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Fermat's interferometric principle is used to compute interior transmission traveltimes τpq from exterior transmission traveltimes τsp and τsq. Here, the exterior traveltimes are computed for sources s on a boundary B that encloses a volume V of interior points p and q. Once the exterior traveltimes are computed, no further ray tracing is needed to calculate the interior times τpq. Therefore this interferometric approach can be more efficient than explicitly computing interior traveltimes τpq by ray tracing. Moreover, the memory requirement of the traveltimes is reduced by one dimension, because the boundary B is of one fewer dimension than the volume V. An application of this approach is demonstrated with interbed multiple (IM) elimination. Here, the IMs in the observed data are predicted from the migration image and are subsequently removed by adaptive subtraction. This prediction is enabled by the knowledge of interior transmission traveltimes τpq computed according to Fermat's interferometric principle. We denote this principle as the ‘traveltime holographic principle’, by analogy with the holographic principle in cosmology where information in a volume is encoded on the region's boundary.

Strong subsurface attenuation leads to distortion of amplitudes and phases of seismic waves propagating inside the earth. Conventional acoustic reverse time migration (RTM) and least-squares reverse time migration (LSRTM) do not account for this distortion, which can lead to defocusing of migration images in highly attenuative geologic environments. To correct for this distortion, we used a linearized inversion method, denoted as [Formula: see text]-LSRTM. During the least-squares iterations, we used a linearized viscoacoustic modeling operator for forward modeling. The adjoint equations were derived using the adjoint-state method for back propagating the residual wavefields. The merit of this approach compared with conventional RTM and LSRTM was that [Formula: see text]-LSRTM compensated for the amplitude loss due to attenuation and could produce images with better balanced amplitudes and more resolution below highly attenuative layers. Numerical tests on synthetic and field data illustrated the advantages of [Formula: see text]-LSRTM over RTM and LSRTM when the recorded data had strong attenuation effects. Similar to standard LSRTM, the sensitivity tests for background velocity and [Formula: see text] errors revealed that the liability of this method is the requirement for smooth and accurate migration velocity and attenuation models.

We show that superresolution imaging in the far-field region of the sources and receivers is theoretically possible if migration of resonant multiples is employed. A resonant multiple is one that bounces back and forth between two scattering points or two neighboring reflectors. For a source with frequency f , N roundtrips in propagating between two scatterers increases the effective frequency to 2N × f and decreases the effective wavelength λ to λ/2N. Thus, interbed multiples can, in principle, be used as high-frequency probes to estimate detailed properties of layers. This is not only applicable to crustal reflections, but also to mantle and core reverberations of interest to earthquake seismologists.

Standard migration images can suffer from (1) migration artifacts caused by an undersampled acquisition geometry, (2) poor resolution resulting from a limited recording aperture, (3) ringing artifacts caused by ripples in the source wavelet, and (4) weak amplitudes resulting from geometric spreading, attenuation, and defocusing. These problems can be remedied in part by least-squares migration (LSM), also known as linearized seismic inversion or migration deconvolution (MD), which aims to linearly invert seismic data for the reflectivity distribution. Given a sufficiently accurate migration velocity model, LSM can mitigate many of the above problems and can produce more resolved migration images, sometimes with more than twice the spatial resolution of standard migration. However, LSM faces two challenges: The computational cost can be an order of magnitude higher than that of standard migration, and the resulting image quality can fail to improve for migration velocity errors of about 5% or more. It is possible to obtain the most from least-squares migration by reducing the cost and velocity sensitivity of LSM.

The timing and amplitudes of arrivals recorded in seismic traces are influenced by velocity variations all along the associated raypaths. Consequently, velocity errors far from the target can lead to blurred imaging of the target body. To partly remedy this problem, we comprehensively reviewed inverting differential traveltimes that satisfied the closure-phase condition. The result is that the source and receiver statics are completely eliminated in the data and velocities far from the target do not need to be known. We successfully used the phase closure equation for traveltime tomography, refraction statics, migration, refraction tomography, and earthquake location, all of which demonstrated the higher resolution achievable by processing data with differential traveltimes rather than absolute traveltimes. More generally, the stationary version of the closure-phase equation is equivalent to Fermat’s principle and can be derived from the equations of seismic interferometry. In summary, the general closure-phase equation is the mathematical foundation for approximately redatuming sources and/or receivers to the target of interest without the need to accurately know the statics or the velocity model away from the target.

