Applications of Parameter Estimation
and Hypothesis Testing
to GPS Network Adjustments
by
Kyle Brian Snow
Report No. 465
Geodetic and GeoInformation Science
Department of Civil and Environmental
Engineering and Geodetic Science
The Ohio State University
Columbus, Ohio 43210-1275
December 2002
It is common in geodetic and surveying network adjustments to treat the rank deficient normal equations in a way that produces zero variances for the so–called “control” points. This is often done by placing constraints on a minimum number of the unknown parameters, typically by assigning a zero variance to the a priori values of these parameters (coordinates). This approach may require the geodetic engineer or analyst to make an arbitrary decision about which parameters to constrain, which may have undesirable effects, such as parameter error ellipses that grow with distance from theconstrained point.
Constraining parameters to a priori values is only one way of overcoming the rank deficiency inherent in geodetic and surveying networks. There are more preferable ways, which this thesis presents, namely Minimum Norm Least–Squares Solution(MINOLESS) and Best Linear MinimumPartial Bias Estimation (BLIMPBE). MINOLESS not only minimizes the weighted norm of the observation error vector but also minimizes the norm of the parameter vector, while BLIMPBE minimizes the bias for a subset of the parameters. In this thesis, these techniques are applied to a geodetic network that serves as a datum access for GPS–buoy work in Lake Michigan. The GPS–buoy has been used extensively in recent years by NOAA, The Ohio State University(OSU), and other organizations to determine lake and ocean surface heights for marine navigation and scientific studies. The work presented in this paper includes 1) parameter estimation using (Weighted) MINOLESS and hypothesis testing for the purpose of determining if recent observations are consistent with published coordinates at an earlierepoch; 2) a discussion of the BLIMPBE estimation technique for three new points to be used as GPS–buoy fiducial stations and a comparison of this technique to the “Adjustment with Stochastic Constraints” method; 3) usage of standardized reliabilitynumbers for correlated observations; 4) a proposal for outlier detection and minimum outlier computation at the GPS–baseline level. The work may also be used as an example to follow for establishing new fiducial points with respect to a geodetic reference frameusing observed GPS baseline vectors.
The results of this work lead to the following conclusions: 1) MINOLESS is the parameter estimation techniques of choice when it is required that changes to all a priori coordinates be minimized while performing a minimally constrained adjustment; 2) BLIMPBE appears to be an attractive alternative for selecting subsets of the parameter vector to adjust. BLIMPBE solutions using various selection–matrix types are worthy of further investigation; 3) outlier detection at the GPS–baseline level permits the entire observed baseline to be evaluated at once, rather than making decisions regarding the hypothesis at the baseline–component level. It is shown that the two approaches can yield different results.
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