from the adjusted line shape and the original recorded line bearing is greater than the tolerance specified, the line is displayed in the adjustment report. If the difference in bearing is three times greater than the specified tolerance, the adjustment fails.

- Distances
The difference in line length. If the difference between the line length computed from the adjusted line shape and the original recorded line length is greater than the tolerance specified, the line is displayed in the adjustment report. If the difference in length is three times greater than the specified tolerance, the adjustment fails.

- Close Points
During the preprocessing phase of the least-squares adjustment, network connectivity is verified by examining the proximity of points and lines to each other. Any two points that are closer than the tolerance specified for Close Points and do not have a dimension between them are flagged as suspect and displayed in the adjustment report. This is a good way to find common points that were not properly joined to each other during the joining process. Using the close points tolerances is also a good way of detecting overlapping points that should be merged into a single point.

Caution:Duplicate points that should be a single point will compromise the connectivity and thus redundancy of the network. Duplicate points should be merged into a single point.

- Line Points
Any line points that do not lie on an adjacent parcel line are flagged and checked. If the distance between the line point and its adjacent parcel line is greater than the tolerance specified for Line Points, the line point is flagged as suspect and displayed in the adjustment report.

The least-squares adjustment will fail to run if a difference detected in a bearing or distance is greater than three times the tolerances specified for Bearings and Distances on the Adjust Coordinates dialog box. This prevents potential blunders from influencing the outcome of the adjustment.

## Adjustment postprocessing: Plan structure constraints

After the least-squares adjustment is completed, a few postprocessing procedures can be applied to enforce geometric constraints. These include enforcing line points and straight lines.

During adjustment or parcel joining, line points may shift off their adjacent parcel lines. When a line point is within the tolerance specified, it will be shifted back onto its parcel line. If it is outside the tolerance, a warning message is written to the adjustment report. This option will remove any slivers or gaps resulting from line points in the parcel fabric. Data inaccuracies are often the cause of line points shifting off their adjacent parcel lines.

Enforcing straight lines retains the original subdivision structure. Often, a series of adjacent lots in a plan requires that front and/or back lot lines have the same bearing, meaning that the individual lot lines are intended to be collinear. Enforcing straight lines detects these plan structures and, if the boundary points are within the specified tolerance, will make these lines collinear.

- Line points tolerance
Any line points found within the specified tolerance will be forced back onto their adjacent parcel lines.

- Straight lines tolerance
Enforces the original subdivision structure. If the points are within the specified tolerances, collinearity will be enforced.

### Including dependent lines

Dependent lines are often used to represent parcel line types that are dependent on parcel boundary lines such as easement lines. In most cases, dependent lines should follow, and be dependent upon, the boundary lines of a parcel. If the Treat dependent lines as standard boundary lines option on the Adjust Coordinates dialog box is unchecked, dependent line dimensions will not participate in and influence the adjustment process. After the adjustment is completed, any dependent lines will receive the same adjustments that were applied to the parcels. If the option is checked, dependent line dimensions will participate in and influence the outcome of the adjustment.

## How the least-squares adjustment handles basis of bearing

In the parcel fabric, dimensions are stored on parcel lines, and these dimensions are never altered by the fabric least-squares adjustment. Parcel line dimensions can only be manually edited.

In the parcel fabric, bearings for the lines in each parcel are assumed to be on an azimuth for that parcel. Furthermore, each parcel may have to be separately rotated and scaled to fit with the datum and projection used in the parcel fabric. If internal angles are used for the traverse entry of a parcel, the angles are stored and bearings are computed for the lines based on an assumed azimuth. Bearings are required because the adjustment uses bearing equations, not angle equations, for the parcel lines.

When a parcel is joined to the fabric, the original dimensions are used to first calculate coordinates for the parcel corners on a local coordinate system. The first point in the parcel is given coordinates of 0.0 east and 0.0 north and the dimensions are used to compute all subsequent points. A Bowditch adjustment is used to distribute the misclose before computing the local coordinates.

During the joining process, unjoined parcel corner points are matched with their corresponding points in the fabric. Transformation parameters are calculated between the coordinates of the parcel and the coordinate system of the fabric. A Helmert transformation (rotation, scale, shift in x, and shift in y) is used. If more than two points are used in the joining, a least-squares procedure is used to determine the parameters. As points are joined, they are transformed into the network of the fabric and the differences or transformation residuals are displayed as dx (change in x) and dy (change in y) on the join dialog box. These residuals are a good indication of how well the parcels being joined are fitting into the surrounding parcel fabric.

After a parcel is joined, the rotation and scale factor (from the transformation) is stored with the parcel and is used by the least-squares adjustment in setting up the bearing equations. In the least-squares adjustment, parcel bearings are treated like a geodetic "direction set." The assumption is that the angles between each parcel line are correct, but the whole group of lines could be rotated slightly (basis of bearing). So the least-squares adjustment solves for corrections for x,y for every point and for a rotation correction or "orientation element" for each parcel.

The adjustment reports a rotation, scale factor, dx (change in x), and dy (change in y) for each parcel adjusted. Within a subdivision plan, the rotation and scale factor should be very similar for each parcel, and the dx and dy give an indication of change of shape for the parcel. If the adjustment is rerun, the rotation and scale for each parcel will be recalculated.