5. Dimensions and Units

a. Angles

When horizontal angles or directions are included in an adjustment, units must be taken into consideration. Distances may be in feet, chains, or meters. Degreee-minute-second (DMS) units are commonly used for angles and directions.

[K] matrix angle and direction elements are computed from measured angles and point coordinates. They are usually small and expressed in seconds. [C] matrix angle elements are in radians because they come from partial derivatives of the tangent trigonometric function.

The [C] and [K] matrices are used to determine the [U] and [V] matrices. To avoid mixed angular units, either the [C] matrix angle elements must be converted to seconds or the [K] matrix's to radians. For easier residual terms and standard error interpretation, [C] elements are converted to seconds multiplying each angular term by 206,265 sec/radians. 

b. Mixed Measurements

Horizontal network adjustment generally includes a mix of distances, angles, and directions. Dimensions of [C] and [K] matrix elements depend on the observation equations used. If [C] and [K] dimensions are mixed, some of the elements in the [U] matrix won't be correct: some may be feet, others could be seconds-feet, which doesn't make sense. To obtain correct dimensions and units, weights must be applied to the adjustment.

As discussed in Chapter E a measurement's weight is the reciprocal of its standard error squared, 1/(SE_i²). This places the measurement's squared dimension in the denominator so units cancel correctly in the [C^TWC] and [C^TWK] matrix products. This makes all [U] matrix elements linear.

If measurement standard errors are unknown, reasonable estimates should be used to ensure dimensions cancel correctly. Otherwise results may not be correct or the adjustment may not converge to a solution.