A Hooke–Kjartansson model of seismic wave propagation in viscoelastic media

Summary The Kjartansson constant-Q model is widely adopted for describing seismic attenuation and velocity dispersion in subsurface viscoelastic media. Nevertheless, this classic formulation suffers a well-known physical defect: its phase velocity falls toward zero as frequency approaches zero, which contradicts realistic static elastic behaviour of geological materials. To address this limitation while preserving the favourable attenuation characteristics of constant-Q behaviour, this paper develops the Hooke–Kjartansson (H–K) viscoelastic model. It combines a Hookean elastic spring element, placed in parallel, with the fractional-order Kjartansson viscoelastic component. This constitutive design retains core attenuation and dispersion features of conventional constant-Q behaviour across the seismic frequency band, yet yields physically plausible finite elastic response at low and zero frequencies. We derive analytical plane-wave solutions for wave propagation in H–K media and interpret the physical meaning of its two governing parameters: parameter q controls elastic contribution and sets the zero-frequency (DC) phase-velocity limit, while fractional exponent β determines how attenuation and dispersion change with frequency. Theoretical analysis proves that the H-K constitutive operator satisfies the minimum-phase condition, which guarantees causal and stable wave responses. We further build a practical Q formulation for the H–K model. Tests against four published laboratory datasets covering shale, dry carbonate and fluid-saturated rock samples demonstrate better fitting accuracy compared with the original Kjartansson constant-Q model, especially for fluid-bearing rocks exhibiting strong frequency-dependent attenuation. This study provides a physically more reasonable and flexible two-parameter rheological framework for simulating seismic wave propagation in subsurface viscoelastic formations.