Abstract. Universal multifractals (UMs) have been widely used to simulate and characterize, with the help of only two physically meaningful parameters, geophysical fields that are extremely variable across a wide range of scales. Such a framework relies on the assumption that the underlying field is generated through a multiplicative cascade process. Derived analysis techniques have been extended to study correlations between two fields not only at a single scale and for a single statistical moment as with the covariance, but across scales and for all moments. Such a framework of joint multifractal analysis is used here as a starting point to develop and test an approach enabling correlations between UM fields to be analysed and approximately simulated. First, the behaviour of two fields consisting of renormalized multiplicative power law combinations of two UM fields is studied. It appears that in the general case the resulting fields can be well approximated by UM fields with known parameters. Limits of this approximation will be quantified and discussed. Techniques to retrieve the UM parameters of the underlying fields as well as the exponents of the combination have been developed and successfully tested on numerical simulations. In a second step tentative correlation indicators are suggested. Finally the suggested approach is implemented to study correlation across scales of detailed rainfall data collected with the help of disdrometers of the Fresnel platform of Ecole des Ponts ParisTech (see available data at https://hmco.enpc.fr/portfolio-archive/taranis-observatory/ , last access: 12 March 2020). More precisely, four quantities are used: the rain rate ( R ), the liquid water content (LWC) and the total drop concentration ( Nt ) along with the mass weighed diameter ( Dm ), which are commonly used to characterize the drop size distribution. Correlations across scales are quantified. Their relative strength (very strong between R and LWC, strong between DSD features and R or LWC, almost null between Nt and Dm ) is discussed.

中文翻译：

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近似多重分形相关和通用多重分形场的乘积，在降雨数据中的应用

摘要。通用多重分形 (UM) 已被广泛用于模拟和表征，仅借助两个具有物理意义的参数，即在各种尺度上变化极大的地球物理场。这样的框架依赖于以下假设：基础场是通过乘法级联过程生成的。派生分析技术已扩展到不仅在单个尺度和单个统计时刻（如协方差），而且在跨尺度和所有时刻研究两个场之间的相关性。这种联合多重分形分析框架在这里用作开发和测试一种方法的起点，该方法能够分析和近似模拟 UM 字段之间的相关性。第一的，研究了由两个 UM 域的重整化乘法幂律组合组成的两个域的行为。看起来在一般情况下，结果字段可以很好地由具有已知参数的 UM 字段近似。这种近似的限制将被量化和讨论。已经开发并成功地在数值模拟上测试了用于检索基础油田的 UM 参数以及组合指数的技术。第二步，建议试探性相关指标。最后，建议的方法用于研究在巴黎理工学院菲涅耳平台的偏差计的帮助下收集的详细降雨数据尺度之间的相关性（请参阅 https://hmco.enpc.fr/portfolio-archive/taranis 上的可用数据-observatory/ ，最后访问：2020 年 3 月 12 日）。更准确地说，使用了四个量：降雨率 (R)、液态水含量 (LWC) 和总液滴浓度 (Nt) 以及质量称重直径 (Dm)，它们通常用于表征液滴尺寸分布. 跨尺度的相关性被量化。讨论了它们的相对强度（R 和 LWC 之间很强，DSD 特征和 R 或 LWC 之间很强，Nt 和 Dm 之间几乎为零）。