Characterizing the degree of amplitude‐variation‐with‐offset nonlinearity in seismic physical modelling reflection data

The nonlinearity of the seismic amplitude‐variation‐with‐offset response is investigated with physical modelling data. Nonlinearity in amplitude‐variation‐with‐offset becomes important in the presence of large relative changes in acoustic and elastic medium properties. A procedure for pre‐processing physical modelling reflection data is enacted on the reflection from a water‐plexiglas boundary. The resulting picked and processed amplitudes are compared with the exact solutions of the plane‐wave Zoeppritz equations, as well as approximations that are first, second, and third order in , , and . In the low angle range of 0°–20°, the third‐order plane‐wave approximation is sufficient to capture the nonlinearity of the amplitude‐variation‐with‐offset response of a liquid‐solid boundary with , , and ρ contrasts of 1485–2745 m/s, 0–1380 m/s, and 1.00–1.19 gm/cc respectively, to an accuracy value of roughly 1%. This is in contrast to the linear Aki–Richards approximation, which is in error by as much as 25% in the same angle range. Even‐order nonlinear corrective terms are observed to be primarily involved in correcting the angle dependence of , whereas the odd‐order nonlinear terms are involved in determining the absolute amplitude‐variation‐with‐offset magnitudes.     

Effects of variations in fluid properties and fracture geometry on dispersion,anisotropy, and reflection in media with planar fractures

Our objective was to discover the effect of variations in fluid properties and fracture geometry on the velocity of seismic wave propagation in fluid‐saturated media with parallel planar fractures. We used numerical models calculated by analytical solutions to examine the behaviour of P‐wave phase velocity dispersion in the normal direction to layering, in non‐porous and porous media with planar fractures. We also examined the anisotropy of low frequency phase and group velocities of fast and slow P‐waves and angular‐dependent reflection coefficients in media with planar fractures, under conditions of saturation by fluids with varying bulk moduli, densities, and fracture apertures. We defined several parametre , , and characterising dispersion, characterising anisotropy, characterising the difference between fast and slow modes, and R₀ and characterising reflection. Our results show that the behaviour of dispersion shows wider stopbands in the case of gas saturation. Concavity indicator of dispersion for gas saturation was greater than that for liquid saturation and is usually greater than one. Anisotropy is more sensitive to bulk modulus contrast than to density contrast between the solid and the fluid, and is more sensitive to density contrast than to bulk modulus contrast. The case of gas saturation usually had a greater negative R₀ and a greater value of compared with those of brine and heavy and light oil saturations. Our results are helpful in distinguishing fluid types saturating geophysical fractures and estimating the aperture and spacing of planar fractures. In seismic exploration, bulk modulus and fluid density can provide useful information in distinguishing among brine, oil, and gas; fracture geometry is important to estimate the permeability of reservoirs.     

Time‐lapse pre‐stack seismic data registration and inversion for CO2 sequestration study at Cranfield

Pre‐stack seismic data are indicative of subsurface elastic properties within the amplitude versus offset characteristic and can be used to detect elastic rock property changes caused by injection. We perform time‐lapse pre‐stack 3‐D seismic data analysis for monitoring sequestration at Cranfield. The time‐lapse amplitude differences of Cranfield datasets are found entangled with time‐shifts. To disentangle these two characters, we apply a local‐correlation‐based warping method to register the time‐lapse pre‐stack datasets, which can effectively separate the time‐shift from the time‐lapse seismic amplitude difference without changing the original amplitudes. We demonstrate the effectiveness of our registration method by evaluating the inverted elastic properties. These inverted time‐lapse elastic properties can be reliably used for monitoring plumes.     

Non‐convex compressed sensing with frequency mask for seismic data reconstruction and denoising

