Multifractal Analysis of Clouds

This repository accompanies the paper Multifractal Analysis for Evaluating the Representation of Clouds in Global Kilometre-Scale Models by Lilli J. Freischem, Philipp Weiss, Hannah M. Christensen, and Philip Stier, submitted to Geophysical Research Letters. Here, we provide Python scripts and Jupyter Notebook examples for conducting multifractal analysis.

Theoretical Overview

For some one-dimensional signal $\theta (x)$, we compute the $q\text{th}$ order structure function as:

$S_q (r)=\langle|\theta(x+r)-\theta(x)|^q \rangle$,

where $r$ is a finite distance in $x$, and $\langle ... \rangle$ indicates a spatial average over all $x$.

Within the scaling range of the signal $\theta(x)$, the magnitude of the structure function is proportional to the distance $r$, raised to some exponent $\zeta_q$:

$S_q (r)\propto r^{\zeta_q}$,

such that $\log S_q (r)$ is approximately linear in $\log r$.

We parameterise the calculated $\zeta_q$ exponents as

$\zeta_q=\frac{aq}{1+aq / \zeta_\infty}$,

using the two-parameter fit introduced by Pierrehumbert, 1996.

Key Functions

multifractalanalysis.py contains the multifractals function which computes structure functions and multifractal scaling parameters from a 2-dimensional cloudfield.

def multifractals(cloudfield, R, Q = np.arange(1, 11)):
    """
    Calculate multifractal scaling exponents of a given cloudfield.

    Parameters:
    - cloudfield: numpy.ndarray
        The cloudfield array to calculate moments for.
    - R: numpy.ndarray
        An array of separation distances to use for calculating moments.
    - Q: numpy.ndarray
        An array of exponents q to use for calculating moments.
    
    Returns:
    - moments: numpy.ndarray
        A 2D array containing the calculated moments for all separation distances R and exponents Q.
    - zetas: numpy.ndarray
        A 1D array of scaling exponents (zeta_q) of the cloudfield for orders Q.
    """

Examples

compute_multifractals.ipynb contains a notebook demonstrating how to use multifractalanalysis.py for computing structure functions, scaling exponents, and multifractal parameters.

Citing

@software{freischem2024,
  author       = {Freischem, Lilli Johanna},
  title        = {Multifractal Analysis of Clouds},
  year         = 2024,
  publisher    = {Zenodo},
  doi          = {10.5281/zenodo.13832882},
  url          = {https://doi.org/10.5281/zenodo.13832882}
}