Extension of Seismic Scanning Tunneling Macroscope to Elastic Waves

Abstract

The theory for the seismic scanning tunneling macroscope is extended from acoustic body waves to elastic body-wave propagation. We show that, similar to the acoustic case, near-field superresolution imaging from elastic body waves results from the O(1/R) term, where R is the distance between the source and near-field scatterer. The higher-order contributions \(R^{-n}\) for \(n>1\) are cancelled in the near-field region for a point source with normal stress.

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Acknowledgements

The research reported in this publication was supported by the King Abdullah University of Science and Technology (KAUST) in Thuwal, Saudi Arabia. For computer time, this research used the resources of the Supercomputing Laboratory and the IT Research Computing group at KAUST. We thank them for providing the computational resources required for carrying out this work. We also thank Mr. Zongcai Feng for checking the equations in this paper.

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Correspondence to Bowen Guo.

Appendices

Appendix A: Geometric Spreading and Evanescent Waves

This section makes the connection between evanescent waves in the \(k_x-k_y-z-\omega\) domain and the superresolution properties of the SSTM in the \(x-y-z-\omega\) domain. The key observation is that, for \(R<<\lambda\), the width of the scattered SSTM profile (solid blue curve) in Fig. 1b is controlled by the 1/R term and not by the frequency dependent sinc function. As we will now show, wavefronts with small radii of curvature and strong geometrical-spreading 1/R are dominated by contributions from the evanescent field.

Fig. 6
figure6

a Plane wave propagating through a slot smaller than a wavelength; b waves scattered by a small scatterer and recorded by near-field receivers in a VSP well; c surface waves generating evanescent waves scattered and propagated as body waves; d scatterer-to-scatterer propagation

Evanescent waves Harmonic plane waves in Fig. 6a propagating through a small opening (with widths \(\Delta x\) and \(\Delta y\)) smaller than the wavelength will generate evanescent waves (de Fornel 2001). In this example, the acoustic plane wave diffracting through the aperture is given by

$$\begin{aligned} P(x,y,z) = \int \int \tilde{P}(k_x,k_y,z=0) e^{i(k_xx+k_yy+k_z z)}\mathrm {d}k_x \mathrm {d}k_y,~~~z>0, \end{aligned}$$
(16)

where c is the propagation velocity and \(\tilde{P}(k_x,k_y,z=0)\) is the spatial Fourier transform of the field at the aperture with area \(\Delta x \Delta y\). The significant contributions to this integral are for the integration domain defined by \(\Delta x \Delta k_x \approx 2 \pi\) and \(\Delta y \Delta k_y \approx 2 \pi\), where the vertical wavenumber \(k_z\) is defined for propagating waves and evanescent waves by

$$\begin{aligned} k_z = \left\{ \begin{array}{c} {\displaystyle } {\displaystyle \sqrt{\omega ^2/c^2 - k_x^2- k_y^2} \quad \text {if} \;(\omega /c)^2>k_x^2+ k_y^2} \\ {\displaystyle i\sqrt{ k_x^2+ k_y^2-\omega ^2/c^2 } \quad \text {if} \;(\omega /c)^2<k_x^2+ k_y^2} \end{array} \right. , \end{aligned}$$
(17)

and \(k=\omega /c\). The rule of thumb is that the smaller the aperture, the larger the aperture of the emerging beam (de Fornel 2001). If the width of the slot is equal to \(\lambda /2\), the emerging beam fills the entire half-space, and if the slot width is less than \(\lambda /2\), then evanescent waves with \(k_x^2+k_y^2>\omega ^2/c^2\) are diffracted from the slot.