Compressed Sensing has recently proved itself as a successful tool to help address the challenges of acquisition and processing seismic data sets. Compressed sensing shows that the information contained in sparse signals can be recovered accurately from a small number of linear measurements using a sparsity‐promoting regularization. This paper investigates two aspects of compressed sensing in seismic exploration: (i) using a general non‐convex regularizer instead of the conventional one‐norm minimization for sparsity promotion and (ii) using a frequency mask to additionally subsample the acquired traces in the frequency‐space () domain. The proposed non‐convex regularizer has better sparse recovery performance compared with one‐norm minimization and the additional frequency mask allows us to incorporate a priori information about the events contained in the wavefields into the reconstruction. For example, (i) seismic data are band‐limited; therefore one can use only a partial set of frequency coefficients in the range of reflections band, where the signal‐to‐noise ratio is high and spatial aliasing is low, to reconstruct the original wavefield, and (ii) low‐frequency characteristics of the coherent ground rolls allow direct elimination of them during reconstruction by disregarding the corresponding frequency coefficients (usually bellow 10 Hz) via a frequency mask. The results of this paper show that some challenges of reconstruction and denoising in seismic exploration can be addressed under a unified formulation. It is illustrated numerically that the compressed sensing performance for seismic data interpolation is improved significantly when an additional coherent subsampling is performed in the domain compared with the domain case. Numerical experiments from both simulated and real field data are included to illustrate the effectiveness of the presented method.     

Experimental quantification of the seismoelectric transfer function and its dependence on conductivity and saturation in loose sand

Under certain circumstances, seismic propagation within porous media may be associated to the conversion of mechanical energy to electromagnetic energy, which is known as a seismo‐electromagnetic phenomenon. The propagation of fast compressional P‐waves is more specifically associated to the manifestations of a seismoelectric field linked to the fluid flows within the pores. The analysis of seismoelectric phenomena, which requires the combination of the theory of electrokinetics and Biot's theory of poroelasticity, provides us with transfer function that links the coseismic seismoelectric field E to the seismic acceleration . To measure the transfer function, we have developed an experimental setup enabling seismoelectric laboratory observation in unconsolidated quartz sand within the kilohertz range. The investigation focused on the impact of fluid conductivity and water saturation over the coseismic seismoelectric field. During the experiment, special attention was given to the accuracy of electric field measurements. We concluded that, to obtain a reliable estimate of the electric field amplitude, the dipole from which the potential differences are measured should be of much smaller length than the wavelength of the propagating seismic field. Time‐lapse monitoring of the seismic velocities and seismoelectric transfer functions were performed during imbibition and drainage experiments. In all cases, the quantitative analysis of the seismoelectric transfer function was in good agreement with theoretical predictions. While investigating saturation variations from full to residual water saturation, we showed that the ratio undergoes a switch in polarity at a particular saturation , which also implies a sign change of the filtration, traducing a reversal of the relative fluid displacement with respect to the frame. This sign change at critical saturation stresses a particular behaviour of the poroelastic medium: the dropping of the coseismic electric field to zero traduces the absence of relative pore/fluid displacements representative of a Biot dynamically compatible medium. We concluded from our experimental study in loose sand that the measurements of the coseismic seismoelectric coupling may provide information on fluid distribution within the pores and that the reversal of the seismoelectric field may be used as an indicator of the dynamically compatible state of the medium.     

A fast algorithm for 3D azimuthally anisotropic velocity scan

The conventional velocity scan can be computationally expensive for large‐scale seismic data sets, particularly when the presence of anisotropy requires multiparameter scanning. We introduce a fast algorithm for 3D azimuthally anisotropic velocity scan by generalizing the previously proposed 2D butterfly algorithm for hyperbolic Radon transforms. To compute semblance in a two‐parameter residual moveout domain, the numerical complexity of our algorithm is roughly as opposed to of the straightforward velocity scan, with N being the representative of the number of points in a particular dimension of either data space or parameter space. Synthetic and field data examples demonstrate the superior efficiency of the proposed algorithm.     

Suppression of near‐surface scattered body‐to‐surface waves: a steerable and nonlinear filtering approach

We present an approach based on local‐slope estimation for the separation of scattered surface waves from reflected body waves. The direct and scattered surface waves contain a significant amount of seismic energy. They present great challenges in land seismic data acquisition and processing, particularly in arid regions with complex near‐surface heterogeneities (e.g., dry river beds, wadis/large escarpments, and karst features). The near‐surface scattered body‐to‐surface waves, which have comparable amplitudes to reflections, can mask the seismic reflections. These difficulties, added to large amplitude direct and back‐scattered surface (Rayleigh) waves, create a major reduction in signal‐to‐noise ratio and degrade the final sub‐surface image quality. Removal of these waves can be difficult using conventional filtering methods, such as an filter, without distorting the reflected signal. The filtering algorithm we present is based on predicting the spatially varying slope of the noise, using steerable filters, and separating the signal and noise components by applying a directional nonlinear filter oriented toward the noise direction to predict the noise and then subtract it from the data. The slope estimation step using steerable filters is very efficient. It requires only a linear combination of a set of basis filters at fixed orientation to synthesize an image filtered at an arbitrary orientation. We apply our filtering approach to simulated data as well as to seismic data recorded in the field to suppress the scattered surface waves from reflected body waves, and we demonstrate its superiority over conventional techniques in signal preservation and noise suppression.     

