The Rényi function plays an important role in the analysis of multifractal random fields. For random fields on the sphere, there are three models in the literature where the Rényi function is known explicitly. The theoretical part of the article presents multifractal random fields on the sphere and develops specific models where the Rényi function can be computed explicitly. For all considered models explicit expressions of their multifractal spectrum are obtained. Properties of the models and dependencies of their characteristics on parameters are investigated. Then these results are applied to the Cosmic Microwave Background Radiation data collected from the Planck mission. The main statistical model used to describe these data in the literature is isotropic Gaussian fields. We present numerical multifractality studies and methodology based on simulating random fields, computing the Rényi function and the multifractal spectrum for different scenarios and actual CMB data. The obtained results can also find numerous potential applications for other geoscience, environmental and directional data.

Rényi函数在多重分形随机场的分析中起着重要作用。对于球体上的随机场，文献中存在三种模型，其中明确已知了Rényi函数。本文的理论部分介绍了球体上的多重分形随机场，并开发了可以明确计算Rényi函数的特定模型。对于所有考虑的模型，均获得了其多重分形谱的显式表达式。研究了模型的属性及其特性对参数的依赖性。然后将这些结果应用于从普朗克任务中收集的宇宙微波背景辐射数据。用于描述文献中这些数据的主要统计模型是各向同性的高斯场。我们提供基于模拟随机场，计算Rényi函数和针对不同场景和实际CMB数据的多重分形谱的数值多重分形研究和方法。获得的结果还可以发现其他地球科学，环境和方向数据的许多潜在应用。