# 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 that 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 (nonconstrained) 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

## Parcel fabric dimension processing in the DynAdjust engine

Use the Analyze Parcels By Least Squares Adjustment tool to run a least-squares adjustment on parcels. In a least-squares adjustment, parcel data is input to the DynAdjust least-squares engine, adjusted using least-squares adjustment and output to adjustment analysis layers. If the results in the adjustment analysis layers are acceptable, the Apply Parcel Least Squares Adjustment tool can be run to apply the adjustment results to the parcel fabric.

See a workflow on how to run a least-squares adjustment on the parcel fabric

### Parcel lines

Parcel line dimensions are input as distances and direction sets to the DynAdjust least-squares engine.

A direction set is composed of an origin point (the from point), a backsight line (reference line), and a foresight line.

Distances and direction sets are processed in the least-squares adjustment as follows:

• The angle formed by the direction set is the measurement that is input to the least-squares 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, 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 are reversed in the direction set.
• Any point in the parcel fabric that has multiple lines connecting to it can have multiple direction sets.
• When there are adjacent records, two direction sets are 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.
• For distances, the from point is stored in the Point 1 Name field and the to point is stored in the Point 2 Name field. There will be a Null value 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.

### Parcel points

Parcel points are input as the following point types to the DynAdjust least-squares engine:

• Free—These are regular parcel points. The point shape geometry is updated when the results of the least-squares adjustment are applied to the parcel fabric.
• Weighted—The point is weighted by its associated accuracy in the XY Accuracy field. The higher the accuracy, the less movement allowed in the point shape geometry.
• Constrained—The spatial location of the point is fixed and does not move when the results of the least-squares analysis are applied. The accuracy of a constrained point is 5 millimeters and overrides any accuracy values entered in the XY Accuracy field.

##### Note:

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

#### Free points

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

Free points can adjust (move) in a least-squares adjustment. When the results of a least-squares adjustment are applied to the parcel fabric, the shape geometries of free points are updated to reflect adjusted spatial locations. Vectors are created for points that moved and are stored in the AdjustmentVectors feature class.

#### Weighted points

To set a point as a weighted 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.

When applying the results of a least-squares adjustment to the parcel fabric, weighted points move based on their given standard deviations (accuracies) and on the influence of line dimensions connected to the point. Weighted points with higher accuracies are expected to adjust less (move less) than weighted points with lower accuracies.

The coordinate values stored in the X, Y, and Z fields of weighted points are converted to geodetic latitude and geodetic longitude measurements and input to the DynAdjust least-squares engine. The adjusted geodetic latitude and geodetic longitude measurements are stored in the AdjustmentLines feature class. Weighted points can be flagged as outliers if their adjusted coordinates do not fit with the adjusted solution of the selected network.

##### Note:
A higher value in the XY Accuracy field of a weighted point gives it a greater allowable range to move, and its coordinates will therefore have a lower influence on the final adjusted coordinates in the solution. A lower value in the XY Accuracy field will have more influence on the final adjusted coordinates of the solution. This means that a higher value in the XY Accuracy field is correlated with a lower weight in the adjustment network, and conversely, that a lower value in the XY Accuracy field is correlated with a higher weight.

The attributed coordinate values of weighted points are processed in the least-squares adjustment as follows:

• If there are no coordinates (Null) in the X, Y, or Z fields of a weighted point, the least-squares analysis uses the shape geometry of the point.
• When the results of a least-squares adjustment are applied to the parcel fabric, coordinate values stored in the X, Y, and Z fields of the weighted point do not change. The adjustment derives an updated spatial location for the point (based on its weight). The adjusted coordinates are stored in the Adjusted X, Adjusted Y, and Adjusted Z fields in the AdjustmentPoints feature class.
• Vectors are created for weighted points that moved and are stored in the AdjustmentVectors feature class.

#### Constrained points

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

Constrained points are input and processed in the least-squares adjustment as follows:

• If there are no coordinates (Null) in the X, Y, and Z fields of a constrained point, the least-squares adjustment uses the shape geometry of the point.
• Constrained points are fixed and do not move. However, if the shape geometry of a constrained point is different from the coordinate values in the X, Y, and Z fields, they are updated to match the attributed coordinates when the results of a least-squares adjustment are applied to a parcel fabric.