# A fast tunable blurring algorithm for scattered data

## TL;DR

This paper introduces a linear-time, tunable blurring algorithm for scattered data that enhances data analysis and assimilation by effectively separating spatial scales in geophysical observations.

## Contribution

The paper presents a novel, efficient blurring method based on Gaussian interpolation and multiresolution convolution, generalizing traditional algorithms to scattered data and enabling new applications.

## Key findings

- Efficient linear-time algorithm for scattered data blurring.
- Effective separation of large-scale and small-scale spatial components.
- Potential to improve data assimilation and analysis in geophysics.

## Abstract

A blurring algorithm with linear time complexity can reduce the small-scale content of data observed at scattered locations in a spatially extended domain of arbitrary dimension. The method works by forming a Gaussian interpolant of the input data, and then convolving the interpolant with a multiresolution Gaussian approximation of the Green's function to a differential operator whose spectrum can be tuned for problem-specific considerations. Like conventional blurring algorithms, which the new algorithm generalizes to data measured at locations other than a uniform grid, applications include deblurring and separation of spatial scales. An example illustrates a possible application toward enabling importance sampling approaches to data assimilation of geophysical observations, which are often scattered over a spatial domain, since blurring observations can make particle filters more effective at state estimation of large scales. Another example, motivated by data analysis of dynamics like ocean eddies that have strong separation of spatial scales, uses the algorithm to decompose scattered oceanographic float measurements into large-scale and small-scale components.