Branch-resolved fixed-band near-constant-Q criteria for variable-order fractional seismic attenuation

Ning Hu, Shuqun Li, Chuyang Hu, Jiayan Sun

We establish a closed-corridor near-constant-Q criterion for variable-order fractional wave operators. In one-dimensional graded media, and in a conditional multidimensional weak-attenuation corridor assuming a standard caustic-free real geometric-optics phase, the branch-normalized local quality factor satisfies \(Q(x,\omega)=Q_0(x)+R(x,\omega)\), \(\|R\|_{L^\infty(U\times[\omega_1,\omega_2])}\le C(\|\nabla\alpha\|_\infty+\|\nabla\beta\|_\infty)\) on each fixed frequency band. The operator studied is \(L_{\alpha,\beta}={}^{C}D_t^{\alpha(x)}+c^2(x)(-\Delta)^{\beta(x)/2}\). This is a fixed-band admissibility theorem, not a universal all-frequency or global variable-order wave-propagation theorem. Its frozen dispersion relation has a distinguished causal-passive logarithmic branch. On this branch the quality factor is exactly frequency independent, the orders producing a prescribed \(Q_0\) form an explicit one-parameter manifold, and the pair \((\alpha,\beta)\) is identifiable when attenuation is combined with the velocity-dispersion exponent. The spatial order is interpreted here as an effective phenomenological coordinate for nonlocal dispersion, not as an independent rock-physics state variable; the causal branch is Kramers–Kronig compatible and the high-\(Q\) reference remains close to the classical \(\beta_{\rm eff}\approx2\) regime. The constant in the estimate depends on the band, the passive-cone margin and the scale separation between wavelength and order variation, so the estimate is local in phase space rather than all-frequency. The general multidimensional WKB calculation is used only as a conditional reduction: it identifies the variable-order logarithmic source once a caustic-free complex phase and a controlled lower-order remainder have been supplied. The result is not intended for multipathing, caustics, sharp order jumps, strongly scattering geological structures, or field-scale global propagation without an additional transmission and parametrix theory. The only variable-order source of frequency dependence is the logarithmic derivative of the spatial symbol. Replacing the orders by regional averages incurs a frozen-\(Q\) error of \(O(\operatorname{osc}\alpha+\operatorname{osc}\beta)\) and can therefore produce order-one errors across sharp contrasts; the corresponding field transfer is conditional, and a nonlocal transmission theorem remains open. Controlled spectral, plane-wave, reduced two-dimensional, and nonseparable small-grid diagnostics test the frozen formula, gradient scaling, averaged-order mechanism, and identifiability. They are finite-dimensional diagnostics, not continuum convergence or field validation. The result provides a geophysical criterion for when smooth heterogeneous fractional orders can be interpreted as a local near-constant-\(Q\) model and when regional averaging is not justified. The numerical experiments are reproducible diagnostics for the formulae and assumptions, not validation of a field-scale propagation model.Reproducibility materials are archived on Zenodo at https://doi.org/10.5281/zenodo.21486530.