Wavelet Stretch Research Articles

Advanced Deep Learning for microseismic events prediction for hydraulic fracture treatment via Continuous Wavelet Transform

Understanding hydraulic fracture propagation events would enable operators to estimate the fracture geometry, understand formation/fracture interactions, and estimate the total microseismic events cloud. This paper introduces a new method for understanding the dynamic fracture propagation events using a signal processing technique. Detection of the dynamic fracture event using continuous wavelet transform (CWT) is used to train a deep learning model with microseismic events to predict this cloud for every hydraulic fracture job. The deep learning model is used for the same formation. The CWT for treatment pressure is a convolution process that involves the application of a wavelet signal to the acquired pressure signal in a continuous manner. The wavelet is stretched, or compressed, and is moved along the acquired pressure signal, acting as a generic microscope to highlight small changes in the treating pressure signal. Using CWT as a mathematical microscope to discern diverse signals has been adopted by numerous engineering and medical applications since the 1990s.A normalized CWT scalogram for fracture-treating pressure is thus created as a unique representation of each fracture propagation mode in the hydraulic fracturing job. The new technique is calibrated with several fracture simulation runs and real field data using the previous techniques like moving reference point (MRP). That previous technique which has been previously validated through fracture simulation, is used as a calibration for the new normalized CWT scalogram technique. The technique was finally calibrated by comparing it with micro-seismic events recorded by the Marcellus Shale Energy and Environment Laboratory (MSEEL).The results of this comparison were exceptionally accurate. It can be used to train a deep learning model that can be used to estimate the cloud of microseismic events for hydraulic fracture jobs in Marcellus shale. The new deep learning framework uses CWT to convert the treating pressure during hydraulic fracture propagation to dimensionless representation (normalized CWT scalogram) which acts as a mathematical microscope for the fracture treating pressure on different wavelet stretch or compressions, enabling the detection of the major changes in the treating pressure then train the deep learning model to estimate microseismic events cloud.

Inversion-based non-stationary normal moveout correction along with prestack high-resolution processing

Seismic wave propagated in viscoelastic media always suffers from amplitude attenuation and velocity dispersion. It not only reduces seismic resolution, but also affects the quality of traditional stationary normal moveout (NMO) correction by distorting wavelets. To address both issues of attenuation and the wavelet stretch simultaneously, we present a non-stationary NMO correction method along with prestack high-resolution processing. Instead of invariant wavelets, spatiotemporal variant wavelets are adopted via an inversion scheme for implementing NMO correction and removing the attenuation effect. In the inversion, two goal-oriented constraints including the temporal sparsity and horizontal coherence of prestack reflectivity are combined into a structural sparsity constraint to impose on the data misfit between the synthetic non-stationary common-midpoint (CMP) gather and the observed one. Both constraints approximately represented by a convex l2,1 norm limit the optimal solution to structural sparsest reflectivity, contributing to construct the flattened corrected gather with invariant wavelets. Synthetic and field data examples illustrate performances of the inversion-based non-stationary NMO correction. Moreover, we compared the corrected gathers obtained by the improved approach with those by the traditional and inversion-based stationary NMO correction approaches, and took the gabor spectrum as a quality control. The proposed method produces more flatten events without stretch and gives consistent frequency components, thus improving prestack seismic resolution.

High-resolution imaging: a wavelet stretch correction in de-absorption prestack time migration approach

ABSTRACT The wavelet stretch effects in pre-stack migration, which is similar to that of NMO stretch, distort the frequency spectrums and taper the high-frequency components at far offsets. The long-offset migrated results in conventional migration methods are muted, resulting in a loss of effective information about the subsurface investigation. Only short-offset data are used to obtain a high-resolution stacked image. However, long-offsets that correspond to high-incident-angle data are critical to velocity inversion and amplitude versus offset (AVO) analysis when estimating lithology and fluid product. Furthermore, attenuation of seismic waves due to the anelasticity of the subsurface medium will cause dissipation of seismic energy and loss of high frequencies, thus broadening the propagating wavelet and degrading the resolution of imaging. Both the wavelet stretch and the absorption will taper the high-frequency components. We developed a wavelet stretch correction and absorption compensation scheme during the migration process by shrinking the wavelet when applying the imaging condition, which matches the stretch effect caused by migration. The correction factor is derived based on the incidence angle associated with many factors. Moreover, the de-absorption pre-stack time migration (QPSTM) is implemented on the effective Q model that makes the Q modelling become easier. Both the application of synthetic and real data examples demonstrate that the proposed anti-stretch QPSTM can be particularly useful recovering the high-frequency components caused by anelasticity and wavelet stretch, and obtain higher-resolution images.

