The troposphere is considered as one of the major error sources in space geodetic techniques. Thus, accurate troposphere delay models are essential to provide high-quality products, such as reference frames, satellite orbits, or Earth rotation parameters. In this paper, a new troposphere delay model for satellite laser ranging, the Vienna Mapping Functions 3 for optical frequencies (VMF3o), is introduced. The model parameters are derived from ray-traced delays generated by an in-house ray-tracing software. VMF3o comprises not only zenith delays and mapping functions, but also linear horizontal gradients, which are not part of the standard SLR analysis yet. The model parameters are dedicated to a signal wavelength of 532 nm. Since some SLR stations operate also with other wavelengths, VMF3o provides a correction formula to transform the model parameters to any requested wavelength between 350 and 1064 nm. A test demonstrates that the correction formula approximates slant delays calculated at different wavelengths very accurately. The remaining error for slant delays at a wavelength of 1064 nm adds up to only a few millimetres at 10∘\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$10^{\circ }$$\end{document} elevation angle. A comparison study of the modelled delays that are derived from VMF3o and ray-traced delays was carried out to examine the quality of the model approach. The remaining differences of modelled and ray-traced delays are expressed as mean absolute error. At 5∘\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$5^{\circ }$$\end{document} elevation angle, the mean absolute error is only a few millimetres. At 10∘\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$10^{\circ }$$\end{document} elevation angle, it is at the 1 mm level. The results of the comparison also reveal that introducing linear horizontal gradients reduces the mean absolute error by more than 80% for low elevation angles.

中文翻译：

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VMF3o：光学频率的维也纳映射函数

对流层被认为是空间大地测量技术的主要误差源之一。因此，准确的对流层延迟模型对于提供高质量的产品至关重要，例如参考系、卫星轨道或地球自转参数。在本文中，介绍了一种新的卫星激光测距对流层延迟模型，即用于光学频率的维也纳映射函数 3 (VMF3o)。模型参数源自内部光线追踪软件生成的光线追踪延迟。VMF3o 不仅包括天顶延迟和映射函数，还包括线性水平梯度，这些还不是标准 SLR 分析的一部分。模型参数专用于 532 nm 的信号波长。由于一些 SLR 站也使用其他波长，VMF3o 提供了一个校正公式，可将模型参数转换为 350 到 1064 nm 之间的任何请求波长。测试表明，校正公式非常准确地近似了在不同波长下计算的斜延迟。1064 nm 波长的倾斜延迟的剩余误差在 10∘\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$10^{\circ }$$\end{document} 仰角。对源自 VMF3o 的建模延迟和光线追踪延迟进行了比较研究，以检查模型方法的质量。建模延迟和光线追踪延迟的剩余差异表示为平均绝对误差。在 5∘\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\ oddsidemargin}{-69pt} \begin{document}$$5^{\circ }$$\end{document} 仰角，平均绝对误差只有几毫米。在 10∘\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\ oddsidemargin}{-69pt} \begin{document}$$10^{\circ }$$\end{document} 仰角，在 1 mm 水平。