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

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.