Although a large body of literature exists on methods for landmark-based navigation of mobile robots, little has been said about their robustness in the presence of changing and wrong landmark correspondences as well as erroneous compass measurements. To this end, we investigate the practical implications of different compass-dependent approaches to 2D Bearing-Only Navigation (2D BON) under such conditions, and we suggest measures to quantify their robustness. The analysis is based on the notion of the Homing Vector Field (HVF), and carried out in simulation, since complete HVFs can not be determined otherwise. However, HVFs provide a generic framework that enables the sound comparison of different 2D BON methods. In addition, we propose a method that is based on the approximation of the gradient of an implicit objective function, which constitutes a gradient descent approach. We present practical results that show the applicability of our approach using natural visual landmarks, omni-directional vision and a compass.