## 1 Introduction  
Worldwide, reference frames with long-term stability and homogeneous consistency play crucial roles in accurate modeling or measuring, e.g., gravity fields, the Earth's rotation, geodynamics, and extensive applications, such as global navigation satellite systems, precise positioning, precision orbit determination for satellites or objects in space, time transfer, etc. The International Terrestrial Reference System (ITRS) and its realization, the International Terrestrial Reference Frame (ITRF) [1], provide a globally unified geometric reference frame with accuracy at the millimeter level. However, a commensurate, highly accurate global physical reference frame that uses the Earth's gravity field currently remains elusive [2]. To establish a consistent and accurate physical reference frame, the International Association of Geodesy (IAG) released IAG Resolution No.1. for the definition and realization of an International Height Reference System (IHRS) in 2015 [3], which is a physical world height system for monitoring gravity field variation. Similar to the geometric reference system and frame, the realization of IHRS is the establishment of the International Height Reference Frame (IHRF). The IHRS is defined by the equipotential surface of the Earth's gravity field, where the geopotential at the surface is the conventional value \(W_{0}=62,636,853.4\,\mathrm{m}^{2}/\mathrm{s}^{2}\) (zero-height level), and the vertical coordinates are geopotential numbers \(C_{p}=W_{0}-W_{P}\)[2,3]. A key concept of realizing the IHRS is unifyinglocal vertical height systems (VHSs, which are at the regional scale) worldwide that are interconnected globally. Since local VHSs are usually based on the mean sea level (MSL) determined by tide gauges, and the MSL is not an equipotential surface, different local VHSs exhibit inconsistencies with each other of up to 1\(\sim\)2 m [4]. Identifying the offsets between the origins (datums) of two arbitrary height systems is the main challenge for realizing the IHRS, and various approaches have been tested and discussed. Currently, four approaches have been discussed extensively and practically applied for height system unification. They are explained briefly as follows with the advantages and drawbacks presented for each. (1) The conventional approach is leveling with gravity reductions. It is mainly used for the realization of local VHS, and the accuracy can reach a submillimeter level between neighboring leveling points [5]. However, leveling is laborious and time-consuming, while errors accumulate over long distances. In addition, the main drawback of leveling is that it cannot connect two continents separated by the ocean, which makes it impractical for the realization of a global VHS [2]. (2) The second method is oceanic leveling [6]. It is suitable for connecting different local height systems separated by oceans. For example, ocean models can provide a mean dynamic topography correction to elevation datums of countries with coastlines, thus realizing unification. However, oceanic leveling is limited to elevation datums near coastlines. The uncertainty of the oceanographic modeling method could be better than one decimeter [7]. Still, it requires years of continuous observation data from tide gauges with an adequate density distribution to reach this level of accuracy [7], and these are unavailable in many places, such as African areas. (3) The third method is estimating the anomalous potential by solving the geodetic boundary value problem (GBVP) [8]. It can provide a global solution for height unification, and the precision in well-surveyed regions reaches several centimeters [9; 10; 11]. However, in sparsely surveyed regions, the precision drops to the decimeter level [5]. Another drawback of the GBVP method is that it requires prior information of potential or height values from various sources (global geopotential model, tide gauge data, gravity observation data, et al.). The errors in this prior information will influence the precision of the GBVP method.