Formulas are derived for the resolution limits of migration-data kernels associated with diving waves, primary reflections, diffractions, and multiple reflections. They are applicable to images formed by reverse time migration (RTM), least squares migration (LSM), and full waveform inversion (FWI), and suggest a multiscale approach to iterative FWI based on multiscale physics. That is, at the early stages of the inversion, events that only generate low-wavenumber resolution should be emphasized relative to the high-wavenumber resolution events. As the iterations proceed, the higher-resolution events should be emphasized. The formulas also suggest that inverting multiples can provide some low- and intermediate-wavenumber components of the velocity model not available in the primaries. Finally, diffractions can provide twice or better the resolution than specular reflections for comparable depths of the reflector and diffractor. The width of the diffraction–transmission wavepath is approximately λ at the diffractor location for the diffraction–transmission wavepath.

We reformulate the equation of reverse-time migration so that it can be interpreted as summing data along a series of hyperbola-like curves, each one representing a different type of event such as a reflection or multiple. This is a generalization of the familiar diffraction-stack migration algorithm where the migration image at a point is computed by the sum of trace amplitudes along an appropriate hyperbola-like curve. Instead of summing along the curve associated with the primary reflection, the sum is over all scattering events and so this method is named generalized diffraction-stack migration. This formulation leads to filters that can be applied to the generalized diffraction-stack migration operator to mitigate coherent migration artefacts due to, e.g., crosstalk and aliasing. Results with both synthetic and field data show that generalized diffraction-stack migration images have fewer artefacts than those computed by the standard reverse-time migration algorithm. The main drawback is that generalized diffraction-stack migration is much more memory intensive and I/O limited than the standard reverse-time migration method.

The theory of least-squares reverse time migration of multiples (RTMM) is presented. In this method, least squares migration (LSM) is used to image free-surface multiples where the recorded traces are used as the time histories of the virtual sources at the hydrophones and the surface-related multiples are the observed data. For a single source, the entire free-surface becomes an extended virtual source where the downgoing free-surface multiples more fully illuminate the subsurface compared to the primaries. Since each recorded trace is treated as the time history of a virtual source, knowledge of the source wavelet is not required and the ringy time series for each source is automatically deconvolved. If the multiples can be perfectly separated from the primaries, numerical tests on synthetic data for the Sigsbee2B and Marmousi2 models show that least-squares reverse time migration of multiples (LSRTMM) can significantly improve the image quality compared to RTMM or standard reverse time migration (RTM) of primaries. However, if there is imperfect separation and the multiples are strongly interfering with the primaries then LSRTMM images show no significant advantage over the primary migration images. In some cases, they can be of worse quality. Applying LSRTMM to Gulf of Mexico data shows higher signal-to-noise imaging of the salt bottom and top compared to standard RTM images. This is likely attributed to the fact that the target body is just below the sea bed so that the deep water multiples do not have strong interference with the primaries. Migrating a sparsely sampled version of the Marmousi2 ocean bottom seismic data shows that LSM of primaries and LSRTMM provides significantly better imaging than standard RTM. A potential liability of LSRTMM is that multiples require several round trips between the reflector and the free surface, so that high frequencies in the multiples suffer greater attenuation compared to the primary reflections. This can lead to lower resolution in the migration image compared to that computed from primaries. Another liability is that the multiple migration image is more down-dip limited than the standard primaries migration image. Finally, if the surface-related multiple elimination method is imperfect and there are strong multiples interfering with the primaries, then the resulting LSRTMM image can be significantly degraded. We conclude that LSRTMM can be a useful complement, not a replacement, for RTM of primary reflections.

A 5.6-km-long line of refraction and reflection seismic data spanning the Pliocene-Pleistocene fill of the Olduvai Basin, Tanzania is presented. The line is oriented along a northwest-southeast profile through the position of Olduvai Gorge Coring Project (OGCP) Borehole 2A. Our aims are to (1) delineate the geometry of the basin floor by tracing bedrock topography of the metaquartzitic and gneissic basement, (2) map synsedimentary normal faults and trace individual strata at depth, and (3) provide context for the sequence observed in OGCP cores. Results with refraction tomography and poststack migration show that the maximum basin depth is around 405 m (±25 m) in the deepest portion, which quadruples the thickness of the basin-fill previously known from outcrops. Variations in seismic velocities show the positions of lower density lake claystones and higher density well-cemented sedimentary sequences. The Bed I Basalt lava is a prominent marker in the refraction seismic results. Bottom-most sediments are dated to >2.2 Ma near where Borehole 2A bottoms out at the depth of 245 m. However, the seismic line shows that the basin-fill reaches a maximum stratigraphic thickness of around 380 m deep at Borehole 2A, in the western basin where the subsidence was greatest. This further suggests that potential hominin palaeoenvironments were available and preserved within the basin-fill possibly as far back as around 4 Ma, applying a temporal extrapolation using the average sediment accretion rate.

