Superresolution with the zero-phase imaging condition

Figure 1.  The phase derivative as a function of $ t $, which represents distance to the scatterer, located at $ t = 0 $. The two black dashed lines bound the possible values of $ \frac{\partial \tilde{\theta}}{\partial \omega} $ in the presence of noise. From this, the operator $ \Gamma(\cdot) $, which acts on the phase derivative $ \frac{\partial \tilde{\theta}}{\partial \omega} $, allows full contribution to the image when $ t < \tau_{_{RES}} -c\sigma $, and zero contribution to the image when $ t > \tau_{_{RES}} + c\sigma $

Figure 2.  The mean resolved size of the scatterer, $ R $, as a function of $ \tau_{_{RES}} $ for various values of $ \sigma $

Figure 3.  The resolved size of the scatterer as a function of $ \sigma $ using the best $ \tau_{_{RES}} $. Green points represent individual samples corresponding to different noise realizations, red points represent the mean for that noise value, and the blue line is fits the data with an $ r^2 $ value of $ 0.988 $. The teal line represents the resolution with the standard imaging condition, which is calculated to be $ 0.273 \;\lambda $ (calculated from footnote 2), where $ \lambda $ is the wavelength. The numerical grid spacing for the experiments was $ 2.5 $ m, which limits the minimum resolvable size of the scatterer

Figure 4.  A velocity model with a scatterer located at $ (200\; m, 1200\; m) $ (left), an image produced with the standard imaging condition (center), and an image produced with the zero-phase imaging condition (right)

Figure 5.  A more complex example using the zero-phase imaging condition using zero-mean oscillatory reflection interfaces. (a) ground truth velocity model; (b) resultant image using the standard imaging condition; (c) resultant image using the zero-phase imaging condition with $ \tau_{_{RES}} = 0.02\;s $; (d) resultant image using the zero-phase imaging condition with $ \tau_{_{RES}} = 0.01\;s $

Figure 6.  A more complex example using the zero-phase imaging condition using non-zero-mean oscillatory reflection interfaces. (a) ground truth velocity model; (b) resultant image using the standard imaging condition; (c) resultant image using the zero-phase imaging condition with $ \tau_{_{RES}} = 0.02\;s $; (d) resultant image using the zero-phase imaging condition with $ \tau_{_{RES}} = 0.01\;s $. This is comparable to Figure 5

Figure 7.  An example introducing caustics into the data. (a) ground truth velocity model, where the migration velocity model includes the lens without the scatterer located at (1000, 0); (b) resultant image using the standard imaging condition; (c) resultant image using the zero-phase imaging condition with $ \tau_{_{RES}} = 0.01 $ s

Figure 8.  The ray through a source located at $ {\mathbf{x}}_s $ and a scatterer located at $ {\mathbf{x}}^* $ indicates where the phase of the integrand of the imaging condtion (Equation 6) will be zero. With a special receiver $ {\mathbf{x}}_{\widetilde{r}} $ antipodal to a given source location $ {\mathbf{x}}_s $, the locus of zero phase when $ \psi_+ ({\mathbf{x}}) = 0 $ is when $ {\mathbf{x}} \in [{\mathbf{x}}^*, {\mathbf{x}}_{\widetilde{r}}] $, and the locus of zero phase for when $ \psi_- ({\mathbf{x}}) = 0 $ is when $ {\mathbf{x}}^* \in [{\mathbf{x}}_s, {\mathbf{x}}^*] $. Therefore, we restrict our receiver array such there is no receiver antipodal to any source location through the scatterer location to avoid including this transmission artifact in the zero-phase imaging condition