Earth Movers Distance (EMD)

Earth Movers Distance answers the question: “How much effort is required to reshape one spectrum into another?”

By treating each spectrum as a distribution of “mass” (reflectance) spread across wavelength “bins,” EMD captures both value and position differences. This makes it well-suited for detecting shifted absorption features—for instance, when a mineral’s diagnostic dip slides a few nanometres due to chemistry, grain size, or moisture.

How it works #

1. Normalise spectra to probability mass $p_i=\frac{R_i}{\sum _{k=1}^n{R}_k}$, $q_i=\frac{S_i}{\sum _{k=1}^n{S}_k}$ Each vector now sums to 1, acting like a discrete probability distribution over n bands.
2. Cumulative mass functions (CMFs) $P_i=\sum _{k=1}^i{p}_k$, $Q_i=\sum _{k=1}^i{q}_k$
3. One-dimensional EMD (for ordered bands) ${\text{EMD}}_{\text{raw}}=\sum _{i=1}^n|P_i-Q_i|$ Intuitively, $|P_i-Q_i|$ is the net “mass” still needing to cross the boundary after band i; summing these across all boundaries gives the minimum total work.
4. Scene-adaptive normalisation (0 – 1) Let ${\text{EMD}}_{99\%}$ be the 99ᵗʰ-percentile raw distance within the analysis area: ${\text{EMD}}_{\text{norm}}=1-\mathrm{clamp}(\frac{{\text{EMD}}_{\text{raw}}}{{\text{EMD}}_{99\%}},0,1)$ Outcome: 1 → best match, 0 → largest practical mismatch.

Key properties #

Property	Benefit
Shift-aware	Detects horizontal displacements of absorption peaks/troughs—not just depth changes.
Global yet sensitive	Considers every band but penalises mis-alignments proportionally to how far mass must move.
Brightness-invariant	Probabilistic scaling removes overall albedo influence, focusing on spectral shape & position.

When to use EMD #

• Hydroxyl group variations: Identify minerals where the OH-feature centre shifts with Al-Fe substitution.
• Moisture content studies: Track wavelength drift of water absorption in soils or vegetation.
• Quality control in calibration: Spot sensors or preprocessing steps that introduce spectral mis-registration.