Geophysical Journal International, 196 (2), ISSN:0956-540X, ISSN:1365-246X

We present the theory and numerical results for interferometrically interpolating 2D and 3D marine surface seismic profiles data. For the interpolation of seismic data we use the combination of a recorded Green's function and a model-based Green's function for a water-layer model. Synthetic (2D and 3D) and field (2D) results show that the seismic data with sparse receiver intervals can be accurately interpolated to smaller intervals using multiples in the data. An up- and downgoing separation of both recorded and model-based Green's functions can help in minimizing artefacts in a virtual shot gather. If the up- and downgoing separation is not possible, noticeable artefacts will be generated in the virtual shot gather. As a partial remedy we iteratively use a non-stationary 1D multi-channel matching filter with the interpolated data. Results suggest that a sparse marine seismic survey can yield more information about reflectors if traces are interpolated by interferometry. Comparing our results to those of f-k interpolation shows that the synthetic example gives comparable results while the field example shows better interpolation quality for the interferometric method.

A plane-wave least-squares reverse-time migration (LSRTM) is formulated with a new parameterization, where the migration image of each shot gather is updated separately and an ensemble of prestack images is produced along with common image gathers. The merits of plane-wave prestack LSRTM are the following: (1) plane-wave prestack LSRTM can sometimes offer stable convergence even when the migration velocity has bulk errors of up to 5%; (2) to significantly reduce computation cost, linear phase-shift encoding is applied to hundreds of shot gathers to produce dozens of plane waves. Unlike phase-shift encoding with random time shifts applied to each shot gather, plane-wave encoding can be effectively applied to data with a marine streamer geometry. (3) Plane-wave prestack LSRTM can provide higher-quality images than standard reverse-time migration. Numerical tests on the Marmousi2 model and a marine field data set are performed to illustrate the benefits of plane-wave LSRTM. Empirical results show that LSRTM in the plane-wave domain, compared to standard reverse-time migration, produces images efficiently with fewer artifacts and better spatial resolution. Moreover, the prestack image ensemble accommodates more unknowns to makes it more robust than conventional least-squares migration in the presence of migration velocity errors.

SUMMARY The theory of supervirtual interferometry is modified so that free-surface related multiple refractions can be used to enhance the signal-to-noise ratio (SNR) of primary refraction events by a factor proportional to √ Ns, where Ns is the number of post-critical sources for a specified refraction multiple. We also show that refraction multiples can be transformed into primary refraction events recorded at virtual hydrophones located between the actual hydrophones. Thus, data recorded by a coarse sampling of ocean bottom seismic (OBS) stations can be transformed, in principle, into a virtual survey with P times more OBS stations, where P is the order of the visible free-surface related multiple refractions. The key assumption is that the refraction arrivals are those of head waves, not pure diving waves. The effectiveness of this method is validated with both synthetic OBS data and an OBS data set recorded offshore from Taiwan. Results show the successful reconstruction of far-offset traces out to a source‐receiver offset of 120km. The primary supervirtual traces increase the number of pickable first arrivals from approximately 1600 to more than 3100 for a subset of the OBS data set where the source is only on one side of the recording stations. In addition, the head waves associated with the first-order free-surface refraction multiples allow for the creation of six new common receiver gathers recorded at virtual OBS station located about half way between the actual OBS stations. This doubles the number of OBS stations compared to the original survey and increases the total number of pickable traces from approximately 1600 to more than 6200. In summary, our results with the OBS data demonstrate that refraction interferometry can sometimes more than quadruple the number of usable traces, increase the source‐receiver offsets, fill in the receiver line with a denser distribution of OBS stations, and provide more reliable picking of first arrivals. Apotentialliabilityofthismethodisthatlong-offsetrefractionarrivalsextractedbyinterferometry might not necessarily be head waves from deeper refraction interfaces. The extracted arrivals might be from a shallower interface, and so only supply redundant information about that portion of the subsurface. Nevertheless, our tomography example shows the value of these arrivals in reducing artefacts and increasing resolution in the tomogram.

