## 1 Introduction  
Over the past few decades, the position coordinates obtained from global positioning system (GPS) devices have found various applications in daily life. However, in hard-to-obtain GPS reception situations, such as urban canyons and narrow alleys, it is difficult to determine the user's location because four satellites cannot be tracked or because of harsh multipath effects. As is well known from the principle of GPS positioning, at least four visible satellites are required to locate themselves. However, it is often impossible to acquire a sufficient number of satellites because of signal blockages caused by nearby buildings. Even when more than four satellites are visible in an urban canyon, there is a high possibility that some of the received signals are from satellites to which direct line-of-sight (LOS) visibility cannot be obtained. This phenomenon is called multipath,meaning that the GPS signals transmitted from the satellite arrive at the receiver antenna through multiple paths. These signals may be reflected by the ground surface or structures around the receiver [1]. According to [2], a multipath signal includes all signals affected by reflection, diffraction, and scattering. In general, to obtain reliable positioning results, one should not use those measurements that were taken in low elevation angles or were bounced from nearby reflecting surfaces. These types of low-quality and corrupted signals deteriorate positioning accuracy. However, in a field with limited satellite visibility, low-quality observations are sometimes required to increase the number of visible satellites and improve satellite geometry. Another reason to include them is to reduce the time required to obtain position fixes, such as the time to first fix (TTFF). In urban canyons, where having the visibility of more than four satellites is not an easy task, one may not be able to determine his/her location if multipath-impacted signals are removed during positioning. Thus, reliable methods should be devised to handle multipath signals effectively. The signal-to-noise ratio (SNR) and satellite elevation angle can be used as indices by which to evaluate the deterioration of the GPS signal quality due to the multipath effect. These indices work as criteria by which to decide which measurement should be selected for positioning and to allocate noise covariance or add weights to the observation. Several weight models have been devised and can be classified as follows: (1) elevation models based simply on the elevation angle of the observed satellite; (2) exponential and SIGMA models utilizing measured SNR values; and (3) modified or combined models of (1) and (2). [PERSON] and [PERSON] [3] introduced the SIGMA-\(\epsilon\) model, in which variances are determined by SNR measurements. [PERSON] et al. [4] developed the SIGMA-\(\Delta\) model, which applies template functions of SNR predictions based on the satellite elevation angle. [PERSON] and [PERSON] [5] realized the limitations of the SIGMA model, modified it using the Danish method, and developed an extended SIGMA model. [PERSON] and [PERSON] [6] optimized the parameters of an exponential model where weights were modeled as an exponential function of SNR and evaluated their performances with respect to the original SIGMA model. [PERSON] and [PERSON] [7] considered SNR and elevation angle together and validated the new scheme in an urban canyon environment. Even though there are slight differences in terms of weighting formulas and the way weights are assigned, the previous studies listed in the previous paragraph adopted a common approach of assigning weights according to satellite elevation angles and SNR values. Thus, they work adequately in an open-sky environment where changes in SNR measurements are very smooth, and a higher elevation angle implies a lower multipath. In urban canyons, where the SNR values fluctuate irregularly owing to the multipath effect, these methods are not efficient.