Instrument and Target Centering Errors

C.4 Instrument and Target Centering Errors

Centering Error values entered in the Instrument Options, cause standard errors for turned angles, distances, directions, delta elevations, and zenith angles to be inflated.

In all following formulas:

I = Horizontal Instrument Centering Error T = Horizontal Target Centering Error

V = Vertical Centering Error at Instrument and at Target As described below, the standard error of horizontal turned angles, directions, azimuths

and horizontal distances are affected only by horizontal centering. The standard error of delta elevations are affected only by vertical centering.

Horizontal Turned Angles: The standard error contribution for an angle due to the horizontal centering error of the instrument is equal to:

Inst StdErr = d3/(d1*d2) * I Where:

d1 and d2 are the horizontal distances to the targets d3 is the horizontal distance between the two targets

The standard error contribution for an angle due to the horizontal centering error of both targets is equal to the following:

2 Target StdErr (Both) = Sqrt(d1 2 +d2 ) / (d1*d2) * T Combining the entered standard error for an angle with the standard error contributons

from the instrument and target centering, we get the total standard error for an angle:

2 2 2 Total Angle Std Err = Sqrt( AngStdErr + InstStdErr + TargetStdErr )

Directions: Total standard error, where “d” is the horizontal distance, is equal to:

2 2 2 2 Total Dir StdErr = Sqrt( DirStdErr 2 +I /d +T /d ) Where:

d is the horizontal distance to the target Azimuths: Same as directions, but using azimuth standard error the in the formula. Horizontal Distances: Total standard error is equal to:

2 2 Total Dist StdErr = Sqrt( DistStdErr 2 +I +T ) Delta Elevations: Total standard error is equal to:

2 2 Total DeltaElev StdErr = Sqrt( DeltaElevStdErr 2 +V +V )

Since slope distances and zenith angles are both “sloped” observations, their total standard errors are each inflated by both horizontal and vertical centering errors.

In the following formulas:

d = Horizontal distance from instrument to target s = Slope distance from instrument to target

e = Elevation difference from instrument to target

Slope Distances: The total standard error is equal to:

2 2 2 2 2 Total Dist StdErr = Sqrt( DistStdErr 2 + (d/s) * (I +T ) + 2*(e/s) *V ) Zenith Angles: The total standard error is equal to:

2 2 2 2 2 Total Zenith StdErr = Sqrt( ZenithStdErr 2 + (e/s) * (I +T ) + 2*(d/s) *V )

Notes:

1. The calculations for angle and direction centering error in the formulas shown on this page and the previous page are carried out in radians.

2. In any formula shown above for a linear observation (slope distance, horizontal distance or delta elevation), the resulting “total” standard error is calculated from the entered standard error of the observation plus any given centering errors. It should be noted that the “entered standard error” for any linear observation is made up of a constant part and PPM. See the discussion of the PPM (Parts per Million) settings starting on page 25 of Chapter 4, “Options.”

3. When standard errors are explicitly entered on an observation data line, these values are by default used as the “total” standard error value that will be used during the network adjustment. They will not be inflated by centering error.

If there are situations where you would like explicitly entered standard errors to be inflated by entered centering errors, you can use the “.ADDCENTERING” inline option as described on page 96 in Chapter 5, “Preparing Data.”

4. Centering errors are not applied to differentially leveled elevation differences.