Single-sided Marchenko focusing of compressional and shear waves

In time-reversal acoustics, waves recorded at the boundary of a strongly scattering medium are sent back into the medium to focus at the original source position. This requires that the medium can be accessed from all sides. We discuss a focusing method for media that can be accessed from one side only. We show how complex focusing functions, emitted from the top surface into the medium, cause independent foci for compressional and shear waves. The focused fields are isotropic and act as independent virtual sources for these wave types inside the medium. We foresee important applications in nondestructive testing of construction materials and seismological monitoring of processes inside the Earth.

Figure 1

Principle of time-reversal acoustics [1]. (a) The response to a source at ${\mathbf{x}}_F$ is represented here by the Green's function $G(\mathbf{x},{\mathbf{x}}_F,t)$. This response is observed at all $\mathbf{x}$ at the boundary of the medium. (b) The time-reversed Green's function $G(\mathbf{x},{\mathbf{x}}_F,-t)$ is emitted from the boundary back into the medium and focuses at ${\mathbf{x}}_F$.

Figure 2

Horizontally layered model.

Figure 3

Forward modeled responses to $P$- and $S$-wave sources.

Figure 4

Time-reversal method: two-sided illumination.

Figure 5

Time-reversal method: single-sided illumination.

Figure 6

Illumination of actual medium from above by ${\mathbf{F}}_1^{+}$.

Figure 7

Illumination from above by ${\mathbf{F}}_1^{+}$ and the time-reversal of $-{\mathbf{F}}_1^{-}$.

Figure 8

Superposition of Fig. 7 and its time-reversal.

Figure 9

Error as a function of the iteration number.