Reverse time migration of transmitted wavefields for salt boundary imaging

The imaging of steep salt boundaries has received much attention with the advent of improved wider azimuth acquisition designs and advanced imaging techniques such as reverse time migration (RTM), for example. However, despite these advancements in capability, there are cases in which the salt boundary is either poorly illuminated or completely absent in the migrated image. To provide a solution to this problem, we have developed two RTM methods for imaging salt boundaries, which use transmitted wavefields. In the first technique, downgoing waves, typically recorded in walkaway vertical seismic profile surveys, are used to image the salt flank via the generation of aplanatic isochrones. This image can be generated in the absence of an explicit interpretation of the salt flank using dual migration velocity models, as demonstrated on a 3D walkaway field data set from the Gulf of Mexico. In the second technique, we extend the basic theory to include imaging of upgoing source wavefields, which are transmitted at the base salt from below, as acquired by a surface acquisition geometry. This technique has similarities to the prism-imaging method, yet it uses transmitted instead of reflected waves at the salt boundary. Downgoing and upgoing methods are shown to satisfactorily generate an image of the salt flank; however, transmission imaging can create artifacts if reflection arrivals are included in the migration or the acquisition geometry is limited in extent. Increased wavelet stretch is also observed due to the higher transmission coefficient. An important benefit of these methods is that transmission imaging produces an opposite depth shift to errors in the velocity model compared with imaging of reflections. When combined with conventional seismic reflection surveys, this behavior can be used to provide a constraint on the accuracy of the salt and/or subsalt velocities.

Least-squares migration — Data domain versus image domain using point spread functions

Conventional amplitude inversion assumes that the migrated image preserves relative-amplitude information. However, illumination effects caused by complex geologic settings, undersampled acquisition geometry, and limited recording aperture pose a challenge to even the most advanced imaging algorithms. In addition, standard depth-migration images can suffer from lack of resolution caused by wavelet stretch, attenuation, and suboptimal deghosting. Least-squares migration (LSM) can mitigate many of these problems and produce better resolved migration images suitable for AVO inversion. However, whether formulated in the data domain or the image domain, LSM is an inversion algorithm and is sensitive to inaccuracies in the source wavelet, velocity model, data preprocessing, and the propagator used. Practical considerations to mitigate these problems under nonideal conditions and cost-reduction strategies differ between the data- and image-domain formulations. The relative merits of each approach are evaluated by using example inversions for complex synthetic models, including free-surface ghost and attenuation effects. When a data-domain implementation of LSM is considered necessary, the image-domain implementation should be considered at the same time, especially when targeting localized reservoir targets under complex overburdens. Application of image-domain least-squares migration on a Gulf of Mexico field data set produces significant improvements in resolution and event continuity in the subsalt target region.

Shallow water demultiple using hybrid multichannel prediction

This paper presents an extension of our previous effort on multiple attenuation in shallow water environment. While our previous workflow, termed as Shallow Water Demultiple (SWD), is robust in suppressing water-layer related multiples (WLRMs) with shallow seafloor, it faces difficulties when the seafloor is too shallow and complex because of the near offset gap related to acquisition. The wavelet stretch resulted from near offset extrapolation causes spectral distortion in multiple model from SWD which leads to sub-optimized subtraction result in shallow part of the data. The new method is a hybrid approach in which shallow WLRMs are handled by using the Greenâ??s function of the seafloor primary reflections, while the rest of the multiples are handled by SWD. The Greenâ??s function in this case is derived from auto-picking the traveltime of the multichannel prediction operator estimated from SWD. The approach combines the strengths of SWD and model-based methods. We show that the multichannel prediction operator estimated in SWD can be used as an accurate kinematic representation of seafloor reflection with high signal-to-noise (S/N) ratio. With this operator, the traveltime of the seafloor event can then be automatically estimated. Making use of this traveltime information, the Green's functions of the water-layer primary reflections can be modeled for tackling the shallow peg-lag multiples that have difficulty handled by SWD. We show the application of our hybrid method for suppressing shallow water multiples on field data acquired from offshore Australia

Stretch-free migration imaging condition

Prestack migration has angle-dependent wavelet stretch effects, which lowers the image resolution at large reflection angles. Most current stretch correction methods operate on the migrated images. We develop a new stretch-free imaging condition, which does a shrink-and-shift operation on the extracted propagation wavelet after extrapolation, but before the imaging condition is applied. The algorithm is illustrated with the excitation amplitude imaging condition; the new images show successful stretch corrections over wide angle apertures, and preserve amplitude and phase.