Robust f‐x projection filtering for simultaneous random and erratic seismic noise attenuation

Linear prediction filters are an effective tool for reducing random noise from seismic records. Unfortunately, the ability of prediction filters to enhance seismic records deteriorates when the data are contaminated by erratic noise. Erratic noise in this article designates non‐Gaussian noise that consists of large isolated events with known or unknown distribution. We propose a robust f‐x projection filtering scheme for simultaneous erratic noise and Gaussian random noise attenuation. Instead of adopting the ?₂‐norm, as commonly used in the conventional design of f‐x filters, we utilize the hybrid ‐norm to penalize the energy of the additive noise. The estimation of the prediction error filter and the additive noise sequence are performed in an alternating fashion. First, the additive noise sequence is fixed, and the prediction error filter is estimated via the least‐squares solution of a system of linear equations. Then, the prediction error filter is fixed, and the additive noise sequence is estimated through a cost function containing a hybrid ‐norm that prevents erratic noise to influence the final solution. In other words, we proposed and designed a robust M‐estimate of a special autoregressive moving‐average model in the f‐x domain. Synthetic and field data examples are used to evaluate the performance of the proposed algorithm.     

Some remarks on Q-compensated sparse deconvolution without knowing the quality factor Q

The subsurface media are not perfectly elastic, thus anelastic absorption, attenuation and dispersion (aka Q filtering) effects occur during wave propagation, diminishing seismic resolution. Compensating for anelastic effects is imperative for resolution enhancement. Q values are required for most of conventional Q-compensation methods, and the source wavelet is additionally required for some of them. Based on the previous work of non-stationary sparse reflectivity inversion, we evaluate a series of methods for Q-compensation with/without knowing Q and with/without knowing wavelet. We demonstrate that if Q-compensation takes the wavelet into account, it generates better results for the severely attenuated components, benefiting from the sparsity promotion. We then evaluate a two-phase Q-compensation method in the frequency domain to eliminate Q requirement. In phase 1, the observed seismogram is disintegrated into the least number of Q-filtered wavelets chosen from a dictionary by optimizing a basis pursuit denoising problem, where the dictionary is composed of the known wavelet with different propagation times, each filtered with a range of possible values. The elements of the dictionary are weighted by the infinity norm of the corresponding column and further preconditioned to provide wavelets of different values and different propagation times equal probability to entry into the solution space. In phase 2, we derive analytic solutions for estimates of reflectivity and Q and solve an over-determined equation to obtain the final reflectivity series and Q values, where both the amplitude and phase information are utilized to estimate the Q values. The evaluated inversion-based Q estimation method handles the wave-interference effects better than conventional spectral-ratio-based methods. For Q-compensation, we investigate why sparsity promoting does matter. Numerical and field data experiments indicate the feasibility of the evaluated method of Q-compensation without knowing Q but with wavelet given.     

Spectral decomposition and de‐noising via time‐frequency and space‐wavenumber reassignment

The reassignment method remaps the energy of each point in a time‐frequency spectrum to a new coordinate that is closer to the actual time‐frequency location. Two applications of the reassignment method are developed in this paper. We first describe time‐frequency reassignment as a tool for spectral decomposition. The reassignment method helps to generate more clear frequency slices of layers and therefore, it facilitates the interpretation of thin layers. The second application is to seismic data de‐noising. Through thresholding in the reassigned domain rather than in the Gabor domain, random noise is more easily attenuated since seismic events are more compactly represented with a relatively larger energy than the noise. A reconstruction process that permits the recovery of seismic data from a reassigned time‐frequency spectrum is developed. Two approaches of the reassignment method are used in this paper, one of which is referred to as the trace by trace time reassignment that is mainly used for seismic spectral decomposition and another that is the spatial reassignment that is mainly used for seismic de‐noising. Synthetic examples and two field data examples are used to test the proposed method. For comparison, the Gabor transform method, inversion‐based method and common deconvolution method are also used in the examples.     

