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OPUS-RS Statistics

Effective June 3, 2007, OPUS-RS will show estimated standard deviations of both the ITRF2000 and NAD 83(CORS96) computed coordinates. These are related to, but are not quite the same as, the peak-to-peak errors shown by regular OPUS.

There are two common ways to determine the standard deviations of the coordinates determined by an adjustment such as that performed by OPUS-RS:

• Formal error propagation.

  In a least squares adjustment, the covariance matrix of the adjusted parameters may be computed by multiplying the variance of unit weight by the inverse of the normal equation coefficient matrix. The variances of the coordinates may be picked out from the diagonal elements of this matrix. This procedure is based on the assumption that the mathematical model reflects physical reality and that only random errors are present in the observations.

  In many applications, including both OPUS and OPUS-RS, this produces standard errors which are far too optimistic (often only a few millimeters). Since they are seldom reliable indicators of the errors in the computed coordinates, these formal errors are not shown on the OPUS and OPUS-RS Reports. For users who really want them, they are available on the extended output.

• Repeated Samples.

  If more than one estimate of a quantity is available, the scatter of those estimates gives a measure of the accuracy of any single one. In both OPUS and OPUS-RS, we compute separate estimates of the coordinates of the rover station by single baselines, each involving a known reference station and the rover station. These are not truly independent estimates, since they all use the data from the rover station; however, they do serve the purpose of isolating errors caused by the adopted coordinates or the observation noise from a particular reference station.

Differences between the approaches of OPUS and OPUS-RS.

In regular OPUS, the computed coordinate is the mean of the coordinates computed by three separate single baseline solutions. This solution is not completely rigorous, since it ignores the fact that the three single baselines are not independent. Furthermore, the range (peak-to-peak) of the three estimates is reported. As shown in (Schwarz, 2006), this range is related to the standard deviation of the mean by the factor 2.93. In practice, the peak-to-peak error has been found to be a useful realistic indicator of the accuracy of the computed coordinate.

In OPUS-RS, the final coordinate is computed by using all the data from the reference stations and the rover station in a single simultaneous least squares adjustment. However, single baseline solutions are also computed as a means of estimating the accuracy. For the most part, the single baseline solutions show if estimated coordinates using a particular reference station fail to agree with the others. This often indicates the presence of non-random errors in the data or adopted coordinates from a particular reference station.

The peak-to-peak range of the single baseline solutions does not have the same meaning in OPUS-RS as it does in regular OPUS. This is because the number of reference stations used in OPUS-RS varies (with a minimum of three and a maximum of six). The algorithm used in OPUS-RS is:

• compute the estimated coordinates of the unknown station (the rover) by single baselines, using each of the reference stations
• compute the final coordinate of the unknown station by a simultaneous least squares adjustment, using the data from all the reference stations and the unknown station
• compute the difference between each single baseline estimate and the final coordinate. Do this both for the X,Y,Z and East, North, Up coordinates.
• compute the standard deviation in each coordinate by squaring the differences, adding them together, dividing by the number of reference stations, and taking the square root.
• Report the resulting number next to the coordinate on the OPUS-RS Report

The numbers reported as standard deviations are valuable because they isolate problems with the reference station coordinates or data. However, it is difficult to assign a probability level to these numbers. Were the single baselines independent of each other and of the final coordinates, these numbers would be the standard deviation of a the coordinates determined by a single baseline. However, neither of these conditions are met, so one can use only an empirical measure. Experiments performed by NGS show that the actual errors in a final coordinate is greater than the number given as a standard deviation in fewer than five percent of the cases.

REFERENCES

Schwarz, Charles R., 2006. Statistics of Range of a Set of Normally Distributed Numbers. ASCE Journal of Surveying Engineering, Vol 132, Number 4, pp. 155-159.