[Paper Review] Analytical Solution for Wave Propagation in Stratified Acoustic/Porous Media. Part II: the 3D Case

Julien Diaz, Abdelaâziz Ezziani|ArXiv.org|Jul 25, 2008

Seismic Imaging and Inversion Techniques9 references3 citations

TL;DR

This paper presents an analytical solution for three-dimensional wave propagation in a bilayered acoustic/poroelastic medium using the Cagniard-de Hoop method. It extends prior 2D work to 3D, deriving explicit expressions for reflected and transmitted waves—including P, S, and Ps types—enabling precise modeling of wave behavior at interfaces, validated through numerical simulations with a point source and Gaussian time function.

ABSTRACT

We are interested in the modeling of wave propagation in an infinite bilayered acoustic/poroelastic media. We consider the biphasic Biot's model in the poroelastic layer. The first part is devoted to the calculation of analytical solution in two dimensions, thanks to Cagniard de Hoop method. In this second part we consider the 3D case.

Motivation & Objective

• To develop a complete analytical solution for transient wave propagation in a 3D stratified medium composed of an acoustic layer over a poroelastic layer.
• To extend the Cagniard-de Hoop method from 2D to 3D for poroelastic wave problems, enabling independent computation of all wave types.
• To derive explicit expressions for the Green’s function and displacement components in both layers, including all wave modes (P, S, Ps, Pf).
• To validate the analytical solution through numerical simulations using a point source with a fifth derivative of a Gaussian time function.
• To provide a benchmark for numerical codes simulating wave propagation in poroelastic media, particularly in complex layered configurations.

Proposed method

• Formulates the 3D wave problem using Biot’s equations in the poroelastic layer and the standard wave equation in the acoustic layer.
• Applies the Cagniard-de Hoop technique to solve the resulting system of partial differential equations in the frequency-wavenumber domain.
• Derives the Green’s function for displacement and pressure by inverting the Laplace-Fourier transform using contour integration and saddle-point methods.
• Computes fictitious and real arrival times for transmitted waves (e.g., Ps) by minimizing travel time across the interface using a variational approach.
• Solves the resulting fourth-order polynomial for the optimal interface reflection point to determine wave path geometry.
• Performs numerical integration using the midpoint quadrature rule and convolves the result with the source time function to obtain time-domain signals.

Experimental results

Research questions

• RQ1 How can the Cagniard-de Hoop method be extended from 2D to 3D for wave propagation in poroelastic media?
• RQ2 What are the analytical expressions for the reflected and transmitted wavefields (P, S, Ps, Pf) in a 3D acoustic/poroelastic bilayered medium?
• RQ3 How do the wave speeds and arrival times of different wave types (especially Ps and Pf) depend on the wavenumber and interface geometry?
• RQ4 What is the role of fictitious velocities in the Cagniard-de Hoop method, and how do they affect the accuracy of the time-domain solution?
• RQ5 How can the analytical solution be numerically validated against a known source function and receiver configuration?

Key findings

• The analytical solution successfully computes all wave types—P, S, Ps, and Pf—separately and accurately in the 3D case, with distinct arrivals observable in time-domain signals.
• The solution is validated numerically using a point source located 500 m above the interface, with a fifth derivative of a Gaussian source function centered at 15 Hz.
• Receivers placed 533 m from the interface in both the acoustic and poroelastic layers show clear separation of wave arrivals, including the Ps wave and Pf wave, as confirmed in Figs. 9 and 10.
• The fictitious arrival time function $\tilde{t}_0(q_y)$ is even and increasing on $\mathbb{R}^+$, and is bijective from $\mathbb{R}^+$ to $\mathbb{R}^+$, ensuring stable inversion in the Cagniard-de Hoop method.
• The method correctly captures the wavefronts and amplitudes of all wave modes, with the Ps wave arriving after the P and S waves due to its lower group velocity.
• The computed displacement components at receivers show distinct wave packets corresponding to incident, reflected, and transmitted waves, confirming the physical consistency of the solution.