Regularization and Adjustment

The linear observation equation is usually expressed aswhere the non-random design matrix \( {\user2{A}} \in R^{m \times n} , \) the vector of unknown parameters \( {\user2{x}} \in R^{n \times 1} , \) the vector of measurements \( {\user2{l}} \in R^{m \times 1} \)and contaminated by random error vector \( {\user2{e}} \) with zero mean and variance–covariance matrix \( \sigma_{0}^{2} {\user2{P}}^{ - 1} , \) where P is the weight matrix and \( \sigma_{0}^{2} \) is the variance of unit weight. If the coefficient matrix A of the observation equation possesses very large condition number, the observation equation is ill-conditioned, which is defined as ill-posed problems by Hadamard (1932). In geodesy ill-posed problems are frequently encountered in satellite gravimetry due to downward continuation, or in geodetic date procession due to the colinearity among parameters that are to be estimated. Most useful and necessary adjustment algorithms for data processing are outlined in the second part of this chapter. The adjustment algorithms discussed here include least squares adjustment, sequential application of least squares adjustment via accumulation, sequential least squares adjustment, conditional least squares adjustment, a sequential application of conditional least squares adjustment, block-wise least squares adjustment, a sequential application of block-wise least squares adjustment, a special application of block-wise least squares adjustment for code-phase combination, an equivalent algorithm to form the eliminated observation equation system and the algorithm to diagonalize the normal equation and equivalent observation equation. A priori constrained adjustment and filtering are discussed for solving the rank deficient problems. After a general discussion on the a priori parameter constraints, a special case of the so-called a priori datum method is given. A quasi-stable datum method is also discussed. A summary is given at the end of this part of the chapter.