Revista Brasileira de Cartografia (Jun 2019)
Two aspects on L1-norm adjustment of leveling networks
Abstract
L1-norm adjustment corresponds to the minimization of the sum of weighted absolute residuals. Unlike Least Squares, it is a robust estimator, i.e., insensitive to outliers. In geodetic networks, the main application of L1-norm refers to the identification of outliers. There is no general analytical expression for its solution. Linear programming is the usual strategy, but it demands decorrelated observations. In the context of Least Squares, it is well known that the application of Cholesky factorization decorrelates observations without changing the results of the adjustment. However, there is no mathematical proof that this is valid for L1-norm. Besides that, another aspect on L1-norm is that equal weights may guarantee maximum robustness in practice. Therefore, it is expected to also provide a better effectiveness in the identification of outliers. This work presents contributions on two aspects concerning L1-norm adjustment of leveling networks, being them: the validity of Cholesky factorization for decorrelation of observations and the effectiveness for identification of outliers of a stochastic model with equal weights for observations. Two experiments were conducted in leveling networks simulated by the Monte Carlo method. In the first one, results indicate that the application of the factorization as previously performed in the literature seems inappropriate and needs further investigation. In the second experiment, comparisons were made between L1 with equal weights and L1 with weights proportional to the inverse of the length of the leveling line. Results show that the first approach was more effective for the identification of outliers. Therefore, it is an interesting alternative for the stochastic model in L1-norm adjustment. Besides providing a better performance in the identification of outliers, the need for observation decorrelation becomes irrelevant if equal weights are adopted.
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