- A Posteriori Variance Factor
- A Priori Variance Factor
- Combined Scale Factor
- Confidence Level
- Confidence Region
- Convergence Criteria
- Degrees of Freedom
- Distance Error
- Grid Scale Factor
- Height Scale Factor
- Observation – No Check ("Side Shot")
- Observation – Redundancy
- Observation – Standardized Residual
- Rotate Bearings
- Standard Deviation
- The a priori standard deviations for some or all observations have been incorrectly estimated. They may be too small or too large.
- The model chosen to relate the observations to the unknown parameters (unknown coordinates) was incomplete or not correct, or the observations contain systematic components (errors) which are not modeled properly. For example: distances may need to be scaled slightly if the EDM is out of calibration; or you may have reduced distances to sea level, but you are still defining them as slope distances; or you may have a rod person not holding the prism correctly – thus introducing a systematic error at each target.
In many cases, the second reason for failure of this test can be confirmed through outlier detection and the appropriate action taken. In the event that the failure cannot be confirmed, we must assume the a priori standard deviations were incorrectly estimated and proceed with further statistical procedures using the a posteriori variance factor, since it is now the only information we have about the scaling of the a priori standard deviation for each observation.
If the a posteriori variance factor is statistically larger than 1.0, the predicted errors (a priori standard deviations) are too small (for the network as a whole or by observation type) and the observations were adjusted more than what was expected. For example: you indicated that your horizontal angles were good to +- 5 seconds (as defined in the standard deviation for each horizontal angle), but many of the horizontal angles were actually adjusted by 20 seconds. This could be the result of a blunder in some observation (bad distance, angle, GPS baseline, etc.) that causes warping of the horizontal angles or it could be unrealistic horizontal angle expected errors (+- 5 seconds).
If the a posteriori variance factor is statistically smaller than 1.0, the predicted errors (a priori standard deviations) are too large (for the network as a whole or by observation type) and the observations were adjusted less than what was expected. For example: you indicated that your slope distances were good to +- 0.010 meter (as defined in the standard deviation for each distance), but many of the distances were actually adjusted by only 0.005 meter. You did better in the field (for this observation type) than you expected to do.
How do you know when the observation variances are too large or too small? You inspect the aposteriori variance factor after adjustment. Often, the aposteriori variance factor will be used as the apriori variance factor for subsequent adjustments. However, you must be careful with this approach. The aposteriori variance factor after an adjustment might be 3.5, but that does not indicate all observation variances in the network should be scaled by that amount on subsequent adjustments. It could be that only your horizontal angle variances (or standard deviations) are too small, while all other observation variances are fine.
This is one of the subjective aspects of network adjustment: using a realistic variance for each observation type. If the network consists of observations which are all the same type (for example, height difference observations in level network), it is rather easy to select the proper variance for each measurement. When there are multiple types of observations in the network (horizontal angle variance, zenith angle variance, chord distance variance, etc.), it becomes harder to model each observation type (variance) optimally.
Ground Distance = Grid Distance / CSFeff
For more detailed information on using Grid Coordinates (State Plane), please check out this excellent article, “Working with Grid Coordinates,” by Richard J. Sincovec, LSI: http://www.ejsurveying.com/Articles/Working_with_Grid_Coordinates.pdf
In general, at the 95 percent confidence level, we can expect the true coordinate positions will fall within the region bounding the computed coordinate value and its confidence region (interval, ellipse or ellipsoid) 95 percent of the time. As the confidence level is lowered from 95 percent to 68 percent, the confidence regions (intervals, ellipses, or ellipsoids) will be smaller. At the 68 percent confidence level, we expect the true position will fall within the computed value and this smaller confidence region. For this reason, greater precision will be reported for distance errors, 2D error ellipses and 3D error ellipsoids at the 68 percent confidence level vs. the 95 percent confidence level.
GSFa – GSF at first station
GSFb – GSF at second station
GSFeff – The effective average GSF along the distance between the two points
Ellipsoid Distance = Grid Distance / GSFeff
A more accurate way to compute the Ellipsoid Distance is by using a better average GSF between stations:
GSFeff = (GSFa + 4 * GSFab + GSFb) / 6.0
GSFab – GSF at the midpoint between each station
Ellipsoid Distance = Grid Distance / GSFeff
Ground Distance = Ellipsoid Distance / HSFeff
If the redundancy number is close to 1, the variance of the adjusted observation will be close to zero.
If the redundancy number is close to zero, then the variance of the residual will be close to zero.
Redundancy values of 1.0 will usually occur between two fixed stations. Since the stations are fixed, the observations are 100% redundant (not needed) and actually inflate the degrees of freedom, making the statistics more optimistic.
Redundancy values of 0.0 can occur when side shots are taken. In this situation, there is no redundancy.
Formula: Standardized Residual = Residual / Residual Standard Deviation
Public Land Survey System (PLSS) records are usually recorded with average bearings between corners. To incorporate these PLSS bearings into an adjustment (for example, 2D), they should be rotated back to true bearings during adjustment. There is a preference setting in Columbus 4 to rotate average bearings to a true bearings during adjustment.
- Compute the mean: (100.0510 + 100.0470 + 100.0480) / 3 = 100.0487
- Compute the residuals for each measurement (measured value – mean value):
100.0510 – 100.0487 = 0.0023
100.0470 – 100.0487 = -0.0017
100.0480 – 100.0487 = -0.0007
- Square all residuals and add them:(0.0023 * 0.0023) + (-0.0017 * -0.0017) + ( -0.0007 * -0.0007) = 0.000008670
- Divide the sum of squared residuals by the degrees of freedom to give you the variance: (3 measurements – 1 ) 0.000008670 / 2 = 0.000004335
The Standard Deviation (0.0021 meters) is the square root of the variance (0.000004335).
The Population Standard Deviation is calculated the same way, but the degrees of freedom is 3 (all measurement counts):
- 0.000008670 / 3 = 0.000002890 – this is the variance
- The Standard Deviation (0.0017 meters) is the square root of the variance.