Nonlinear seismic amplitude versus offset inversion using the exact Zoeppritz equation

TL;DR

We address nonlinear AVO inversion using the exact Zoeppritz equations to recover $v_p$, $v_s$, and $\rho$ in multilayer media. An explicit analytical gradient is derived via the adjoint-state method and integrated into a nonlinear limited-memory quasi-Newton (L-BFGS) optimization, with a logistic map enforcing parameter bounds. The method is validated on 1D synthetic, 2D Marmousi-like, and Troll field data, showing stable convergence and accurate elastic-property estimates even with noise. The exact Zoeppritz-based gradients overcome limitations of linearized or weak-contrast models, enabling robust inversion across strong impedance contrasts and wide-angle responses.

Abstract

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 presents an explicit analytical gradient formulation of the exact Zoeppritz equation, discretized for multilayer media and derived using the adjoint-state method. The resulting expressions provide the gradient of a convolution-based objective function with respect to P-wave velocity, S-wave velocity, and density. 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 least squares target function is minimized using a nonlinear limited-memory quasi-Newton algorithm. We demonstrate the effectiveness of the analytical gradient solution of the exact Zoeppritz equations in seismic inversion problems involving P-wave and S-wave velocity and density models. The inversion methodology is validated using 1D well logs-based and 2D synthetic seismic data with varying noise levels. 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.

Figures (12)

• Figure 1: Flow diagram summarizing the stepwise structure of the inversion methodology.
• Figure 2: Inverted P- and S-wave velocities and density shown alongside the initial and true models.
• Figure 3: Observed, inverted, and initial seismic traces plotted for three angles of incidence (15$\degree$, 30$\degree$, and 45$\degree$), illustrating data fit achieved by the inversion.
• Figure 4: Results of inversion: predicted elastic properties (P- and S-wave velocities and density) compared with the initial and true models.
• Figure 5: Convergence plot of the inversion in (a) example 1 and (b) example 2 showing $\log_{10}$ of the objective function versus iteration numbers.