Super-Resolution Via Weighted Time-ReversalManuel Alejandro Jaimes Caballero∗ and Roel Snieder, Colorado School of Mines

SUMMARY

We formulate super-resolution in the context of time-reverseimaging as a modified Backus-Gilbert inverse problem anddetermine the optimal complex weights that produce spatio-temporal focusing with resolution better than the typicaldiffraction limit. The optimal weights give the wrong sourcelocation if the velocity is not well-known, and noisy data de-grades the quality of focusing. The optimal weights also pro-duce sidelobes if the area used in the time-reversal processis larger than the one for which the weights are calculated.However, there are also advantages to applying weighted time-reversal. The optimization corrects for receiver geometry. Ad-ditionally, our method is able to handle variations in the sourcelocation. If we accurately know the location of a controlpoint in the subsurface we can use the corresponding optimalweights to achieve super-resolved focusing in locations sur-rounding the control point.

INTRODUCTION

Locating and imaging seismic sources has long been of in-terest in different areas of quantitative seismology. Methodswidely used to achieve source focusing include inverse scat-tering (Broggini et al.,2012), time-reversal (Fink,1997), phaseconjugation (Parvulescu,1961), and the inverse filter (Tanteret al.,2000; 2001). Seismologists typically use time-reversal(TR) techniques for source focusing. Applications of TR meth-ods in seismology include: Study of earthquake source mecha-nisms and location estimation (e.g., Lokmer et al.,2009); mon-itoring of nuclear explosions (e.g., Larmat et al.,2006); envi-ronmental applications (e.g., Larmat et al.,2010); microseis-mic event location (e.g., Lu et al.,2008); reservoir monitoring(e.g., Shapiro,2008); and reversed time migration (e.g., Schus-ter,2002).

Optimally, the wave that one sends back to the medium shouldfocus at the location where the seismic event originated at timet = 0 provided one has an accurate knowledge of the velocityof the medium. However, even under a perfect set up, the fo-cusing is spatially limited by the dominant wavelength of theseismic data. This limit is the wave diffraction limit, com-monly known to be λ/2 (Fink,1997).

(Francia,1952) proposed the idea of imaging beyond the diffrac-tion limit but was not applicable at the time due to practicallimitations. New tests in the field of optics (Rogers and Zhe-ludev,2013) have shown that it is possible to go beyond thisfocusing limit to achieve super-resolution, which is of impor-tance in the inverse source problem as well as in conventionalseismic imaging.

In the seismological community Schuster et al.(2012) intro-duce the idea of a seismic scanning tunneling macroscope which

recovers evanescent waves. To overcome the near-field limita-tion, Guo et al.(2016) show that in the far field one can use res-onant multiples in data migration to achieve sub-wavelengthresolution.

To understand how we can achieve super-resolution withoutevanescent waves or a strongly heterogeneous medium we posethe following question: Can we find frequency-dependent com-plex weights for each of the seismic receivers such that aftertime reversal the focal spot at the source location has a widthsmaller than the diffraction limit? We formulate this questionas a modified Backus-Gilbert(BG) problem in the sense thatwe search optimal finite weights that allow for reconstructionof a band-limited delta function in space and time, robust toerrors in the velocity model and estimated source location.

Our approach is linked to the work of: Anderson et al.(2015)who achieve temporal focusing via deconvolution,which is im-plemented as an inverse filter in the frequency domain; andBazargani and Snieder(2016) who minimize the difference be-tween the backpropagated wavefield and the time-reversed dis-placement field in the near source region by searching for op-timal signals to send into the medium.

THEORY

Consider the time-reversal imaging condition in the time do-main where the signal recorded by each receiver, G(xi, ti,x0, t0)∗S(t), is time reversed (i.e. G(xi,−t0,x0,−ti)∗S(−t)) and sentback into the medium as a new source (Fink and Prada,2001):

fi(t) = G(x, t,xi, ti)∗ (G(xi,−t0,x0,−ti)∗S(−t)). (1)

G(x, t,xi, ti) corresponds to the Green’s function between eachreceiver and the points in the medium where the signal propa-gates. x0 and t0, are the origin location and time, respectively.For convenience we use t0 = 0. The index i specifies a par-ticular receiver, the symbol ”∗” implies temporal convolution,and x specifies the location of the grid point in the field ofview(FOV) where time-reversal takes place.

We use the following convention for the Inverse Fourier Trans-form

κ(t) =∫ +∞

−∞

K(ω)eiωtdω. (2)

Ideally for a point impulsive source, we would expect the time-reversed wavefield to focus in time and space, namely

∑i

fi(t) = δ (x−x0)δ (t). (3)

Using the Fourier representation in equation 2, we write equa-tion 3 as

Super-Resolution Via Weighted Time-Reversal

∫ +∞

−∞

∑i

Fi(ω)eiωtdω =

∫ +∞

−∞

δ (x−x0)eiωtdω, (4)

from which we have

∑i

Fi(ω) = δ (x−x0). (5)

To improve the localization of the time-reversed field we nowconsider frequency-dependent complex weights σi,

∑i

σi(ω)Fi(ω) = δ (x−x0), (6)

notice that we want to achieve spatial focusing at each fre-quency under consideration.

