**4. Discussion**

Using pseudorange measurements for positioning in the eLoran system can make full use of the available eLoran stations, thereby expanding the coverage of the eLoran system and improving the positioning accuracy of the system. An important problem with eLoran pseudorange positioning, however, is that the geometric distribution of available eLoran transmitting stations may cause the positioning problem to be non-convex. This makes the existing pseudorange positioning algorithms such as NR algorithms extremely dependent on the selection of initial value. In practical positioning applications, it is difficult for the receiver to obtain reliable initial values in many cases. Therefore, conventional positioning algorithms may converge to wrong solutions due to lack of reliable initial values. At present, there is no literature to study the eLoran pseudorange localization initialization problem.

We transformed the eLoran pseudorange positioning into a nonlinear least squares problem with box constraints and proposed the shrink-branch-bound algorithm (SBB), a global optimization algorithm that can achieve accurate positioning without any initial value. The SBB algorithm first obtains the shrunk region of the estimator through the shrink method. The positioning problem is then solved within this shrunk feasible region using a branch-and-bound algorithm, where a trust region reflective algorithm is used for each bound process. We verified the performance of this method through simulation experiments. The results show that the success rate of the SBB algorithm to solve the position is more than 99.5%, when no initial value is available. However, the success rate of other conventional nonlinear least squares algorithms (such as LM algorithm, Dogleg algorithm) in this case is only around 50%. These results confirm that our proposed SBB algorithm can help the receiver to obtain correct positioning results when no initial value is available.

For the eLoran receiver, both the accuracy of the positioning algorithm and the computational complexity need to be considered. The computational complexity of the SBB algorithm is comparable to traditional Newton-based methods or Cauchy-related methods, which means that it can be implemented in the receiver.

#### **5. Conclusions**

eLoran is the ideal backup and supplement to GNSS systems. The improved accuracy of time synchronization between eLoran stations provides conditions for eLoran pseudorange positioning. We proposed a shrink-branch-bound (SBB) algorithm to solve the eLoran pseudorange positioning problem when the receiver has no initial value available. We verified the performance of the SBB algorithm through simulation experiments. The

results show that the success rate of SBB algorithm in converging to the correct result without initial value is over 99.5%, which is more than 40% higher than that of conventional nonlinear least squares algorithms such as LM algorithm and Dogleg algorithm.

The proposed SBB algorithm is expected to make up for the defect that the existing eLoran pseudorange localization algorithm may converge to wrong results when no initial value is available, so it can be used as a cold-start algorithm for eLoran receivers. Therefore, the focus of follow-up research is to combine the SBB algorithm with the existing highprecision positioning algorithms, which is expected to further improve the positioning accuracy and reliability of the eLoran system under high dynamic conditions

**Author Contributions:** Conceptualization, K.L. and J.Y.; methodology, K.L.; software, K.L. and W.Y.; validation, K.L., J.Y. and W.Y.; formal analysis, S.L. and C.Y.; investigation, K.L.; resources, Y.H.; data curation, W.G.; writing—original draft preparation, K.L.; writing—review and editing, K.L.; visualization, W.Y.; supervision, J.Y.; project administration, S.L.; funding acquisition, C.Y., J.Y. All authors have read and agreed to the published version of the manuscript.

**Funding:** This work was supported by Chinese Academy of Sciences "Light of West China" Program (Grant No. E017YR1R10) and "Youth Innovation Promotion Association CAS" (GrantNo. 1188000YCZ).

**Data Availability Statement:** Restrictions apply to the availability of these data. The ownership of data belongs to the National Time Service Center (NTSC). These data can be available from the corresponding author with the permission of NTSC.

**Acknowledgments:** The authors would like to thank their colleagues for testing of the data provided in this manuscript. We are also very grateful to our reviewers who provided insight and expertise that greatly assisted the research.

**Conflicts of Interest:** The authors declare no conflict of interest.
