Fixed-band criteria for local constant-Q models in heterogeneous fractional seismic attenuation

Ning Hu, Shuqun Li, Chuyang Hu, Jiayan Sun

Local constant-\(Q\) models are widely used to describe seismic attenuation, but their validity is unclear when attenuation and dispersion vary spatially over the wavelengths being modelled. Variable-order fractional wave models can represent such heterogeneity, yet no quantitative criterion relates spatial order variation to the departure from local constant-\(Q\) behaviour. We derive a branch-resolved fixed-band criterion for a space--time fractional attenuation model with spatially varying temporal order \(\alpha(x)\) and spatial order \(\beta(x)\). On the causal-passive branch the frozen medium has an exactly frequency-independent local quality factor \(Q_0(x)\). In one-dimensional graded media, and in a weak-attenuation multidimensional setting built on a caustic-free real geometric-optics phase, the local deviation satisfies
\[
Q(x,\omega)=Q_0(x)+R(x,\omega),\qquad
\|R\|\le C(\|\nabla\alpha\|+\|\nabla\beta\|)
\]over a prescribed seismic band. The wavelength-normalized order gradient \(\mathcal I_U\) thus serves as the controlling parameter for local constant-\(Q\) validity. The same branch relation shows that \(Q\) alone leaves a one-parameter trade-off between \(\alpha\) and \(\beta\), whereas adding phase-velocity dispersion identifies the two-order description. Regional averaging is controlled for small order oscillations but can produce \(O(1)\) local-\(Q\) errors across sharp contrasts. One-dimensional experiments recover the predicted first-order scaling, and nonseparable two-dimensional tests show the same gradient-controlled behaviour in a smooth low-\(Q\) anomaly. The criterion provides a screen for deciding where local constant-\(Q\) modelling is defensible; sharp interfaces and global caustic propagation remain outside the theory.