The evanescent waves are described by

$$\begin{aligned} \tilde{P}(k_x,k_y,z)= & {} P_0 e^{- z/d_p}e^{i(k_xx+k_yy)}, \end{aligned}$$
(18)

where \(P_0=\tilde{P}(k_x,k_y,z=0)\) is the wavenumber spectrum at the slot and \(d_p\) is the penetration depth associated with the evanescent waves:

$$\begin{aligned} d_p= & {} (k_x^2 +k_y^2 -\omega ^2/c^2)^{-1/2} . \end{aligned}$$
(19)

These evanescent waves propagate only along the \(x-y\) plane and not along the z-axis, and are related to the high spatial frequencies of the slot. However, evanescent waves carry high-wavenumber information about the slot’s horizontal features because the horizontal wavenumber \(\sqrt{k_x^2 +k_y^2}\) can be arbitrarily large in order to satisfy the evanescent dispersion relationship for large values of \(k_z\):

$$\begin{aligned} k_x^2 +k_y^2 -k_z^2= & {} \omega ^2/c^2. \end{aligned}$$
(20)

If \(|z|=\epsilon\) is much smaller than a wavelength, then the exponentially decaying term in Eq. 16 significantly contributes to the integration over the evanescent wavenumbers such that \(\epsilon \sqrt{k_x^2+k_y^2-(\omega /c)^2} < 1\). This implies that the dominant \(1/R=1/|\mathbf{{x}}_o-\mathbf{{s}}'|\) term in the SSTM Eq. 4 requires greater evanescent contributions as \(R \approx |z|\) becomes smaller. The rapid decay of the evanescent waves from the object means that the measuring instruments should be much closer than a wavelength of the evanescent wave generator, also known as the near-field region of the object. Examples of such generators include local impedance variations that are smaller than a wavelength in Fig. 6b, surface waves interacting with a near-field scatterer in Fig. 6c, wave guides that leak evanescent energy between neighboring layers (Cragg and So 2000; de Fornel 2001;Ben-Aryeh 2004; Jia et al. 2010) or sources near reflecting boundaries (de Fornel 2001).

Appendix B: Far-field and Near-field approximations

From Eqs. 6 and 7, we have

$$\begin{aligned}&\lambda +2\mu = \rho \alpha ^2 = \rho \frac{\omega ^2}{{k_\alpha }^2}, \nonumber \\&\mu = \rho \beta ^2 = \rho \frac{\omega ^2}{{k_\beta }^2}. \end{aligned}$$
(21)

In the far-field region of the source, \(k_\alpha R \gg 1\) and \(k_\beta R \gg 1\) (Snieder 2002). The 1/R term will dominate in this region and the \(1/R^2\) and \(1/R^3\) terms can be neglected. Using this approximation and the relation in Eq. 21, the Green’s functions \(\mathbf{G_\alpha }\) and \(\mathbf{G_\beta }\) in Eqs. 9 and 10 can be approximated as

$$\begin{aligned}&{\tilde{\mathbf{G}}}_{\alpha } = \frac{e^{-ik_\alpha R}}{4\pi \rho \omega ^2} \frac{1}{R} {k_\alpha }^2 {\hat{\mathbf{R}}}{\hat{\mathbf{R}}}^T, \end{aligned}$$
(22)
$$\begin{aligned}&{\tilde{\mathbf{G}}}_{\beta } = \frac{e^{-ik_\beta R}}{4\pi \rho \omega ^2} \frac{1}{R} {k_\beta }^2 (\mathbf{I}- {\hat{\mathbf{R}}}{\hat{\mathbf{R}}}^T), \end{aligned}$$
(23)

where \({\tilde{\mathbf{G}}}_{\alpha }\) and \({\tilde{\mathbf{G}}}_{\beta }\) are the far-field approximations of \(\mathbf{{G}_\alpha }\) and \(\mathbf{{G}_\beta }\), respectively. Plugging Eqs. 22 and 23 into Eq. 5a, the Green’s tensor takes the following form in the far-field region:

$$\begin{aligned} \mathbf{G}_{\text {Far-field}}=\, & {} {\tilde{\mathbf{G}}}_{\alpha }+{\tilde{\mathbf{G}}}_{\beta }, \nonumber \\= & {} \frac{1}{4\pi \rho \omega ^2} \frac{1}{R} \left( k_\alpha ^2 e^{-ik_\alpha R} {\hat{\mathbf{R}}}{\hat{\mathbf{R}}}^T + k_\beta ^2 e^{-ik_\beta R} (\mathbf{I}- {\hat{\mathbf{R}}}{\hat{\mathbf{R}}}^T) \right) \;. \end{aligned}$$
(24)

