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

Numerous geophysical fields exhibit intermittent features with sharp fluctuations across all scales, skewed probability distribution and long-range
correlations. A common framework to analyse and simulate such fields is multifractals. The underlying idea of this framework is that these fields are
the result of an underlying multiplicative cascade process. It is physically based in the sense that it is assumed the fields inherit the scale-invariant properties of the governing Navier–Stokes equations and hence should exhibit scale invariant features as well. The reader is referred to
the reviews by

Much less work has been devoted to the analysis of the correlations and couplings between two fields exhibiting multifractal properties. A framework was
originally presented by

Actually the previously discussed frameworks have only been implemented for log-normal cascades, for which computations basically boil down to a single parameter and correlation functions are represented by linear ones. Furthermore only two specific cases have been primarily studied, either a proportional or a power law relation between the two studied fields. In this paper, we suggest relying on this theoretical framework and extending its use to UMs and to relations between fields consisting of multiplicative power law combinations.

In Sect. 2, the theoretical framework of UM and joint multifractal analysis is presented. Its theoretical consequences on the analysis of multiplicative power law combinations of UM fields are explored in Sect. 3. Numerical simulations are used to confirm the validity of the suggested analysis techniques. A new indicator of correlation is presented in Sect. 4 and its limitations discussed. Finally the framework is implemented on rainfall data to study the correlation between rain rate, liquid water content and quantities characterizing the drop size distribution.

The goal is to represent the behaviour of a field

Let us consider two fields,

Let us consider two independent UM fields

Let us first discuss intuitively the influence of the parameters

In order to evaluate the expected multifractal behaviour of

Figure

Illustration of the scaling moment functions

In this sub-section an empirical technique to estimate the UM parameters of

The approach presented above is tested on numerical simulations obtained with discrete in-scale cascades. It consists of iteratively repeating
a cascade step with a non-infinitesimal scale ratio in which a “parent” structure is divided into “daughter” structures whose affected value
is the one of the “parent” structure multiplied by a random factor, ensuring that Eqs. (

A set of 10 000 realizations of 512 long 1D discrete cascades is used, and analyses are carried out on ensemble averages.

Before starting, let us clarify the objective of this section.

The parameters used for these simulations are

Results of this analysis are displayed in Fig.

In order to estimate

Results of numerical analysis with

Finally, let us discuss the uncertainties in the estimates of

Estimate of

Let us consider two fields

More precisely, the consequences of describing each field as a multiplicative power law combination of the other and an independent one will be
explored. The notations are

If both lines of Eq. (

Estimates of

Both sides of Eq. (

Figure

Illustration of the relations between the various parameters characterizing the correlation across scales between two UM fields in the case

In Sect. 4.1, limitations of this fully symmetric framework are highlighted. However, it is possible to suggest a rather intuitive indicator enabling
most of the information obtained from the joint multifractal correlation analysis (i.e. the computation of

Plot of

The rainfall data used in this paper were collected by a OTT Parsivel

In this paper four quantities are studied:

LWC, the liquid water content (

Multifractal analyses are carried out on ensemble analyses, i.e. on average over various samples. Once rainfall events (an event is defined as a rainy
period during which more than 1

Dyadic sample sizes are simpler to use for multifractal analysis, which results in some data not being used. With the process described above, 63, 52, 38 and
22 % of the data is actually not used for sample sizes of 32, 64, 128 and 256 respectively. A size of 32 time steps, corresponding to 16

Illustration of the four studied rainfall quantities corresponding to a 32 min sample (i.e. 64 time steps) that occurred on 15 January 2018.

Results of joint multifractal analysis for

Let us first discuss the results of the joint multifractal analysis carried out between

The main curves are shown in Fig.

UM parameters for the studied fields.

The joint multifractal analysis (Eq.

Numerical output of the joint multifractal analysis of the four studied fields. For each box, using the notations of
Eq. (

Similar qualitative results are found for the other combinations, and numerical values are reported in Table

In this paper, we used the framework of joint multifractal analysis to characterize the correlation across scales between two multifractal fields. We
extended the existing framework to universal multifractals and also to analyse the correlations between two fields consisting of renormalized
multiplicative power law combinations of two known UM fields. In general, the resulting fields can be well approximated by UM fields. Estimates of the
corresponding pseudo-UM parameters can be theoretically computed by focusing on the behaviour for moments close to one. These estimates remain valid
for a range of moments between

In a second step, this analysis was used to develop an innovative framework to investigate the correlations between two UM fields. It basically
consists of looking at the best parameters, enabling one field to be written as a power law multiplicative combination of the other field and a random
one. In this context, a good candidate for a simple indicator of the strength of the correlation (called IC) is the proportion of
intermittency of a field explained by the other one. In the general case, this framework is not symmetric, which is a limitation. However when the

Finally this was implemented on rainfall data collected by a disdrometer installed on the roof the Ecole des Ponts ParisTech. More precisely the
correlations between

Further investigations on other fields in various contexts should be carried out to confirm the ability of this framework to both characterize and simulate correlations across scales between two multifractal fields. In future work, this framework should also be extended to more than two fields.

The data and python scripts which have been used for this paper have been made available on a public repository. It can be accessed at

AG designed the study and wrote the paper. The joint analysis by the authors of the obtained results shaped the paper into its actual form.

The authors declare that they have no conflict of interest.

Authors gratefully acknowledge partial financial support from the chair “Hydrology for Resilient Cities” (endowed by Veolia) of Ecole des Ponts ParisTech and the Île-de-France region RadX@IdF Project.

This paper was edited by Stéphane Vannitsem and reviewed by two anonymous referees.