The hybrid quasi-geoid of North Macedonia: MK_HREF2022

The MK_HREF2022 quasi-geoid model is a height reference surface for North Macedonia, referred to the GRS80 ellipsoid. It was computed by the Agency for Real Estate Cadastre of the Republic of North Macedonia, in cooperation with the Norwegian Mapping Authority (Kartverket), and it is given in the form of a regular grid, with resolution of 0.01° × 0.01°, within the limits of 40.8° < latitude < 42.4° and 20.4° < longitude < 23.1°. The model is based on a set of relatively new gravity data measured in 2014 (along the high accuracy levelling lines and in the frame of the regional gravimetric survey), in total 2470 points in mean-tide system. Stokes/Molodensky equations and the Fast Fourier transform technique were applied in the calculations, with use of the global geopotential model EGM2008. The computed quasi-geoid surface was fitted to 77 high quality GNSS/levelling points, fairly distributed throughout the territory of the country. The resulting height reference surface allows the conversion between ellipsoidal heights (ETRS89, epoch 1989.0) and heights in the official national height system (NVT1, datum Trieste 1875.0, normal orthometric heights). The inner accuracy of the calculated quasi-geoid model is 4.0 cm, while the outer accuracy (based on 1035 control GNSS/levelling measurements along levelling lines) is up to 10 cm. MK_HREF2022 is the official quasi-geoid model for North Macedonia since November 28, 2025.

The geoid model is provided in ISG format 2.0 (ISG Format Specifications), while the file in its original data format is available at the model ISG webpage.

MK_HREF2022 似大地水准面模型(quasi-geoid model)是北马其顿的高程基准面，基准参考为GRS80椭球体(GRS80 ellipsoid)。该模型由北马其顿共和国不动产登记局与挪威测绘局（Kartverket）联合计算完成，以规则格网形式存储，格网分辨率为0.01°×0.01°，覆盖范围为纬度40.8°至42.4°、经度20.4°至23.1°。该模型基于2014年测得的一批较新的重力数据（涵盖高精度水准路线测量及区域重力测量项目），共包含平均潮汐系统下的2470个测点。计算过程中采用了斯托克斯/莫洛坚斯基方程与快速傅里叶变换(Fast Fourier transform)技术，并使用了全球重力位模型EGM2008。计算得到的似大地水准面与均匀分布于全国境内的77个高质量全球导航卫星系统(GNSS)/水准联测点进行了拟合。该高程基准面可实现椭球高（ETRS89，历元1989.0）与官方国家高程系统（NVT1，的里雅斯特1875.0基准面，正常正高）之间的转换。本次计算得到的似大地水准面模型的内符合精度为4.0厘米，外符合精度（基于沿水准路线获取的1035个GNSS/水准联测控制点）可达10厘米。MK_HREF2022自2025年11月28日起成为北马其顿官方的似大地水准面模型。
该似大地水准面模型以ISG格式2.0（ISG Format Specifications）提供，原始数据格式的文件可在该模型的ISG官方网页获取。