# Fusion of Land-Based and Satellite-Based Localization Using Constrained Weighted Least Squares
## 1 Introduction  
Source localization has important applications in wireless sensor networks, radar, navigation, and other fields [1; 2; 3; 4]. Typical localization parameters include direction of arrival (DOA), time difference of arrival (TDOA), frequency difference of arrival (FDOA), and so on. Over-the-horizon (OTH) localization is very common for emitters such as incarfts and warships in the military field. For important military targets, it is very necessary to utilize multiple means and multiple platforms to locate and surveillance them, and the positioning accuracy is higher compared to single means. For targets such as ships, the signals they send can usually be received by land-based shortwave stations as well as satellites. Therefore, it is necessary to study accurate and effective multi-platform localization algorithms [5]. For short-wave stations, the frequency range of the received signal is from 3 MHz to 30 MHz, and the signal usually arrives at the observatory station after reflection from the ionosphere. The state of the ionosphere has a great influence on the localization results, so it is necessary to use an appropriate ionosphere model. Recently, TDOA-based localization for short waves as a new method has attracted the interest of many scholars. A mathematical solution for short-wave TDOA-based localization was proposed in the study of [PERSON] [6]. [PERSON] [7] proposed a gradient-type algorithm based on the Quasi-Parabolic (QP) ionosphere model. Although the QP model can describe the ionospheric state more accurately and can give analytic equations for a ray path, it involves a number of ionospheric parameters, which brings more errors. Although the ionospheric virtual height (IVH) reflection model [8; 9; 10] is relatively simple, it contains the ionospheric reflection height as the main parameter, which can accurately reflect the state of the ionosphere and can effectively simplify the localization problem. However, ionospheric reflection height errors can also have a significant impact on the localization results, so the influence on the localization algorithm needs to be considered. The time-varying property of the ionosphere leads to a large error in the estimation of the TDOA and thus low accuracy in TDOA-based localization for short-wave sources. In contrast, DOA-based localization for short-wave emitters has gained maturity, and the localization results are more stable. [PERSON] [9] proposed a new algorithm for short-wave sources using orthogonal triangular decomposition (QR decomposition) based on the IVH model. The positioning accuracy of this algorithm can reach the corresponding Cramer-Rao Lower Bound (CRLB). In addition, satellite-based localization is a high-precision method that typically utilizes the TDOA. A closed-form algorithm for TDOA-AOA hybrid localization was proposed in [11] based on weighted least squares (WLS). [PERSON]. [12] analyzes the effect of earth constraints on multi-satellite localization and obtains the final result by weighting the positioning results under different constraints through weighting coefficients. However, the position of satellites is usually inaccurate due to orbital errors, altitude errors, and other reasons [13]. Therefore, satellite position errors need to be considered in order to improve the localization accuracy [14]. Ref. [15] corrects for satellite errors using the calibration source and proposed a Lagrange algorithm. Ref. [16] proposed a moving horizon estimation (MHE)-based technique to achieve source tracking using TDOA-FDOA measurements from multiple satellites. The surveillance of enemy warships is an important application in the military field, which usually utilizes short waves or satellites for communication. Therefore, high-precision positioning is of great significance. However, short-wave positioning is greatly affected by ionospheric interference, while satellite positioning is affected by orbital errors.