The rotated Cartesian coordinate method to remove the axial singularity of cylindrical coordinates in finite‐difference schemes for elastic and viscoelastic waves

When modelling the propagation of 3D non‐axisymmetric elastic and viscoelastic waves in cylindrical coordinates using the finite‐difference time‐domain method, a mathematical singularity occurs due to the presence of terms in the elastic and viscoelastic wave equations. For many years, this issue has been impeding the accurate numerical solution near the axis. In this work, we propose a simple but effective method for the treatment of this numerical singularity problem. By rotating the Cartesian coordinate system around the z‐axis in cylindrical coordinates, the numerical singularity problems in both 2D and 3D cylindrical coordinates can be removed. This algorithm has three advantages over the conventional treatment techniques: (i) the excitation source can be directly loaded at , (ii) the central difference scheme with second‐order accuracy is maintained, and (iii) the stability condition at the axis is consistent with the finite‐difference time‐domain in Cartesian coordinates. This method is verified by several 3D numerical examples. Results show that the rotating the Cartesian coordinate method is accurate and stable at the singularity axis. The improved finite‐difference time‐domain algorithm is also applied to sonic logging simulations in non‐axisymmetric formations and sources.     

Measuring changes in fracture properties from temporal variations in anisotropic attenuation of microseismic waveforms

We investigate fracture‐induced attenuation anisotropy in a cluster of events from a microseismic dataset acquired during hydraulic fracture stimulation. The dataset contains 888 events of magnitude ?3.0 to 0.0. We use a log‐spectral‐amplitude‐ratio method to estimate change in over a half‐hour time period where fluid is being injected and an increase in fracturing from S‐wave splitting analysis has been previously inferred. A Pearson's correlation analysis is used to assess whether or not changes in attenuation with time are statistically significant. P‐waves show no systematic change in during this time. In contrast, S‐waves polarised perpendicular to the fractures show a clear and statistically significant increase with time, whereas S‐waves polarised parallel to the fractures show a weak negative trend. We also compare between the two S‐waves, finding an increase in with time. A poroelastic rock physics model of fracture‐induced attenuation anisotropy is used to interpret the results. This model suggests that the observed changes in t* are related to an increase in fracture density of up to 0.04. This is much higher than previous estimates of 0.025 ± 0.002 based on S‐wave velocity anisotropy, but there is considerably more scatter in the attenuation measurements. This could be due to the added sensitivity of attenuation measurement to non‐aligned fractures, fracture shape, and fluid properties. Nevertheless, this pilot study shows that attenuation measurements are sensitive to fracture properties such as fracture density and aspect ratio.     

Identifying gravity anomalies caused by granitic intrusions in Nanling mineral district,China: a multifractal perspective

Several power‐law relationships of geophysical potential fields have been discussed recently with renewed interests, including field value–distance () and power spectrum–wavenumber () models. The singularity mapping technique based on the density/concentration–area (C–A) power‐law model is applied to act as a high‐pass filter for extracting gravity and magnetic anomalies regardless of the background value and to detect the edges of gravity or magnetic sources with the advantage of scale invariance. This is demonstrated on a synthetic example and a case study from the Nanling mineral district, Southern China. Compared with the analytic signal amplitude and total horizontal gradient methods, the singularity mapping technique provides more distinct and less noisy boundaries of granites than traditional methods. Additionally, it is efficient for enhancing and outlining weak anomalies caused by concealed granitic intrusions, indicating that the singularity method based on multifractal analysis is a potential tool to process gravity and magnetic data.     

Surface‐wave inversion for a P‐velocity profile with a constant depth gradient of the squared slowness