We thank the 2016 sponsors of Center for Subsurface Imaging and Fluid Modeling (CSIM) at King Abdullah University of Science and Technology (KAUST) for their support. We also send the appreciation to KAUST Supercomputing Laboratory.

Wadi aquifers in Saudi Arabia historically have been recharged primarily by channel loss (infiltration) during floods. Historically, seasonal groundwater levels fluctuated from land surface to about 3 m below the surface. Agricultural irrigation pumping has lowered the water table up to 35 m below the surface. The geology surrounding the fluvial system at Wadi Qidayd consists of pelitic Precambrian rocks that contribute sediments ranging in size from mud to boulders to the alluvium. Sediments within the wadi channel consist of fining upward, downstream-dipping beds, causing channel floodwaters to pass through several sediment sequences, including several mud layers, before it can reach the water table. Investigation of the wadi aquifer using field observation, geological characterization, water-level monitoring, geophysical profiles, and a hypothetical model suggests a critical water level has been reached that affects the recharge of the aquifer. The wetted front can no longer reach the water table due to the water uptake in the wetting process, downstream deflection by the clay layers, and re-emergence of water at the surface with subsequent direct and diffusive evaporative loss, and likely uptake by deep-rooted acacia trees. In many areas of the wadi system, recharge can now occur only along the channel perimeter via fractured rocks that are in direct horizontal hydraulic connection to the permeable beds above and below the water table.

SUMMARY We propose a seismic scanning tunnelling macroscope (SSTM) that can detect subwavelength scatterers in the near-field of either the source or the receivers. Analytic formulas for the time reverse mirror (TRM) profile associated with a single scatterer model show that the spatial resolution limit to be, unlike the Abbe limit of λ/2, independent of wavelength and linearly proportional to the source-scatterer separation as long as the scatterer is in the near-field region. This means that, as the scatterer approaches the source, imaging of the scatterer with super-resolution can be achieved. Acoustic and elastic simulations support this concept, and a seismic experiment in an Arizona tunnel shows a TRM profile with super-resolution adjacent to the fault location. The SSTM is analogous to the optical scanning tunnelling microscopes having subwavelength resolution. Scaled to seismic frequencies, it is theoretically possible to extract 100 Hz information from 20 Hz data by the imaging of near-field seismic energy.

Least‐squares migration has been shown to improve image quality compared to the conventional migration method, but its computational cost is often too high to be practical. In this paper, we develop two numerical schemes to implement least‐squares migration with the reverse time migration method and the blended source processing technique to increase computation efficiency. By iterative migration of supergathers, which consist in a sum of many phase‐encoded shots, the image quality is enhanced and the crosstalk noise associated with the encoded shots is reduced. Numerical tests on 2D HESS VTI data show that the multisource least‐squares reverse time migration (LSRTM) algorithm suppresses migration artefacts, balances the amplitudes, improves image resolution and reduces crosstalk noise associated with the blended shot gathers. For this example, the multisource LSRTM is about three times faster than the conventional RTM method. For the 3D example of the SEG/EAGE salt model, with a comparable computational cost, multisource LSRTM produces images with more accurate amplitudes, better spatial resolution and fewer migration artefacts compared to conventional RTM. The empirical results suggest that multisource LSRTM can produce more accurate reflectivity images than conventional RTM does with a similar or less computational cost. The caveat is that the LSRTM image is sensitive to large errors in the migration velocity model.

