# Least squares adjustments and the parcel fabric

The parcel fabric is a redundant measurement network. Parcel lines connect parcel corner points to form a measurement network. Lines connect at common points and have dimensions, which define geometric distance and angle relationships with other points.

A least squares adjustment can be run on parcels. The adjustment uses dimensions on redundant parcel lines to estimate best-fit coordinates (x,y,z) for parcel fabric points. The adjustment uses network redundancy to identify lines with potential dimension errors and lines with dimensions that do not fit with the rest of the network (outliers).

To summarize, a least-squares adjustment works on a parcel fabric as follows:

• The adjustment uses direction and distance dimensions on both current and historic parcel boundary lines.
• Points connected to boundary or connection lines are also used as measurements in the adjustment.
• Line dimensions and point coordinates can be weighted in the adjustment. Coordinates and dimensions with higher accuracy are given higher weights; that is, they are given less allowance for change. They will therefore have a greater influence on the outcome of the overall adjustment results by holding closer to their original position or dimension.

Different types of adjustments can be run on the parcel fabric depending on whether you are evaluating or improving spatial accuracy.

• Free network adjustment—The measurement network is not constrained by control points, and measurements are checked for mistakes.
• Constrained adjustment—Two or more control points are included in the adjustment to constrain the measurement network and compute updated coordinates of free points.

### Consistency check using a free network adjustment

A consistency check runs a free network adjustment on the input parcels to ensure parcel lines do not contain mistakes in their dimensions. For example, a consistency check can be run after new parcels have been manually entered from a new record.

A consistency check evaluates the dimensions of the input lines, and dimensions that do not fit with the solution are identified as outliers or possible blunders.

A weighted least squares adjustment uses control points and parcel line dimensions to estimate updated, more spatially accurate coordinates for parcel fabric points. A weighted least squares adjustment can be run to evaluate and improve the overall spatial accuracy of the parcel fabric. Control points are points with known x,y,z coordinates. Control points constrain the adjustment and are used to compute updated coordinates for free (non-constrained) points.

In a weighted least squares adjustment, line dimensions and control points can be weighted based on their accuracies. Control point accuracies are known and weights can range from completely constrained (highest accuracy and x,y,z does not change) to lower weights (lower accuracies) that allow more movement. Dimension accuracies are generally based on the legal parcel record. Parcel dimension from more recent parcel records generally have higher accuracies and thus higher weights in the least squares adjustment. Lines and control points with higher weights have a greater influence on the outcome of the least squares adjustment.

A weighted least squares adjustment also can be used to update the coordinates of lower-weighted control points and identify areas in the parcel network where more control is needed.

## When to run a least-squares adjustment on the parcel fabric

A least-squares adjustment can be run on the parcel fabric in the following scenarios:

• When entering data from a new parcel record—Run a consistency check using the Analyze Parcels By Least Squares Adjustment geoprocessing tool on the newly entered data to identify potential mistakes or outlier measurements.
• After new data has been added to the parcel fabric—Run a weighted least squares analysis using the Analyze Parcels By Least Squares Adjustment geoprocessing tool to evaluate how newly added data affects the spatial accuracy of the parcel fabric.
• When there is sufficiently accurate data to improve the spatial accuracy of the parcel fabric—Apply the results of a weighted least squares analysis using the Apply Parcel Least Squares Adjustment geoprocessing tool to update and improve the accuracy of parcel fabric points.

The parcel fabric uses the DynAdjust least squares adjustment engine. DynAdjust is a least squares application that adjusts coordinates of both small and large geodetic networks. DynAdjust uses a phased adjustment approach in which large networks are adjusted in sequential blocks. The DynAdjust engine can scale to adjust small engineering surveys to large, national geodetic networks.

