The ionosphere is one of the largest error sources in GNSS (Global Navigation Satellite Systems) applications and can cause up to several meters of error in positioning. Especially for single-frequency users, who cannot correct the ionospheric delay, the information about the state of the ionosphere is mandatory. Dual- and multi-frequency GNSS users, on the other hand, can correct the ionospheric effect on their observations by linear combination. However, real-time applications such as autonomous driving or precision farming, require external high accuracy corrections for fast convergence. Mostly, this external information is given in terms of grids or coefficients of the vertical total electron content (VTEC). Globally distributed GNSS stations of different networks, such as the network of the International GNSS Services (IGS), provide a large number of multi-frequency observations which can be used to determine the state of the ionosphere. These data are used to generate Global Ionosphere Maps (GIM). Due to the inhomogeneous global distribution of GNSS real-time stations and especially due to the large data gaps over oceanic areas, the global VTEC models are usually limited in their spatial and spectral resolution. Most of the GIMs are mathematically based on globally defined radial basis functions, i.e., spherical harmonics (SH), with a maximum degree of 15 and provided with a spatial resolution of 2.5°×5° in latitude and longitude, respectively. Regional GNSS networks, however, offer dense clusters of observations, which can be used to generate regional VTEC solutions with a higher spectral resolution. In this study, we introduce a two-step model (TSM) comprising a global model as the first step and a regional model as the second step. We apply polynomial and trigonometric B-spline functions to represent the global VTEC. Polynomial B-splines are used for modelling the finer structures of VTEC within selected regions, i.e., the densification areas. The TSM provides both, a global and a regional VTEC map at the same time. In order to study the performance, we apply the developed approach to hourly data of the global IGS network as well as the EUREF network of the European region for St. Patrick storm in March 2015. For the assessment of the generated maps, we use the dSTEC analysis and compare both maps with different global and regional products from the IGS Ionosphere Associated Analysis Centers, e.g., the global product from CODE (Berne, Switzerland) and from UPC (Barcelona, Spain), as well as the regional maps from ROB (Brussels, Belgium). The assessment shows a significant improvement of the regional VTEC representation in the form of the generated TSM maps. Among all other products used for comparison, the developed regional one is of the highest accuracy within the selected time span. Since the numerical tests are performed using hourly data with a latency of one to two hours, the presented procedure is seen as an intermediate step for the generation of high precision regional real-time corrections for modern applications.

中文翻译：

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使用两步B样条曲线方法的全球和区域高分辨率VTEC建模

电离层是GNSS（全球导航卫星系统）应用程序中最大的误差源之一，并且可能导致多达几米的定位误差。特别是对于无法校正电离层延迟的单频用户，必须提供有关电离层状态的信息。另一方面，双频和多频GNSS用户可以通过线性组合来校正电离层对其观测的影响。但是，诸如自动驾驶或精确农业之类的实时应用需要外部高精度校正才能快速收敛。通常，此外部信息是根据垂直总电子含量（VTEC）的网格或系数给出的。不同网络（例如国际GNSS服务（IGS）的网络）的全球分布的GNSS站，提供了大量的多频观测值，可用于确定电离层的状态。这些数据用于生成全球电离层贴图（GIM）。由于GNSS实时站的全球分布不均匀，尤其是由于海洋区域的巨大数据空白，全球VTEC模型通常在空间和频谱分辨率上受到限制。大多数GIM在数学上基于全局定义的径向基函数，即球形谐波（SH），最大程度为15，并且在纬度和经度上分别具有2.5°×5°的空间分辨率。但是，区域GNSS网络提供了密集的观测值簇，可用于生成具有更高光谱分辨率的区域VTEC解决方案。在这个研究中，我们引入了两步模型（TSM），其中包括全局模型（第一步）和区域模型（第二步）。我们应用多项式和三角B样条函数来表示全局VTEC。多项式B样条用于在选定区域（即致密化区域）内对VTEC的精细结构进行建模。TSM同时提供全球和区域VTEC地图。为了研究性能，我们将开发的方法应用于2015年3月圣帕特里克风暴的全球IGS网络以及欧洲地区的EUREF网络的小时数据。对于生成的地图，我们使用dSTEC进行分析，并将这两个地图与IGS电离层关联分析中心的不同全球和区域产品进行比较，例如，CODE（Berne，瑞士），UPC（西班牙巴塞罗那）以及ROB的区域地图（比利时布鲁塞尔）。该评估以生成的TSM图的形式显示了区域VTEC表示形式的显着改善。在所有其他用于比较的产品中，开发的区域产品在选定的时间跨度内具有最高的精度。由于数值测试是使用每小时数据进行的，延迟时间为一到两个小时，因此所提出的过程被视为生成用于现代应用的高精度区域实时校正的中间步骤。在选定的时间范围内，发达的地区之一具有最高的准确性。由于数值测试是使用每小时数据进行的，延迟时间为一到两个小时，因此所提出的过程被视为生成用于现代应用的高精度区域实时校正的中间步骤。在选定的时间范围内，发达的地区之一具有最高的准确性。由于数值测试是使用每小时数据进行的，延迟时间为一到两个小时，因此所提出的过程被视为生成用于现代应用的高精度区域实时校正的中间步骤。