A Mexican Hat with holes: calculating low resolution power spectra from data with gaps

TL;DR

This paper introduces a modified $ ext{Delta}$-variance method using a Mexican Hat filter to accurately compute low-resolution power spectra from data with complex gaps, applicable across 1D, 2D, and 3D datasets.

Contribution

A novel approach that corrects for data gaps in power spectrum calculations by representing the Mexican Hat filter as a difference of Gaussian filters, improving accuracy in irregularly masked data.

Findings

Effectively compensates for complex data gaps

Applicable to 1D, 2D, and 3D data

Enhances power spectrum estimation accuracy

Abstract

A simple method for calculating a low-resolution power spectrum from data with gaps is described. The method is a modification of the $\Delta$-variance method previously described by Stutzki and Ossenkopf. A Mexican Hat filter is used to single out fluctuations at a given spatial scale and the variance of the convolved image is calculated. The gaps in the image, defined by the mask, are corrected for by representing the Mexican Hat filter as a difference between two Gaussian filters with slightly different widths, convolving the image and mask with these filters and dividing the results before calculating the final filtered image. This method cleanly compensates for data gaps even if these have complicated shapes and cover a significant fraction of the data. The method was developed to deal with problematic 2D images, where irregular detector edges and masking of contaminating sources compromise the power spectrum estimates, but it can also be straightforwardly applied to 1D timing analysis or 3D data cubes from numerical simulations.