# WaveSim: A Wavelet-based Multi-scale Similarity Metric for Weather and Climate Fields

arXiv: https://arxiv.org/abs/2512.14656 · HTML: https://mlsanity.com/paper/2512.14656/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 4 of its 49 references that are in the map.
- First posted on arXiv 16 Dec 2025.

## Builds on (most cited first, 4 of 4)

- [FuXi: A cascade machine learning forecasting system for 15-day global weather forecast](https://mlsanity.com/paper/2306.12873/) (2023)
- [WeatherBench 2: A benchmark for the next generation of data-driven global weather models](https://mlsanity.com/paper/2308.15560/) (2023)
- [Adaptive wavelet distillation from neural networks through interpretations](https://mlsanity.com/paper/2107.09145/) (2021)
- [DSSIM: a structural similarity index for floating-point data](https://mlsanity.com/paper/2202.02616/) (2022)

## Built on by (most cited first, 1 of 1)

- [Wavelet Flow Matching for Multi-Scale Physics Emulation](https://mlsanity.com/paper/2605.16573/) (2026)

## Abstract (arXiv)

We introduce WaveSim, a multi-scale similarity metric for the evaluation of spatial fields in weather and climate applications. WaveSim exploits wavelet transforms to decompose input fields into scale-specific wavelet coefficients. The metric is built by multiplying three orthogonal components derived from these coefficients: Magnitude, which quantifies similarities in the energy distribution of the coefficients, i.e., the intensity of the field; Displacement, which captures spatial shift by comparing the centers of mass of normalized energy distributions; and Structure, which assesses pattern organization independent of location and amplitude. Each component yields a scale-specific similarity score ranging from 0 (no similarity) to 1 (perfect similarity), which are then combined across scales to produce an overall similarity measure. We first evaluate WaveSim using synthetic test cases, applying controlled spatial and temporal perturbations to systematically assess its sensitivity and expected behavior. We then demonstrate its applicability to physically relevant case studies of key modes of climate variability in Earth System Models. Traditional point-wise metrics lack a mechanism for attributing errors to physical scales or modes of dissimilarity. By operating in the wavelet domain and decomposing the signal along independent axes, WaveSim bypasses these limitations and provides an interpretable and diagnostically rich framework for assessing similarity in complex fields. Additionally, the WaveSim framework allows users to place emphasis on a specific scale or component, and lends itself to user-specific model intercomparison, model evaluation, and calibration and training of forecasting systems. We provide a PyTorch-ready implementation of WaveSim, along with all evaluation scripts, at: https://github.com/gabrieleaccarino/wavesim.