In the near-field region of the source, \(k_\alpha R \ll 1\) and \(k_\beta R \ll 1\). Using the Taylor series expansion, we have

$$\begin{aligned}&e^{-ik_\alpha R}=1-ik_\alpha R-\frac{1}{2}(k_\alpha R)^2 ..., \end{aligned}$$
(25)
$$\begin{aligned}&e^{-ik_\beta R}=1-ik_\beta R-\frac{1}{2}(k_\beta R)^2 .... \end{aligned}$$
(26)

Inserting Eq. 25 into Eq. 9, the \(1/R^2\) and \(1/R^3\) terms together, expressed as \({\hat{\mathbf{G}}}_{\alpha }\), are

$$\begin{aligned} {\tilde{\mathbf{G}}}_{\alpha }&=\frac{(k_\alpha ^2-ik_\alpha ^3 R-\frac{1}{2}k_\alpha ^4 R^2)}{4\pi \rho \omega ^2} \left( - \frac{1}{ik_\alpha R^2} (\mathbf{I}-3 {\hat{\mathbf{R}}}{\hat{\mathbf{R}}}^T) - \frac{1}{(ik_\alpha )^2 R^3} (\mathbf{I}-3 {\hat{\mathbf{R}}}{\hat{\mathbf{R}}}^T) \right), \end{aligned}$$
(27)

where \({\tilde{\mathbf{G}}}_{\alpha }\) is the near-field approximation of \(\mathbf{{G}_\alpha }\). Similarly the near-field approximation of \(\mathbf{{G}_\beta }\), \({\tilde{\mathbf{G}}}_{\beta }\), can be written as

$$\begin{aligned} {\tilde{\mathbf{G}}}_{\beta } =&\frac{(k_\beta ^2-ik_\beta ^3 R-\frac{1}{2}k_\beta ^4 R^2)}{4\pi \rho \omega ^2} \Big ( \frac{1}{ik_{\beta }R^2} (\mathbf {I} - 3{\hat{\mathbf{R}}}{\hat{\mathbf{R}}}^T) + \frac{1}{(ik_{\beta })^2 R^3} (\mathbf {I} - 3 {\hat{\mathbf{R}}}{\hat{\mathbf{R}}}^T) \Big ). \end{aligned}$$
(28)

The near-field Green’s tensor in Eq. 5a can now be written as

$$\begin{aligned} \mathbf{G}_{\text {Near-field}}&= {\tilde{\mathbf{G}}}_{\alpha }+{\tilde{\mathbf{G}}}_{\beta }, \nonumber \\&=\frac{-k_\beta ^2}{8\pi \rho \omega ^2} \frac{1}{R} \left( 1-\frac{k_\alpha ^2}{k_\beta ^2} \right) (\mathbf{I}-3 {\hat{\mathbf{R}}}{\hat{\mathbf{R}}}^T). \end{aligned}$$
(29)

It can be seen that the \(1/R^2\) and \(1/R^3\) terms are cancelled in the near-field case. One can also say that these terms have been eliminated due to a cancellation between shear and compressional motion in the near field. Similar to the far-field case in Eq. 24, there is also a 1/R dependence in the near-field Green’s tensor.