Multisource migration of phase-encoded supergathers has shown great promise in reducing the computational cost of conventional migration. The accompanying crosstalk noise, in addition to the migration footprint, can be reduced by least-squares inversion. But the application of this approach to marine streamer data is hampered by the mismatch between the limited number of live traces/shot recorded in the field and the pervasive number of traces generated by the finite-difference modelling method. This leads to a strong mismatch in the misfit function and results in strong artefacts (crosstalk) in the multisource least-squares migration image. To eliminate this noise, we present a frequency-division multiplexing (FDM) strategy with iterative least-squares migration (ILSM) of supergathers. The key idea is, at each ILSM iteration, to assign a unique frequency band to each shot gather. In this case there is no overlap in the crosstalk spectrum of each migrated shot gather m(x, ωi), so the spectral crosstalk product m(x, ωi)m(x, ωj) =δi, j is zero, unless i=j. Our results in applying this method to 2D marine data for a SEG/EAGE salt model show better resolved images than standard migration computed at about 1/10th of the cost. Similar results are achieved after applying this method to synthetic data for a 3D SEG/EAGE salt model, except the acquisition geometry is similar to that of a marine OBS survey. Here, the speedup of this method over conventional migration is more than 10. We conclude that multisource migration for a marine geometry can be successfully achieved by a frequency-division encoding strategy, as long as crosstalk-prone sources are segregated in their spectral content. This is both the strength and the potential limitation of this method.

SUMMARY Inverting for the subsurface velocity distribution by refraction traveltime tomography is a well-accepted imaging method by both the exploration and earthquake seismology communities. A significant drawback, however, is that the recorded traces become noisier with increasing offset from the source position, and so accurate picking of traveltimes in far-offset traces is often prevented. To enhance the signal-to-noise ratio (SNR) of the far-offset traces, we present the theory of supervirtual refraction interferometry where the SNR of far-offset head-wave arrivals can be theoretically increased by a factor proportional to ; here, N is the number of receiver or source positions associated with the recording and generation of the head-wave arrival. There are two steps to this methodology: correlation and summation of the data to generate traces with virtual head-wave arrivals, followed by the convolution of the data with the virtual traces to create traces with supervirtual head-wave arrivals. This method is valid for any medium that generates head-wave arrivals recorded by the geophones. Results with both synthetic traces and field data demonstrate the feasibility of this method. There are at least four significant benefits of supervirtual interferometry: (1) an enhanced SNR of far-offset traces so the first-arrival traveltimes of the noisy far-offset traces can be more reliably picked to extend the useful aperture of the data, (2) the SNR of head waves in a trace that arrive later than the first arrival can be enhanced for accurate traveltime picking and subsequent inversion by later-arrival traveltime tomography, (3) common receiver-pair gathers can be analysed to detect the presence of diving waves in the first arrivals, which can be used to assess the nature of the refracting boundary, and (4) the source statics term is eliminated in the correlation operations so that the timing of the virtual traces is independent of the source excitation time. This suggests the possibility of applying this method to earthquake data recorded by receivers that are inline with the refraction paths and source locations.

Least-squares migration (LSM) has been shown to be able to produce high-quality migration images, but its computational cost is considered to be too high for practical imaging. We have developed a multisource least-squares migration algorithm (MLSM) to increase the computational efficiency by using the blended sources processing technique. To expedite convergence, a multisource deblurring filter is used as a preconditioner to reduce the data residual. This MLSM algorithm is applicable with Kirchhoff migration, wave-equation migration, or reverse time migration, and the gain in computational efficiency depends on the choice of migration method. Numerical results with Kirchhoff LSM on the 2D SEG/EAGE salt model show that an accurate image is obtained by migrating a supergather of 320 phase-encoded shots. When the encoding functions are the same for every iteration, the input/output cost of MLSM is reduced by 320 times. Empirical results show that the crosstalk noise introduced by blended sources is more effectively reduced when the encoding functions are changed at every iteration. The analysis of signal-to-noise ratio (S/N) suggests that not too many iterations are needed to enhance the S/N to an acceptable level. Therefore, when implemented with wave-equation migration or reverse time migration methods, the MLSM algorithm can be more efficient than the conventional migration method.

Time reversal mirrors can be used to backpropagate and refocus incident wavefields to their actual source location, with the subsequent benefits of imaging with high-resolution and super-stacking properties. These benefits of time reversal mirrors have been previously verified with computer simulations and laboratory experiments but not with exploration-scale seismic data. We now demonstrate the high-resolution and the super-stacking properties in locating seismic sources with field seismic data that include multiple scattering. Tests on both synthetic data and field data show that a time reversal mirror has the potential to exceed the Rayleigh resolution limit by factors of 4 or more. Results also show that a time reversal mirror has a significant resilience to strong Gaussian noise and that accurate imaging of source locations from passive seismic data can be accomplished with traces having signal-to-noise ratios as low as 0.001. Synthetic tests also demonstrate that time reversal mirrors can sometimes enhance the signal by a factor proportional to the square root of the product of the number of traces, denoted as N and the number of events in the traces. This enhancement property is denoted as super-stacking and greatly exceeds the classical signal-to-noise enhancement factor of . High-resolution and super-stacking are properties also enjoyed by seismic interferometry and reverse-time migration with the exact velocity model.

