Robert J. Kelly
We discuss two topics: the notion of a "mixed" model and the application of the linear model to the LAAS ground station. Engineers are usually not familiar with the mixed model concept. The statistics literature addresses three kinds of linear models: the "fixed model, the "random effects" model and the "mixed" model. The mixed model is a combination of fixed and random effects. In many aircraft navigation systems the linear model is a fixed model. That is the unknown parameters are fixed quantities over the measurement period. In the navigation application the fixed parameter can be the aircraft x,y,z positions. In a random effects model, the unknown parameters are unobservable parameters that can only be defined by variance components. In the LAAS application random effects are selective availability and tropospheric errors. Both models contain an unobservable model error. We emphasize that random effects are not the model errors.
In the second topic we derive the B-values from first principles and compare them to the GPS Local Area Augmentation System (LAAS) B-values. We begin with the linear model for the reference receiver architecture of the ground reference station. This model is a mixed model. Using the mixed model we derive the optimum estimates and optimum tests. The optimum B-values are the residuals of the linear model. Using the minimum mean square error (MSE) as the performance criteria, allows us to calculate the optimum residuals. We call these optimum linear model residuals, the linear model B-values. We also show that the difference in the MSE between the linear model B-values and the LAAS B-values is not significant. Thus the linear model B-values supports the choice of the LAAS B-value algorithm. Most surprisingly, the LAAS B-value algorithm for cov(e) = s2I is identical to the residual test statistics used by the RAIM algorithm in the unaugmented GPS. Thus there is a common integrity-monitoring theme that connects unaugmented GPS, WAAS GOPS and LAAS GPS.
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