A theoretical contribution to the 1D inverse problem of reflection seismograms

Least-squares full-waveform inversion (FWI) is considered in the frequency domain for a set of noise-free observations of time length T at the surface obeying the 1D wave equation, with a known source. The initial model is of constant velocity. The first iteration, which equals the constant-velocity migration inversion (CVMI), is thoroughly analyzed. In CVMI, for the unit source power spectrum, it is within reach to analytically derive and interpret the mathematical formulas of the first-order partial derivatives of the modeled observations (Jacobian), and the gradient and Gauss-Newton Hessian of the objective function, and learn what information the calculation requires to obtain a successful physical result (i.e., velocity update). We recognize the gradient elements, except the last one, to be sums of reflection-amplitude weighted band-limited sign functions and the Hessian elements, except along the last column and row, to be band-limited, diagonal-centered triangle functions, which for infinite bandwidth reduces to the Kronecker delta function. When the fundamental frequency 1/T is lacking in the observations, the gradient loses information of the low-wavenumber trend of the velocity update. The Hessian becomes close to singular, and any stabilized inverse has no chance to repair the deficiencies of the gradient caused by any missing low frequency in the observations. FWI is started by applying CVMI. First, Jacobians are modeled by classic reflectivity modeling. Second, the diagonal Hessians can be used for estimating discrete velocity updates. Third, the Jacobian can be modeled in the first-order Wentzel-Kramers-Brillouin-Jeffreys (WKBJ) approximation and by neglecting transmission effects. Finally, single-frequency and low-frequency seismograms can be inverted by using broadband Hessians. The main mathematical findings are developed by simple numerical models and data.

Automated summary

Least-squares full-waveform inversion in the frequency domain is applied to noise-free observations obeying the 1D wave equation with a known source and constant velocity initial model. The analysis of the first iteration, equivalent to constant-velocity migration inversion, allows for derivation of mathematical formulas for gradient and Hessian elements required for successful velocity updates. Missing fundamental frequency in observations results in loss of low-wavenumber trend information in gradient and close to singular Hessian.