Refraction surveys are a well-established method of imaging subsurface velocities, both in terms of the deep crustal structure at global scales and in the shallow near surface. These surveys generally involve deploying an array of receivers on the surface (or water bottom) and recording arrivals from a seismic source initiated at or near the surface.

Formulas are derived that relate the strength of the crosstalk term in multisource imaging to the variance of the random static shifts in supergathers. Analysis reveals that the crosstalk term is exponentially reduced with increasing variance of the static shift and increasing source frequency. Applying n types of statics to the traces, e.g., for n = 3 types of statics such as random time shifts, location perturbations, or amplitude changes, will lead to an n-fold reduction in the crosstalk strength. In the case of random source locations, comparison to the Quasi-Monte Carlo migration method suggests the possibility of the crosstalk term becoming stronger with increasing depth of the imaging point. These conclusions should be valid for migration, least squares migration, or waveform inversion of multisource supergathers.

An antialiasing formula has been derived for interferometric redatuming of seismic data. More generally, this formula is valid for numerical implementation of the reciprocity equation of correlation type, which is used for redatuming, extrapolation, interpolation, and migration. The antialiasing condition can be, surprisingly, more tolerant of a coarser trace sampling compared to the standard antialiasing condition. Numerical results with synthetic vertical seismic profile (VSP) data show that interferometry artifacts are effectively reduced when the antialiasing condition is used as a constraint with interferometric redatuming.

A recently developed time-domain multiscale waveform tomography (MWT) method is applied to synthetic and field marine data. Although the MWT method was already applied to synthetic data, the synthetic data application leads to a development of a hybrid method between waveform tomography and the salt flooding technique commonly use in subsalt imaging. This hybrid method can overcome a convergence problem encountered by inversion with a traveltime velocity tomogram and successfully provides an accurate and highly resolved velocity tomogram for the 2D SEG/EAGE salt model. In the application of MWT to the field data, the inversion process is carried out using a multiscale method with a dynamic early-arrival muting window to mitigate the local minima problem of waveform tomography and elastic effects. With the modified MWT method, reasonably accurate results as verified by comparison of migration images and common image gathers were obtained. The hybrid method with the salt flooding technique is not used in this field data example because there is no salt in the subsurface according to our interpretation. However, we believe it is applicable to field data applications.

Least-squares migration (LSM) is a linearized waveform inversion for estimating a subsurface reflectivity model that, relative to conventional migration, improves spatial resolution of migration images. The cost, however, is high because LSM typically requires 10 or more iterations, which is at least 20 times more than the CPU cost of conventional migration. To alleviate this expense, we offer a deblurring filter that can be used in a regularization scheme or a preconditioning scheme to give acceptable LSM images with less than one-third the cost of the standard LSM method. Our results in applying deblurred LSM to synthetic data and field data support this claim. In particular, a Marmousi2 model test shows that the data residual for preconditioned deblurred LSM decreases rapidly in the first iteration, which is equivalent to 10 or more iterations of LSM. Empirical results suggest that regularized deblurred LSM after three iterations is equivalent to about 10 iterations of LSM. Applying deblurred LSM to 2D marine data gives a higher-resolution image compared to those from migration or LSM with three iterations. These results suggest that LSM combined with a deblurring filter allows LSM to be a fast, practical tool for improved imaging of complicated structures.

This efficient multiscale method for time-domain waveform tomography incorporates filters that are more efficient than Hamming-window filters. A strategy for choosing optimal frequency bands is proposed to achieve computational efficiency in the time domain. A staggered-grid, explicit finite-difference method with fourth-order accuracy in space and second-order accuracy in time is used for forward modeling and the adjoint calculation. The adjoint method is utilized in inverting for an efficient computation of the gradient directions. In the multiscale approach, multifrequency data and multiple grid sizes are used to overcome somewhat the severe local minima problem of waveform tomography. The method is applied successfully to 1D and 2D heterogeneous models; it can accurately recover low- and high-wavenumber components of the velocity models. The inversion result for the 2D model demonstrates that the multiscale method is computationally efficient and converges faster than a conventional, single-scale method.

