The invention discloses a viscoelastic acoustic wave full-waveform inversion method and apparatus. The method comprises the following steps of determining a time domain first order speed-stress viscoelastic acoustic wave equation according to the viscoelastic fluctuation equation of a GSLS model, wherein, the GSLS model is composed of a plurality of Maxwell bodies; according to the time domain first order speed-stress viscoelastic acoustic wave equation and a 2-norm objective function, determining a relaxation modulus gradient calculating formula and a longitudinal wave speed gradient calculating formula corresponding to the equation; determining a forward propagation wave field and a residual error inverse time back propagation wave field by adopting a high-order staggered grid finite difference method according to the time domain first order speed-stress viscoelastic acoustic wave equation; determining a relaxation modulus gradient and a longitudinal wave speed gradient according to the forward propagation wave field, the residual error inverse time back propagation wave field, the relaxation modulus gradient calculating formula and the longitudinal wave speed gradient calculating formula; and adopting a local optimization method to perform two-dimension viscoelastic acoustic wave full-waveform inversion according to the longitudinal wave speed gradient. The constant Q characteristic of the earth medium in an earthquake frequency band range can be better approximated. Great numeric dispersion inhibition capabilities are achieved.