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Hybrid L1/L2 misfit (Bube–Langan)

Definition

\[ \rho_{\delta}(r) \;=\; \delta^2 \Bigl(\sqrt{1 + (r/\delta)^2}-1\Bigr), \qquad r = d_{\mathrm{syn}} - d_{\mathrm{obs}}. \]

Asymptotics:

  • \(|r|\ll\delta\)    \(\Rightarrow\) \(\rho_\delta(r)\approx \tfrac{1}{2}r^2\) (L2 regime).
  • \(|r|\gg\delta\)    \(\Rightarrow\) \(\rho_\delta(r)\approx \delta|r|-\delta^2\) (L1 regime).

This is numerically identical to the pseudo-Huber smoothing but is named after Bube & Langan (1997) who introduced it for seismic tomography. Ha, Chung & Shin (2009) used it for elastic FWI and showed that it can recover inversions where pure L2 fails on noisy field data.

API

from sweep_loss import HybridL1L2Loss
loss = HybridL1L2Loss(delta=0.5)(syn, obs)

Tests

tests/test_hybrid_l1l2.py checks:

  • numerical agreement with PseudoHuberLoss for several \(\delta\),
  • L2 limit for \(|r|\ll\delta\),
  • L1 limit for \(|r|\gg\delta\),
  • gradient is uniformly bounded by \(\delta\),
  • module == functional alias.

References

  • Bube, K. P. & Langan, R. T. (1997). Hybrid L1/L2 minimisation with applications to tomography. Geophysics 62 (4), 1183-1195. doi:10.1190/1.1444219
  • Ha, T., Chung, W. & Shin, C. (2009). Waveform inversion using a back-propagation algorithm and a Huber function norm. Geophysics 74 (3), R15-R24. doi:10.1190/1.3112572