Direct requested-time modelling of space--time fractional seismic attenuation on fixed operators

Space--time fractional attenuation models represent dissipative memory and effective nonlocal dispersion in seismic wave propagation, but Caputo history marching is expensive when only selected receiver--time responses are required. Matrix Mittag--Leffler representations are classical; their use as a costed fixed-model response-query workflow for seismic attenuation is less developed. We formulate the space--time Mittag--Leffler eigen-evaluator (ST--MLEE) for a fixed positive self-adjoint semidiscrete seismic operator \(A_\beta\), evaluating selected data as \(d_r(t)=r^T E_\alpha(-t^\alpha A_\beta)b\) instead of advancing through all preceding time levels. Modal implementations reuse the eigensystem across times, receivers and dense source weights. Standard sparse Lanczos projection is cheaper for large operators, but its basis is source-dependent. The comparison is reported through \(q_*\), \(\rho_t\) and \(\rho_r\), which measure setup amortisation, requested-time sparsity and receiver sparsity. Same-equation L1/Grunwald--Letnikov history solvers validate the receiver traces within the 1.5% waveform tolerance on matched two-dimensional and controlled three-dimensional tests. On the largest irregular operator, a source-perturbation scan places the first single-vector Lanczos reuse failure at \(\eta_b=0.015\). Smooth variable-order matrices can be queried in the same fixed-operator form, whereas sharp geological contacts and PML-type absorbing boundaries require separate interface or non-self-adjoint matrix-function treatments. For repeated fixed-operator queries with \(q\ge q_*\) and sparse output fractions \(\rho_t,\rho_r\ll1\), matrix-function evaluation can bypass full Caputo time stepping while preserving matched waveform accuracy.