Wavelet distortion correction due to domain conversion

Converted-wave (PS) data, when converted to PP time, develop time- and location-varying compression of the seismic wavelet due to a variable subsurface [Formula: see text] [Formula: see text]. The time-dependent compression distorts the wavelet in a seismic trace. The lack of a consistent seismic wavelet in a domain-converted PS volume can eventually lead to an erroneous joint PP/PS inversion result. Depth-converted seismic data also have wavelet distortion due to velocity-dependent wavelet stretch. A high value of seismic velocity produces more stretch in a seismic wavelet than a low value. Variable wavelet stretch renders the depth data unsuitable for attribute analysis. A filtering scheme is proposed that corrects for distortion in seismic wavelets due to domain conversions (PS to PP time and time-to-depth) of seismic data in an amplitude-preserving manner. The method uses a Fourier scaling theorem to predict the seismic wavelet in the converted domain and calculates a shaping filter for each time/depth sample that corrects for the distortion in the wavelet. The filter is applied to the domain-converted data using the method of nonstationary filtering. We provide analytical expressions for the squeeze factor [Formula: see text] that is used to predict the wavelet in the converted domain. The squeeze factor [Formula: see text] for PS to PP time conversion is a function of the subsurface [Formula: see text] whereas for PP time-to-depth conversion [Formula: see text] is dependent on subsurface P-wave velocity. After filtering, the squeezed wavelets in domain-converted PS data appear to have resulted from a constant subsurface [Formula: see text], which we denote as [Formula: see text]. Similarly, the filtered depth-converted data appear to have resulted from a constant subsurface P-wave velocity [Formula: see text].

Wavelets approach to time-varying functional differential equations

The analysis of time-varying functional differential equations via wavelets is proposed in this paper. At first, based upon some useful properties of wavelets, a special product matrix and a related coefficient matrix are applied to deal with the time-varying part. Then the wavelet stretch matrix is introduced to deal with the functional part. The unknown wavelet coefficient matrix will be found in the generalized Lyapunov equation. The local property of wavelets is completely applied to shorten the calculation process in the task.

Ray-based stochastic inversion of prestack seismic data for improved reservoir characterization

Trace inversion for reservoir parameters is affected by angle averaging of seismic data and wavelet distortion on the migration image. In an alternative approach to stochastic trace inversion, the data are inverted prestack before migration using 3D dynamic ray tracing. This choice makes it possible to interweave trace inversion with Kirchhoff migration. The new method, called ray-based stochastic inversion, is a generalization of current amplitude versus offset/amplitude versus angle (AVO/AVA) inversion techniques. The new method outperforms standard stochastic inversion techniques in cases of reservoir parameter estimation in a structurally complex subsurface with substantial lateral velocity variations and significant reflector dips. A simplification of the method inverts the normal-incidence response from reservoirs with approximately planar layering at the subsurface target locations selected for inversion. It operates along raypaths perpendicular to the reflectors, the direction that offers optimal resolution to discern layering in a reservoir. In a test on field data from the Gulf of Mexico, reservoir parameter estimates obtained with the simplified method, the estimates found by conventional stochastic inversion, and the actual values at a well drilled after the inversion are compared. Although the new method uses only 2% of the prestack data, the result indicates it improves accuracy on the dipping part of the reservoir, where conventional stochastic inversion suffers from wavelet stretch caused by migration.

Improving lateral and vertical resolution of seismic images by correcting for wavelet stretch in common-angle migration

Long-offset or high-incident-angle seismic reflections provide us with improved velocity resolution, better leverage against multiples, less contamination by ground roll, and information that is often critical when estimating lithology and fluid product. Unfortunately, high-incident-angle seismic reflections suffer not only from nonhyperbolic moveout but also from wavelet stretch during imaging, resulting in lower-resolution images that mix the response from adjacent lithologies. For an arbitrary acoustic medium, wavelet stretch from prestack migration depends only on the cosine of the reflection angle, such that the amount of wavelet stretch will be the same for all samples of a common-reflection-angle migrated trace. Thus, we are able to implement a wavelet stretch correction by applying a simple stationary spectral shaping operation to common-angle migrated traces. We obtain such traces directly by a prestack Kirchhoff migration algorithm. Correcting for stretch effectively increases the fold of imaged data, far beyond that achieved in conventional migration, resulting in improved signal-to-noise ratio of the final stacked section. Increasing the fidelity of large incident angles results in images with improved vertical and lateral resolution and with increased angular illumination, valuable for amplitude variation with angle (AVA) and amplitude variation with offset (AVO) analysis. Finally, such large-angle images are more sensitive to and therefore provide increased leverage over errors in velocity and velocity anisotropy. These ideas were applied to prestack time migration on seismic data from the Fort Worth basin, in Texas.