Appendix C: Elastic SSTM Equations

Using the near- and far-field Green’s tensors in Eqs. 24 and 29, we can write

$$\begin{aligned} \mathbf{G}({\mathbf{{g}}}|{\mathbf{{x}}_o})= & {} \frac{1}{4 \pi \rho \omega ^2}\frac{1}{|{\mathbf{{g}}-\mathbf{{x}}_o}|} \left( k_\alpha ^2 e^{-ik_\alpha {|\mathbf{{g}}-\mathbf{{x}}_o|}} \mathbf {M}({\mathbf{{g}},\mathbf{{x}}_o}) + k_\beta ^2 e^{-ik_\beta {|\mathbf{{g}}-\mathbf{{x}}_o|}} (\mathbf{I}- \mathbf {M}({\mathbf{{g}},\mathbf{{x}}_o}) \right) \nonumber , \\ \mathbf{G}({\mathbf{{x}}_o}|{\mathbf{{s}}})= & {} \frac{-k_\beta ^2}{8\pi \rho \omega ^2} \frac{1}{|{\mathbf{{x}}_o-\mathbf{{s}}}|} \left( 1-\frac{k_\alpha ^2}{k_\beta ^2} \right) \left( \mathbf{I}-3 \mathbf {M}({\mathbf{{x}}_o,\mathbf{{s}}})\right), \end{aligned}$$
(30)

where \(\mathbf {M}(\mathbf{r,r_0})={\hat{\mathbf{R}}}{\hat{\mathbf{R}}}^T =\frac{(\mathbf{r-r_0})(\mathbf{r-r_0})^T}{|\mathbf{r-r_0}|^2}\). As shown in Fig. 1, the scatterer at \(\mathbf{{x}}_o\) is in the near-field region of the source at \(\mathbf{{s}}\) while the receiver \(\mathbf{{g}}\) is in the far-field region of the scatterer. Substituting Eq. 30 in Eq. 13 we get

$$\begin{aligned} {\mathbf{m}}({\mathbf{s^{\prime}}},{\mathbf{s}}) & = \frac{1}{{1024\pi ^{4} \rho ^{4} }}(1 - \frac{{k_{\alpha }^{2} }}{{k_{\beta }^{2} }})^{2} \int_{{ - \omega _{0} }}^{{\omega _{0} }} {\text{d}} \omega \int_{{ - L}}^{L} {\text{d}} g\frac{1}{{|{\mathbf{g}} - {\mathbf{x}}_{o} |^{2} }}\frac{1}{{|{\mathbf{x}}_{o} - {\mathbf{s}}||{\mathbf{x}}_{o} - {\mathbf{s}}^{\prime } |}} \\ & \quad \times \frac{1}{{\omega ^{8} }}[{\mathbf{I}} - 3{\mathbf{M}}({\mathbf{x}}_{o} ,{\mathbf{s}})][{\mathbf{I}} - 3{\mathbf{M}}({\mathbf{x}}_{o} ,{\mathbf{s}}^{\prime } )] \\ & \quad \times \left\{ {k_{\alpha }^{4} {\mathbf{M}}({\mathbf{g}},{\mathbf{x}}_{o} )^{2} + k_{\beta }^{4} [{\mathbf{I}} - {\mathbf{M}}({\mathbf{g}},{\mathbf{x}}_{o} )]^{2} + k_{\alpha }^{2} k_{\beta }^{2} e^{{ - i|{\mathbf{g}} - {\mathbf{x}}_{o} |(k_{\alpha } - k_{\beta } )}} {\mathbf{M}}({\mathbf{g}},{\mathbf{x}}_{o} )[{\mathbf{I}} - {\mathbf{M}}({\mathbf{g}},{\mathbf{x}}_{o} )]} \right. \\ & \quad + \left. {k_{\alpha }^{2} k_{\beta }^{2} e^{{ - i|{\mathbf{g}} - {\mathbf{x}}_{o} |(k_{\beta } - k_{\alpha } )}} [{\mathbf{I}} - {\mathbf{M}}({\mathbf{g}},{\mathbf{x}}_{o} )]{\mathbf{M}}({\mathbf{g}},{\mathbf{x}}_{o} )} \right\}. \\ \end{aligned}$$
(31)

