# Robust Estimators in Geodetic Networks Based on a New Metaheuristic: Independent Vortices Search
This strategy has been applied in several studies in geodesy, as it may lead to better results than classical methods [15; 16; 17; 18; 19; 20; 21]. In MH research, the particle swarm optimization (PSO) [22] has been widely applied, followed by the artificial bee colony [23; 24; 25] and ant colony optimization methods [26], more recently [27]. Another solution is to apply a data-snooping procedure [28; 29], a statistical test which takes place after the LS computation. However, studying statistical tests is not the purpose of this paper. Researchers have also applied MHs in others areas of geoscience, such as remote sensing [30; 31]. For the adjustment of geodetic networks, MHs compute the unknown parameters (point co-ordinates) within a pre-defined search space and check them by evaluating the objective function for the chosen RE. The estimate is then used for the goodness-of-fitting evaluation. The process is repeated, following the exploration strategy adopted by the MH, until either an acceptable solution or the computational limit is reached [20]. Applying MHs in geodetic networks adjustment has not been widely explored. Most studies have been limited to working with only one or two REs, often in simple cases. For example, they may not have considered the generation of multiple error scenarios [18; 19; 21], or may have omitted the consideration of random errors [20]. It is important to explore adjustment solutions where random errors can be simulated, getting as close to a real situation as possible. Still, several error scenarios need to be tested with a variety of REs to obtain a representative set and measure the reliability of the strategy. Furthermore, an MH needs to present stable results when minimizing the RE functions and present good strategies to avoid local minima. Therefore, this work has two main goals: First, developing better MHs by presenting the Independent Vortices Search (IVS) based on the Vortex Search method [32], which brings advances to mitigate the limitations found in the original algorithm [33]. We compare its performance with the original method, other modifications, and a competing algorithm. The second goal is improving quality control in geodetic networks by combining smart and robust methods. By separating the results into three scenarios (with no outliers, small outliers, and large outliers), we classify the REs according to the demand and situation. Then, the behaviors of several known REs when adjusting a geodetic network are tested: Least Trimmed Squares (LTS) [34], Least Median of Squares (LMS) [34], LTS-RC adding a constraint [20], Sign-Constrained Robust Least Squares (SRLS) [35], Least Trimmed Absolute deviations (LTA) [36], and Iteratively Reweighted Least Squares (IRLS) [37]. We generated observations from the official co-ordinates of a station and standard deviations from GNSS signal processing. Several error scenarios were simulated in the observations, which are tested by all REs, and the results are compared with the conventional LS. In addition, different median and trim positions in residual vectors of some REs are investigated. We analyzed the results by: (1) the identification outliers, analyzing the residuals vector, and (2) by the estimated parameters, comparing them with the official co-ordinates of the network. Additionally, for problems based on the Gauss-Markov model, we present a new proposal for the search space definition when using MHs. Furthermore, a comprehensive analysis of the results achieved with different trim and median position in some REs is given, showing which equation is best in each situation. This paper is organized into six sections: In Section 2, we present a theoretical overview of the LS and RE methods applied in the experiments. Section 3 presents the first contribution of the study, showing the Independent Vortices Search algorithm. In Section 4, we present details of the experimental setup, how we analyzed the quality of the solutions, and the implementation of the REs and IVS.