Imaging of subsalt sediments is a challenge for traditional migration methods such as Kirchhoff and one-way wave-equation migration. Consequently, the more accurate two-way method of reverse-time migration (RTM) is preferred for subsalt imaging, but its use can be limited by high computation cost. To overcome this problem, a 3D target-oriented reverse-time datuming (RTD) method is presented, which can generate redatumed data economically in target areas beneath complex structures such as salt domes. The redatumed data in the target area then can be migrated inexpensively using a traditional migration method. If the target area is much smaller than the acquisition area, computation costs are reduced significantly by the use of a novel bottom-up strategy to calculate the extrapolated Green’s functions. Target-oriented RTD is tested on 2D and 3D SEG/EAGE synthetic data sets and a 3D field data set from the Gulf of Mexico. Results show that target-oriented RTD combined with standard migration can image sediments beneath complex structures accurately with much less calculation effort than full volume RTM. The requirement is that the area over the target zone is smaller than that of the acquisition survey.

Deadly mine collapses have recently occurred in West Virginia (January 2007, two miners killed); Russia (March 2007, 106 miners killed); Utah (August 2007, six miners and another three from the rescue team killed); Colombia (October 2007, 24 miners killed); and many other places. Locating trapped miners as soon as the collapse occurs will save lives and avoid dangerous searches in the wrong places.

Surface waves are a form of coherent noise that can obscure valuable reflection information in exploration records. It is sometimes difficult to eliminate these surface waves by traditional filtering approaches, such as an [Formula: see text] filter, without damaging the useful signals. As a partial remedy, we propose an interferometric method to predict and subtract surface waves in seismic data. The removal of surface waves by the proposed interferometric method consists of three steps: (1) remove most of the surface waves by a nonlinear local filter; (2) predict the residual surface waves by the interferometric method; (3) separate the residual surface waves from the result of step 2 by a nonlinear local filter, and remove the residual surface waves by a matched filter from the result of step 1. Field data tests for 2D and 3D data show that the method effectively suppresses surface waves and preserves the reflection information. Results suggest that the effectiveness of this method is sensitive to the parameter selection of the nonlinear local filter.

Interferometric migration of free-surface multiples in vertical-seismic-profile (VSP) data has two significant advantages over standard VSP imaging: (1) a significantly larger imaging area compared to migrating VSP primaries and (2) less sensitivity to velocity-estimation and static errors than other methods for migration of multiples. In this paper, we present a 3D wave-equation interferometric migration method that efficiently images VSP free-surface multiples. Synthetic and field data results confirm that a reflectivity image volume, comparable in size to a 3D surface seismic survey around the well, can be computed economically. The reflectivity image volume has less fold density and lower signal-to-noise ratio than that obtained by a conventional 3D surface seismic survey because of the relatively weak energy of multiples and the limited number of geophones in the well. However, the efficiency of this method for migrating VSP multiples suggests that it might sometimes be a useful tool for 4D seismic monitoring where reflectivity images can be computed quickly for each time-lapse survey.

An interferometric form of Fermat’s principle and traveltime tomography is used to invert ground-penetrating radar (GPR) data for the subsurface velocity distribution. The input data consist of GPR traveltimes of reflections from two buried interfaces, [Formula: see text] (reference) and [Formula: see text] (target), where the data are excited and recorded by GPR antennas at the surface. Fermat’s interferometric principle is then used to redatum the surface transmitters and receivers to interface [Formula: see text] so the associated reflection traveltimes correspond to localized transit times between interfaces [Formula: see text] and [Formula: see text]. The overburden velocity model above interface [Formula: see text] is not required. The result after tomographic inversion is a high-resolution estimate of the velocity between interfaces [Formula: see text] and [Formula: see text] that does not depend on the velocity model above interface [Formula: see text]. A motivation for introducing interferometric traveltime tomography is that typical layer-stripping approaches will see the slowness error increase with depth as the layers are inverted. This suggests that near-surface statics errors are propagated and amplified with depth. In contrast, the interferometric traveltime tomography method largely eliminates statics errors by taking the difference between reflection events that emanate from neighboring layer interfaces. Slowness errors are not amplified with depth. However, the method is sensitive to the estimation accuracy for the geometry of the reference interface. Both synthetic and real field data are used successfully to validate the effectiveness of this interferometric technique.