Using Eq. 6, the above expression can be modified as

$$\begin{aligned} {\mathbf{m}}({\mathbf{s}}^{\prime } ,{\mathbf{s}}) & = \frac{1}{{1024\pi ^{4} \rho ^{4} }}(1 - \frac{{\beta ^{2} }}{{\alpha ^{2} }})^{2} \int_{{ - \omega _{0} }}^{{\omega _{0} }} {\text{d}} \omega \int_{{ - L}}^{L} {\text{d}} g\frac{1}{{|{\mathbf{g}} - {\mathbf{x}}_{o} |^{2} }}\frac{1}{{|{\mathbf{x}}_{o} - {\mathbf{s}}||{\mathbf{x}}_{o} - {\mathbf{s}}^{\prime } |}} \\ & \quad \times \frac{1}{{\omega ^{4} }}[{\mathbf{I}} - 3{\mathbf{M}}({\mathbf{x}}_{o} ,{\mathbf{s}})][{\mathbf{I}} - 3{\mathbf{M}}({\mathbf{x}}_{o} ,{\mathbf{s}}^{\prime } )]\left\{ {\frac{1}{{\alpha ^{4} }}{\mathbf{M}}({\mathbf{g}},{\mathbf{x}}_{o} )^{2} + \frac{1}{{\beta ^{4} }}[{\mathbf{I}} - {\mathbf{M}}({\mathbf{g}},{\mathbf{x}}_{o} )]^{2} + \frac{2}{{\alpha ^{2} \beta ^{2} }}} \right. \\ & \quad \left. { \times \cos \left[ {\omega \left( {\frac{{\beta - \alpha }}{{\alpha \beta }}} \right)}|{\mathbf{g}} - {\mathbf{x}}_{o} | \right] [{\mathbf{I}} - {\mathbf{M}}({\mathbf{g}},{\mathbf{x}}_{o} )]{\mathbf{M}}({\mathbf{g}},{\mathbf{x}}_{o} )} \right\}, \\ & = \frac{1}{{1024\pi ^{4} \rho ^{4} }}(1 - \frac{{\beta ^{2} }}{{\alpha ^{2} }})^{2} \int_{{ - \omega _{0} }}^{{\omega _{0} }} {\frac{{{\text{d}}\omega }}{{\omega ^{4} }}} \int_{{ - L}}^{L} {\text{d}} g\;{\mathcal{Y}}\left\{ {\frac{1}{{\alpha ^{4} }}{\mathbf{M}}({\mathbf{g}},{\mathbf{x}}_{o} )^{2} + \frac{1}{{\beta ^{4} }}[{\mathbf{I}} - {\mathbf{M}}({\mathbf{g}},{\mathbf{x}}_{o} )]^{2} + \frac{2}{{\alpha ^{2} \beta ^{2} }}} \right. \\ & \quad \left. { \times \cos \left[ {\omega \left( {\frac{{\beta - \alpha }}{{\alpha \beta }}} \right)|{\mathbf{g}} - {\mathbf{x}}_{o} |} \right][{\mathbf{I}} - {\mathbf{M}}({\mathbf{g}},{\mathbf{x}}_{o} )]{\mathbf{M}}({\mathbf{g}},{\mathbf{x}}_{o} )} \right\}, \\ \end{aligned}$$
(32)

where

$$\begin{aligned} \mathbf{\mathcal{Y}}=\frac{1}{|{\mathbf{{g}}-\mathbf{{x}}_o}|^2}\frac{1}{|{\mathbf{{x}}_o-\mathbf{{s}}}||{\mathbf{{x}}_o-\mathbf{{s}}'}|} [\mathbf{I} -3\mathbf {M}({\mathbf{{x}}_o,\mathbf{{s}}})] [\mathbf{I} -3\mathbf {M}({\mathbf{{x}}_o,\mathbf{{s}}'})]. \end{aligned}$$
(33)

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Tarhini, A., Guo, B., Dutta, G. et al. Extension of Seismic Scanning Tunneling Macroscope to Elastic Waves. Pure Appl. Geophys. 175, 207–216 (2018). https://doi.org/10.1007/s00024-017-1692-x

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Keywords

  • Superresolution Imaging
  • Propagating Body Waves
  • Acoustic Case
  • Point Scatterers
  • Actual Source Position