## 1 Introduction  
Passive radar technology, which allows operators to detect and localize potential targets using already existing transmitters such as commercial frequency modulation (FM) broadcast/digital audio broadcast (DAB)/terrestrial digital video broadcast (DVB-T) [1; 2] and non-cooperative radar transmission [3; 4], has been interesting to both civilian and military fields in the last few decades [5]. This sort of radar, compared to active radar technology, offers numerous advantages including lower cost, lower power usage and more covert surveillance capability, which suggests the possibility of employing passive radars on a wide range of concerned applications such as homeland security, costal surveillance and early warning system for vehicles detection, etc. One of the remarkable characters of passive radar is the deployment of two receiving channels, with one for capturing the direct path signal from the transmitter and the other for collecting the potential target echoes [6]. By performing a cross-correlation operation between the direct path signal and the target echoes, the time delay (TD) could be measured, which holds information about the unknown target position. By multiplying with the signal propagation speed, the TD can be directly converted into the bistatic range (BR) [7]. Each BR measurement traces out an ellipsoid equation, with its foci located at the transmitter and the receiver positions. Theoretically, for multi-static passive radar (MPR), if enough BR measurements with respect to multiple transmitter-receiver pairs are available, the target position can be determined by solving the set of nonlinear ellipsoid equations. However, due to the high nonlinearity implied in the BR measurement equations, determining the target position from the BR measurements obtained at a single time instant is not a trivial topic. In recent years, numerous algorithms have been developed to address this challenging topic, mainly including iterative methods [8; 9] and closed-form solution methods [10; 11; 12; 13; 14]. Iterative methods [8; 9] rely on an initial position guess close to the true solution but such a good guess may not always be available in reality. By contrast, closed-form solution methods [10; 11; 12; 13; 14] have always been more attractive to researchers due to their computational efficiency, independence on initial guess and absence of divergence problem. Illuminated by [PERSON] and [PERSON]'s two-step weighted least squares (2 WLS) idea [15], these closed-form solution methods [10; 11; 12; 13; 14] generally follow the basic two-step framework below: in the first step, the non-linear BR equations are linearized into pseudo-linear ones by introducing some proper nuisance parameters and a coarse estimate of target position can be obtained from the pseudo-linear equation set via weighted least squares (WLS) minimization; then in the second step, the function relation between the target position and the introduced nuisance parameters is explored to refine the initial estimate. Nevertheless, all the aforementioned methods are designed based on the assumption that the positions of the transmitters and receivers are exactly known, but such exact priori knowledge may not be available in reality. In fact, the positions of the transmitters and receivers are inevitably perturbed by errors to some extent, and these errors (also referred to as position uncertainties) are often non-negligible, especially when the antennas are mounted on moving platforms [16; 17] or the exploited transmitters are highly non-cooperative (such as the hostile radar radiation whose position could usually only roughly determined by electronic reconnaissance technique [18; 19]).