Some of the capabilities of the DynAdjust least squares adjustment engine include the following:

• Adjustment of coordinates in three dimensions (x,y,z)
• Support of multiple measurement types, for example, horizontal angles and geodetic azimuths
• Minimally constrained or free network adjustments
• Estimation of precision of adjusted coordinates
• Statistical analyses of adjustment results

## How parcel fabric dimensions are processed in the DynAdjust engine

COGO dimensions on parcel lines and coordinates of parcel points are input as measurements into the DynAdjust least squares adjustment engine.

### Parcel lines

Parcel line dimensions are input as the following measurement types into the DynAdjust least squares adjustment engine:

• Distances
• Direction sets
• Geodetic longitude
• Geodetic latitude

#### Direction sets and distances

Parcel line dimensions are input as direction sets and distances into the DynAdjust least squares adjustment engine. A direction set is composed of an origin point (the from point), a backsight line (reference line), and a foresight line.

To run a least squares analysis on parcels, use the Analyze Parcels By Least Squares Adjustment geoprocessing tool.

Direction sets and distances are processed in the least squares analysis as follows:

• The angle formed by the direction set is the measurement that is input to the least squares adjustment engine. The angle is derived from the COGO direction values of the backsight and foresight lines.
• In the image above, point 3762 is the origin point of the direction set. The backsight or reference direction is the line from point 3762 to point 3186. The foresight direction is the line from point 3762 to 3763.
• In the least squares adjustment analysis, the angle is adjusted and applied to the foresight direction to obtain an adjusted foresight direction for the line. The least squares adjustment returns an adjusted direction and distance for the foresight line.
• If the directions on the backsight or foresight lines are in the opposite direction, they will be reversed for the direction set.
• There can be multiple direction sets for any point in the parcel fabric that has multiple lines connecting to it. For example, in the image above, there are two direction sets for origin point 3762.
• When there are adjacent records two direction sets will be created for the same origin point. This is done to account for the possibility of different bases of bearings (rotations) being used for different records.
• Least squares adjustment inputs and results are stored in the AdjustmentLines feature class as follows:
• The origin point of a direction set is stored in the Point 1 Name field. The end point of the backsight line is stored in the Point 2 Name field. The end point of the foresight line is stored in the Point 3 Name field.
• The direction set angle or distance of the foresight line is stored in the Measurement field. The Measurement Type field uses a subtype that indicates whether the measurement is an angle or a distance.
• The adjusted COGO direction or adjusted distance of the foresight line is stored in the Adjusted Measurement field.
• The difference between the adjusted foresight dimension and original dimension is stored in the Measurement Correction field.

#### Geodetic latitude and geodetic longitude

In a weighted least squares analysis, the x, y coordinates of control points (weighted and constrained) are converted to and input as a geodetic latitude and geodetic longitude measurements into the DynAdjust least squares adjustment engine. The coordinate values are converted into latitudes and longitudes with associated standard deviations and if there is a shift in the latitidue and longitude values because of the adjustment, they will be output as measurements in the AdjustmentLines feature class.

### Parcel points

Parcel fabric points are input as the following point types into the DynAdjust least squares adjustment engine:

• Free—These are regular parcel points. The point is considered to be floating and the coordinates can be updated in the adjustment.
• Weighted —The point is weighted by its associated accuracy in the XY Accuracy field. The higher the accuracy, the less change is allowed in the coordinates.
• Constrained— The point is fixed and coordinates do not change in the adjustment. The accuracy of a constrained point is 5mm and overrides any accuracy values entered into the XY Accuracy field.

A parcel fabric point is free when its Adjustment Constraint attribute is set to XY free, Z Constrained This is the default.

To set a point as constrained the least squares adjustment, set the Adjustment Constraint attribute to XYZ Constrained.

To set a point as a weighted control point in the least squares adjustment, set the Adjustment Constraint attribute to XY free, Z Constrained and add an a priori accuracy estimate to the XY Accuracy field.

##### Note:

The Fixed Shape field on the parcel fabric points feature class is not used by the DynAdjust least squares adjustment engine. Set the Fixed Shape field to yes when you want to anchor a point in editing processes such as alignment.