## Glossary

A Posteriori Variance Factor (variance of unit weight and standard error of unit weight)

When a network is adjusted, a mathematical quantity known as the a posteriori variance factor is computed (variance of unit weight). It is a statistical measure of how well your adjustment results matched your expectations – as described by your a priori standard errors (standard deviations) for the observations. Another way of expressing this number is by taking its square root, which is commonly referred to as the standard error of unit weight.The a posteriori variance factor for the entire network includes the results from all observations. It is a single value to describe the network as a whole. The a posteriori variance factor for individual observation types, is only for those observation types. The sum total of all a posteriori variance factors for all observation types is equal to the a posteriori variance factor for the entire network.If the a posteriori variance factor is significantly different from 1.0, we have statistical evidence for suspecting one of the following:

1. The a priori standard deviations for some or all observations have been incorrectly estimated. They may be too small or too large.
2. 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.

In general:

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.

A Priori Variance Factor

The apriori variance factor is used to scale each observation variance just prior to network adjustment. By default, it is set to 1.0. This number is usually modified when historical adjustment results suggest that each observation variance (or standard deviation) is either too large or too small. If the observation variances are too small, you might set the apriori variance factor to a value greater than 1.0. If the observation variances are too large, you might set the apriori variance factor to a value less than 1.0.

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.

Combined Scale Factor

The combined scale factor (CSF) can be used to scale a grid distance to the ground distance between points (based on their known height). The CSF is simply GSF * HSF.CSFeff = (CSFa + 4 * CSFab + CSFb) / 6.0CSFab – CSF at the midpoint between each station

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

Confidence Level

The Interval within which we have a specified degree of confidence (expressed as a percentage) that an actual value lies. For example, if we know the theoretical mean and standard deviation of our distance measurements (mean and sigma), the 95 percent confidence interval for a random distant measurement is (mean – 2 * sigma, mean + 2 * sigma). We would expect any random distance measurement to fall within this computed interval 95% of the time.

Confidence Region

The Region within which we have a specified degree of confidence (expressed as a percentage) that an actual value lies. For normally distributed random errors in two dimensions, a confidence region is bounded by an ellipse. Therefore, assuming these random errors, a confidence region indicating accuracy of horizontal control survey coordinates is bounded by an ellipse. Confidence regions for surveys with positions reported in three dimensions would be bounded by an ellipsoid.

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.

Convergence Criteria

At the start of a network adjustment (2D and 3D), approximate coordinates are used as an initial solution. On each adjustment iteration, these coordinates are modified to reflect the observations and their weights. As each iteration progresses, changes to the previous iteration coordinates get smaller and smaller. When the coordinates from the previous iteration and the current iteration differ by less than the Convergence Criteria (for example, set to 0.001m), the solution is said to have converged and adjustment stops.

Degrees Of Freedom

The number of measurements above and beyond the minimum required to define the sample. When measuring the slope distance between A and B, only one measurement is required. However, if we measure the distance three times, then the degrees of freedom is:3 measured – 1 required = 2

Distance Error

An estimate of the precision (at the 68.3% confidence level) of the adjusted chord (slope) distance between two stations in a network. A 1D expansion factor is applied to this value to scale it to various confidence levels (e.g., 95%).

Grid Scale Factor

The grid scale factor (GSF) can be used to scale a grid distance to a distance along the ellipsoid surface (roughly mean sea level). When the grid is below the ellipsoid, the GSF will be less than 1.0. When the grid is above the ellipsoid, the GSF will be greater than 1.0. The GSF is exactly 1.0 at all points where the projection surface intersects the ellipsoid.Average GSF between two grid coordinates:GSFeff = (GSFa + GSFb) / 2.0

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

Height Scale Factor

The height scale factor (HSF) can be used to scale an ellipsoidal distance to the ground distance between points (based on their known height). Similar to GSF, the HSF can be used this way:HSFeff = (HSFa + 4 * HSFab + HSFb) / 6.0HSFab – HSF at the midpoint between each station

Ground Distance = Ellipsoid Distance / HSFeff

Iterations

2D and 3D network adjustments are non-linear and, therefore, require one or more iterations to find the optimal solution. Each iteration results in better coordinates than the previous iteration (unless the adjustment diverges due to bad or inconsistent data). Note: 1D networks (leveling) are linear and require only one iteration to find the optimal heights.

Mean

The simple mathematical average of a set of two or more numbers. If we measure the slope distance (meters) from A to B three times (100.0510, 100.0470, 100.0480), the mean measurement is:(100.0510 + 100.0470 + 100.0480) / 3 = 100.0487

Observation – No Check ("Side Shot")

An adjusted observation with a residual value of zero (or very near zero). These are usually created when side shots are made off the main network. If only one horizontal angle, zenith angle and slope distance are measured to a non-fixed point (i.e., 3D network), there will be no redundancy to that point (the redundancy will be zero). The point will be calculated during adjustment, but each observation will have a residual of zero.

Observation – Redundancy

For each adjusted observation in the network, a redundancy number is calculated. The redundancy number ranges from 0 to 1. Observation redundancy numbers are a measure of how close the variance of the residual is to the variance of the observation.

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.

Observation – Standardized Residual

The standardized residual of an observation is a unit-less value that can be used to directly compare different observation types (for example, zenith angles and slope distances). Observations with large standardized residuals (positive or negative) may be flagged as possible outliers.

Formula: Standardized Residual = Residual / Residual Standard Deviation

Residual

The difference between the actual measurement and the adjusted measurement (or mean measurement). See Standard Deviation for sample.

Rotate Bearings

When bearings are used in a network adjustment, it may be necessary to rotate them to true azimuths. If the bearing is really just an azimuth, no rotation is needed. If the bearing is the average bearing between points, it should be rotated to a true azimuth to get the best adjustment results.

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.

Standard Deviation

A measure of the dispersion of a set of data from its mean. The more spread apart the data, the larger the standard deviation.Given the following slope distance measurements in meters (100.0510, 100.0470, 100.0480), the standard deviation of the measurements is calculated this way:

1. Compute the mean: (100.0510 + 100.0470 + 100.0480) / 3 = 100.0487
2. 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
3. Square all residuals and add them:(0.0023 * 0.0023) + (-0.0017 * -0.0017) + ( -0.0007 * -0.0007) = 0.000008670
4. 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.

Variance

The Standard Deviation squared.