Multiples contain valuable information about the subsurface, and if properly migrated can provide a wider illumination of the subsurface compared to imaging with VSP primary reflections. In this paper we review three different methods for migrating multiples. The first method is model‐based, and it is more sensitive to velocity errors than primary migration; the second method uses a semi‐natural Green's function for migrating multiples, where part of the traveltimes are computed from the velocity model, and part of the traveltimes (i.e., natural traveltimes) are picked from the data to construct the imaging condition for multiples; the third method uses cross‐correlation of traces. The last two methods are preferred in the sense that they are significantly less sensitive to velocity errors and statics because they use “natural data” to construct part of the migration imaging conditions. Compared with the interferometric (i.e., crosscorrelation) imaging method the semi‐natural Green's function method is more computationally efficient and is sometimes less prone to migration artifacts. Numerical tests with 2‐D and 3‐D VSP data show that a wider subsurface coverage, higher‐fold and more balanced illumination of the subsurface can be achieved with multiple migration compared with migration of primary reflections only. However, there can be strong interference from multiples with different orders or primaries when multiples of high order are migrated. One possible solution is to filter primaries and different orders of multiples before migration, and another possible solution is least squares migration of all events. A limitation of multiple migration is encountered for subsalt imaging. Here, the multiples must pass through the salt body more than twice, which amplifies the distortion of the image.

Seismic resonant multiples, which can yield resolutions more than twice the classical resolution limit, are used for far-field super-resolution imaging., We demonstrate for the first time that seismic resonant multiples, usually considered as noise, can be used for super-resolution imaging in the far-field region of sources and receivers. Tests with both synthetic data and field data show that resonant multiples can image reflector boundaries with resolutions more than twice the classical resolution limit. Resolution increases with the order of the resonant multiples. This procedure has important applications in earthquake and exploration seismology, radar, sonar, LIDAR (light detection and ranging), and ultrasound imaging, where the multiples can be used to make high-resolution images.

We present a least-squares reverse time migration (LSRTM) method using a factorization-free priorconditioning approach to overcome the low signal-to-noise (SNR) problem arising out of using severely undersampled data. Priorconditioning is a technique where the prior information is incorporated directly into the forward operator and into the solution space of the problem. The prior information that is used in this work is that the inverted reflectivity is sparse in the radon domain. The proposed method is factorization-free since the forward mapping is defined through the action of a sparse operator on a vector. The priorconditioning method is shown to produce reliable images with good SNR and free from aliasing artifacts when using very sparse shots for both synthetic and field data.

We present a method for inverting seismic reflections using full-waveform inversion (FWI) with inaccurate starting models. For a layered medium, near-offset reflections (with zero angle of incidence) are unlikely to be cycle-skipped regardless of the low-wavenumber velocity error in the initial models. Therefore, we use them as a starting point for FWI, and the subsurface velocity model is then updated during the FWI iterations using reflection wavepaths from varying offsets that are not cycle-skipped. To enhance low-wavenumber updates and accelerate the convergence, we take several passes through the non-linear GaussSeidel iterations, where we invert traces from a narrow range of near offsets and finally end at the far offsets. Every pass is followed by applying smoothing to the cumulative slowness update. The smoothing is strong at the early stages and relaxed at later iterations to allow for a gradual reconstruction of the subsurface model in a multiscale manner. Applications to synthetic and field data, starting from inaccurate models, show significant low-wavenumber updates and flattening of common-image gathers after many iterations.

We propose an interferometric least-squares migration method that can significantly reduce migration artifacts due to statics and errors in the near-surface velocity model. We first choose a reference reflector whose topography is well known from the, e.g., well logs. Reflections from this reference layer are correlated with the traces associated with reflections from deeper interfaces to get crosscorrelograms. These crosscorrelograms are then migrated using interferometric least-squares migration (ILSM). In this way statics and velocity errors at the near surface are largely eliminated for the examples in our paper.

We propose sparse least-squares reverse time migration (LSRTM) using seislets as a basis for the reflectivity distribution. This basis is used along with a dip-constrained preconditioner that emphasizes image updates only along prominent dips during the iterations. These dips can be estimated from the standard migration image or from the gradient using plane-wave destruction filters or structural tensors. Numerical tests on synthetic datasets demonstrate the benefits of this method for mitigation of aliasing artifacts and crosstalk noise in multisource least-squares migration.