To construct a delta function given a finite set of measurementswe use a modified version of the Backus-Gilbert (BG) method(Backus and Gilbert,1968). We seek to find the linear com-bination of the individual time-reversed wavefields that bestresembles a delta function. To this end consider the followingobjective function

J(x,x0,ω) =∑

i

(x−x0)2|σi(ω)Fi(ω)|2

+λ

(∑i

σi(ω)Fi(ω)−δ (x−x0)

).

(7)

We aim to minimize the objective function J for all points xin FOV rather than minimizing the sum of objective functionsover the domain of interest, as it is typically done in BG theory.

We want to minimize this objective function in terms of thecomplex weights σi and the lagrange multiplier λ . By mini-mizing the first term of equation 7 we force the energy of theweighted TR wavefield to be zero everywhere except at thesource location. The second term guarantees the amplitude ofthe TR wavefield to be non-zero only at the source, where welet the delta function be unity.

Now we consider the partial derivatives of J with respect to theoptimization parameters

∂J∂σ∗i

=∑

j

(x−x0)2(σ j(ω)Fj(ω)F∗i (ω))+λF∗i (ω), (8)

∂J∂λ

=∑

j

(σ j(ω)F∗j (ω))−δ (x−x0), (9)

here the superscript ∗ implies complex conjugation. The sub-scripts i, j specify receivers.

To minimize the objective function we set the partial deriva-tives in equations 8 and 9 equal to zero and solve this systemof equations simultaneously for all grid points in FOV. These

equations are written as

(x1−x0)2F1

1 F∗11 · · · (x1−x0)

2F1N F∗1

1 F∗11

.

.

....

.

.

....

(x1−x0)2F1

1 F∗N1 · · · (x1−x0)

2F1N F∗N

1 F∗N1

F∗11 · · · F∗N

1 0...

.

.

....

.

.

.(xk −x0)

2Fk1 F∗1

k · · · (xk −x0)2Fk

N F∗1k F∗1

k

.

.

....

.

.

....

(xk −x0)2Fk

1 F∗Nk · · · (xk −x0)

2FkN F∗N

k F∗Nk

F∗1k · · · F∗N

k 0...

.

.

....

.

.

.(xM −x0)

2FM1 F∗1

M · · · (xM −x0)2FM

N F∗1M F∗1

M

.

.

....

.

.

....

(xM −x0)2FM

1 F∗NM · · · (xM −x0)

2FMN F∗N

M F∗NM

F∗1M · · · F∗N

M 0

︸ ︷︷ ︸

A

σ1......

σNλ

︸ ︷︷ ︸

χ

=

0.........

1

...

0

...

...

...

,

︸ ︷︷ ︸b(10)

here 1≤ k ≤M, k is the grid point x, M is the number of gridpoints in the field of view, and N is the number of receivers.Thesuperscript in F specifies the corresponding grid point whereasthe subscript specifies the receiver number. The only non-zeroelement in the vector on the right-hand side of equation 10corresponds to setting x = x0 in equation 9. Ultimately, welook for solutions to equation 10 that minimize ||Aχ −b|| inthe least-squares sense.

NUMERICAL IMPLEMENTATION

To simulate a typical seismic scenario consider the array inFig. 1. This set-up is challenging because the signal recordedby each receiver has low-frequency content and the array illu-minates the source at limited angles. For simplicity we con-sider a homogeneous medium. For the simulation we use 11receivers in a medium with v = 2000 m/s, frequency rangingfrom 0.32 Hz to 50 Hz with a spacing of 0.32 Hz,and a sourcewavelet with dominant frequency of 12.73 Hz.

Figure 1: Set-up for the short planar geometry. The red boxindicates the field of view. The blue star corresponds to thesource location.

We expect the weights to amplify the high frequency contentof the signal, and correct for incomplete wavefield samplingand illumination angles. To better appreciate the challenges

Super-Resolution Via Weighted Time-Reversal

imposed by the geometry consider the optimal and standardtime-reversed wavefields with the cross-sections at the focus-ing level.

Figure 2: Comparison of standard TR (left) and optimal TRimages (right). Notice the improvement of the focusing withthe optimal weights.

Figure 3: Cross section at source level for standard and optimalfocusing. Half-width is below λ/2∼ 39.72 m.