Improving seismic data quality in the Gippsland Basin (Australia)

Deep seismic exploration in the Gippsland Basin is hindered by strong noise below the Latrobe Group coal sequence. The reflectivity method provides a means for constructing detailed and accurate synthetic seismograms, often from little more than a partial sonic log. The noise contributions to the synthetics can then be interpreted using additional synthetics computed from variations upon the depth model and by exercising control over the wave types modeled. This approach revealed three types of persistent noise in progressively deeper parts of the subcoal image: (1) mode‐converted interbed multiples (generated within the coal sequence), (2) S-wave reflections and long‐period multiples (generated between the coal sequence and the Miocene carbonates), and (3) surface‐related multiples. The noise interpretation can also be performed upon semblance analyses of the elastic synthetics to guide a velocity analysis away from a well. This procedure helped to avoid picking the interformation long‐period multiples, whose stacking velocities were only 5 to 10% below those of the weak target zone primaries. An improve subcoal image was obtained by making full use of the versatile noise suppression offered by a τ-p domain processing stream. By separating the strong linear events at the far offsets, it is possible to stack a larger portion of the target zone reflections, provided hyperbolic velocity filtering (HVF) is applied to suppress the transform artifacts. Hyperbolic velocity filtering can be incorporated into a point‐source τ-p transform to suppress S-wave reflections and guided waves while preserving plane‐wave amplitudes to assist the subsequent deconvolution of the mode‐converted interbed multiples. Stacking in the τ-p domain is achieved using an elliptical moveout correction that reduces wavelet stretch and approximates the exact reflection traveltime better than NMO. Two regional seismic lines were reprocessed in this manner and cointerpreted with the modeling studies performed at nearby wells to avoid the noise events that still remained. Several new events appeared in the immediate target zone, passing the low‐frequency character expected following transmission through a coal sequence.

Resolution in seismic imaging: Is it all a matter of perspective?

Classical wave‐equation migration of seismic data combines both acoustic wave propagation and an imaging condition to produce a subsurface section. Integral methods implicitly combine high‐frequency ray asymptotics of the wave equation with a ray‐dependent imaging condition. The inherent limits on subsurface resolution imposed by acoustic‐wave generation (e.g., source bandwidth), subsurface propagation (e.g., attenuation), and recording (e.g., receiver arrays and aperture) have been studied extensively. This article studies how (or whether) the choice of imaging condition affects resolution. We will see in particular that the phenomenon of migration wavelet stretch, wherein the vertical resolution of steeply dipping reflectors is preferentially lowered by migration, is neither an artifact of an imaging condition nor a reduction of resolution on steeply dipping reflectors. In fact, we will discover that in seismic imaging of prestack data, steeply dipping reflectors are ultimately better resolved than horizontal beds. Finally, the concept of “superresolution” from seismic inversion, a generalization of thin‐bed tuning analysis, is explored and explicated with a simple example of multiple reflections.

Processing Through to Stack in the τ-ρ Domain

Improved seismic sections can be obtained by processing through to stack in the τ-ρ domain. This approach provides significant improvements over routine processing, using hyperbolic velocity filtering (HVF) to suppress the τ-ρ transform artefacts. HVF is a form of time and offset variant filtering that only allows each point in the t-x domain to contribute to a small number of p traces during a τ-ρ transform. HVF can be incorporated into a point-source τ-ρ transform, extending previous implementations that applied HVF during the slant stack. This assists gap deconvolution by providing plane-wave amplitudes in a τ-ρ gather where reverberations are exactly periodic along p traces and where S-wave reflections and long period multiples are suppressed. The τ-ρ domain also offers improved velocity analysis through the use of wide-angle reflections and because primary reflection ellipses never cross each other. The τ-ρ transform of a CMP gather can then be stacked with further advantages resulting from the elliptical moveout correction, which minimises wavelet stretch and approximates the exact reflection traveltime better than NMO. Examples from the Carnarvon and Gippsland Basins confirm that the cumulative theoretical advantages of τ-ρ domain processing are achieved in practice.

Amplitude‐versus‐offset measurement errors in a finely layered medium

Recently Spratt (1987) showed how amplitude‐versus‐offset analysis (AVO) can be sensitive to small residual velocity errors. However, even when the velocity is determined perfectly, serious AVO distortions remain due to normal‐moveout stretch, differential tuning as a function of offset, spherical divergence, and source and receiver directivity patterns. I have found that all of these errors can be expanded in a Taylor series about the zero‐offset event time, assuming it is much larger than the wavelet width. The first term of this series represents the residual velocity error term found by Spratt, while the second term encompasses the remaining effects mentioned. In practice, either term can be larger than the underlying amplitude variations being estimated. For example, Ricker wavelet stretch leads to a peak AVO error which is 61 percent of the peak zero‐offset reflectivity, even though the velocity field is uniform and correct. This result is independent of the wavelet frequency, and the range of incidence angles used in the analysis. Positive gradients in moveout velocity amplify this error, while narrowband filtering of the data prior to AVO analysis greatly widens its temporal extent. Aligning a particular event with static shifts instead of normal‐moveout correction can eliminate stretch, but not differential tuning error, in a finely layered target zone whose wavelets overlap.