Seismic amplitude-variation-with-offset inversion using an algebraically equivalent formulation of the exact Zoeppritz equations and a matrix-free adjoint-state framework

The amplitude-variation-with-offset inversion techniques are formulated to estimate elastic properties by fitting modeled seismic responses to observed data.

Solving inverse seismic problems requires minimizing a target objective function for which gradient-based methods are frequently adopted.

However, the efficiency and accuracy of these methods depend significantly on the approach used to compute the gradient of the target function.

This work first derives an algebraically equivalent reformulation of the exact Knott-Zoeppritz PP equations, yielding a differentiable forward model suitable for gradient-based optimization.

The resulting formulation is then used to derive an explicit adjoint-state gradient of a convolution-based objective function for discretized multilayer media, providing an efficient framework for exact nonlinear inversion of P- and S-wave velocities and density, including the consistent treatment of post-critical-angle reflections through the complex-valued Zoeppritz response.

The adjoint state-based solution improves computational efficiency by avoiding numerical approximations while maintaining high accuracy in calculating the gradient for seismic inversion.

Additionally, using the exact Zoeppritz equation helps overcome the limitations associated with weak elastic property contrasts across subsurface layers.

The inversion methodology is validated using 1D well-log-based and 2D synthetic seismic data at varying signal-to-noise ratios, including a 500-member ensemble test of sensitivity to the starting model and a Marmousi comparison of Zoeppritz versus linearized Aki-Richards reflectivity at high contrast and wide angles.

Then it is applied to a 2D field data set from the Troll oil and gas field in the Norwegian North Sea.

The results demonstrate that the proposed inversion framework provides stable and reliable estimates of elastic property models.