The weights achieve improved localization of the source. Figs.2 and 3 show side lobes resulting from the experiment geom-etry and the low number of receivers. Now we consider theoptimal and standard time-domain waveforms to visualize theoptimal and standard signals.

Figure 4: Comparison of original and optimal signals for re-ceiver at x = 1000 m. The focusing time at the source locationcorresponds to t=0.

Fig 4 shows compression of the direct pulse, which results insharper focusing. Periodic reverberations appear before andafter the direct pulse. These reverberations have a differenttemporal spacing at different receivers and are essential for im-proving the resolution.

SENSITIVITY ANALYSES

To study the robustness of the optimal focusing we considervariations in the background velocity, and source location. We

also consider the effect of adding noise to the recorded sig-nal before time-reversal. Lastly, we investigate how focusingchanges as the field of view is increased.

Velocity perturbationFirst, we calculate the weights for the true velocity of the medium.We then perform the back-propagation using a perturbed ve-locity. Denoting γ as the percentage variation of the true ve-locity, we look at γ = 0.5, 1.0, 2.0, and 5.0% .

Figure 5: Optimal time-reversed images for different levelsof velocity perturbation. The blue star still indicates the truesource location.

Notice from Fig 5 that the quality of focusing degrades withincreasing velocity perturbation. An erroneous velocity givesthe wrong source location and increases the width of the focalspot. This means that if we want to achieve accurate focusing(both in terms of location and size of the focal spot) we need toknow the velocity of the medium, a consequence of the time-reversal process.

True vs estimated source locationNow suppose that the estimated and true source locations aredifferent. To study this scenario we first calculate the weightscorresponding to a fixed source location. Using those weightswe perform the time-reversal process for events which origi-nate at other locations within the field of view.

Figure 6: Optimal time-reversed images for different sourcelocations using the weights corresponding to the originalsource location (x0,z0) = (0,1500).

Super-Resolution Via Weighted Time-Reversal

Fig. 6 shows that the optimal weights allow for focusing ofevents at locations different than the true location for whichthe weights were calculated. Artifacts on the boundary of FOVarise when the distance between the new and original sourcelocations increases.

Noisy inversionNow we consider the effect of adding noise to the recordeddata. We add band-limited random noise to the recorded sig-nal. One would expect the weights to amplify the noise, andconsequently the optimal TR image to be degraded.

Figure 7: Optimal time-reversed wavefield for different noise.

Figure 8: Cross sections of optimal time-reversed wavefieldsfrom Figure 7.

Figs. 7 and 8 show noise degrades the quality of focusing. Anoise level up to 10% allows for a half-width below the diffrac-tion limit. Thus, the weights are robust to noise.

Size of field-of-viewTo analyze whether we achieve local or global focusing wecalculate the weights for the fixed FOV in Fig. 1 and performthe time-reversal process in a larger FOV.

Fig. 9 shows that as the size of FOV increases the sidelobesbecome more significant. Notice the focusing at the sourcelocation(x=0) becomes very small relative to the wavefield atother locations. This implies that we are only able to achievelocal focusing in the vicinity of the source rather than every-where in the medium.

Figure 9: Cross sections of optimal focusing at source level forFOV 4 times larger than original FOV (left) and FOV 16 timeslarger than original FOV (right).

This finding is not new, Rogers and Zheludev(2013) show thatsuper-oscillatory lenses, which are developed to achieve opti-mal focusing, allow for imaging beyond the diffraction limitbut often produce adjacent side-bands as the field of view isincreased.

CONCLUSIONS

It is possible to achieve super-resolution in a homogeneousmedium by applying a complex filter to the signal before time-reversal. This is similar to spatial light modulation in optics(e.g., Chung and Kim,1999) where one modifies the amplitudeand phase of the source signal to achieve imaging of a point inspace upon propagation through an imaging system.

To calculate the optimal weights we use a modified Backus-Gilbert approach which guarantees the focal spot to be local-ized at the source. Neither evanescent waves nor resonant mul-tiples are needed to achieve super-resolution if we know howthe original signal must be modulated. We find geometricalartifacts remain in the optimal TR images. The weights par-tially compensate for the limited geometry by modulating thefrequency spectrum of the signal.

The weights are robust to errors in the source location. Thisis important because we do not always know the true sourcelocation. The fact that we obtain proper focusing in locationsdifferent from the true source implies that if we have full in-formation of a control point in the field of view we can achievefocusing in other points within the FOV. To achieve proper fo-cusing we also need to know the velocity of the medium ac-curately, this drawback arises from the nature of time-reversal.In addition, using noisy data in the optimization results in de-graded focusing. Lastly, even with optimization techniques,only local focusing is possible.

ACKNOWLEDGMENTS

This work was supported by the Consortium Project on Seis-mic Inverse Methods for Complex Structures at the Center forWave Phenomena (CWP).

Super-Resolution Via Weighted Time-Reversal

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