Surface waves are often used to estimate a near‐surface shear‐velocity profile. The inverse problem is solved for the locally one‐dimensional problem of a set of homogeneous horizontal elastic layers. The result is a set of shear velocities, one for each layer. To obtain a P‐wave velocity profile, the P‐guided waves should be included in the inversion scheme. As an alternative to a multi‐layered model, we consider a simple smooth acoustic constant‐density velocity model, which has a negative constant vertical depth gradient of the squared P‐wave slowness and is bounded by a free surface at the top and a homogeneous half‐space at the bottom. The exact solution involves Airy functions and provides an analytical expression for the dispersion equation. If the ratio is sufficiently small, the dispersion curves can be picked from the seismic data and inverted for the continuous P‐wave velocity profile. The potential advantages of our model are its low computational cost and the fact that the result can serve as a smooth starting model for full‐waveform inversion. For the latter, a smooth initial model is often preferred over a rough one. We test the inversion approach on synthetic elastic data computed for a single‐layer P‐wave model and on field data, both with a small ratio. We find that a single‐layer model can recover either the shallow or deeper part of the profile but not both, when compared with the result of a multi‐layer inversion that we use as a reference. An extension of our analytic model to two layers above a homogeneous half‐space, each with a constant vertical gradient of the squared P‐wave slowness and connected in a continuous manner, improves the fit of the picked dispersion curves. The resulting profile resembles a smooth approximation of the multi‐layered one but contains, of course, less detail. As it turns out, our method does not degrade as gracefully as, for instance, diving‐wave tomography, and we can only hope to fit a subset of the dispersion curves. Therefore, the applicability of the method is limited to cases where the ratio is small and the profile is sufficiently simple. A further extension of the two‐layer model to more layers, each with a constant depth gradient of the squared slowness, might improve the fit of the modal structure but at an increased cost.     

Transmission + reflection anisotropic wave-equation traveltime and waveform inversion

A transmission + reflection wave-equation traveltime and waveform inversion method is presented that inverts the seismic data for the anisotropic parameters in a vertical transverse isotropic medium. The simultaneous inversion of anisotropic parameters and ε is initially performed using transmission wave-equation traveltime inversion method. Transmission wave-equation traveltime only provides the low-intermediate wavenumbers for the shallow part of the anisotropic model; in contrast, reflection wave-equation traveltime estimates the anisotropic parameters in the deeper section of the model. By incorporating a layer-stripping method with reflection wave-equation traveltime, the ambiguity between the background-velocity model and the depths of reflectors can be greatly mitigated. In the final step, multi-scale full-waveform inversion is performed to recover the high-wavenumber component of the model.  We use a synthetic model to illustrate the local minima problem of full-waveform inversion and how transmission and reflection wave-equation traveltime can mitigate this problem. We demonstrate the efficacy of our new method using field data from the Gulf of Mexico.     

Effect of micro‐inhomogeneity on the effective stress coefficients and undrained bulk modulus of a poroelastic medium: a double spherical shell model

Although most rocks are complex multi‐mineralic aggregates, quantitative interpretation workflows usually ignore this complexity and employ Gassmann equation and effective stress laws that assume a micro‐homogeneous (mono‐mineralic) rock. Even though the Gassmann theory and effective stress concepts have been generalized to micro‐inhomogeneous rocks, they are seldom if at all used in practice because they require a greater number of parameters, which are difficult to measure or infer from data. Furthermore, the magnitude of the effect of micro‐heterogeneity on fluid substitution and on effective stress coefficients is poorly understood. In particular, it is an open question whether deviations of the experimentally measurements of the effective stress coefficients for drained and undrained elastic moduli from theoretical predictions can be explained by the effect of micro‐heterogeneity. In an attempt to bridge this gap, we consider an idealized model of a micro‐inhomogeneous medium: a Hashin assemblage of double spherical shells. Each shell consists of a spherical pore surrounded by two concentric spherical layers of two different isotropic minerals. By analyzing the exact solution of this problem, we show that the results are exactly consistent with the equations of Brown and Korringa (which represent an extension of Gassmann's equation to micro‐inhomogeneous media). We also show that the effective stress coefficients for bulk volume α, for porosity n_? and for drained and undrained moduli are quite sensitive to the degree of heterogeneity (contrast between the moduli of the two mineral components). For instance, while for micro‐homogeneous rocks the theory gives n_? = 1, for strongly micro‐inhomogenous rocks, n_? may span a range of values from –∞ to ∞ (depending on the contrast between moduli of inner and outer shells). Furthermore, the effective stress coefficient for pore volume (Biot–Willis coefficient) α can be smaller than the porosity ?. Further studies are required to understand the applicability of the results to realistic rock geometries.     

Extended reflectivity method for modelling the propagation of diffusive–viscous wave in dip‐layered media

The reflectivity method plays an important role in seismic modelling. It has been used to model different types of waves propagating in elastic and anelastic media. The diffusive–viscous wave equation was proposed to investigate the relationship between frequency dependence of reflections and fluid saturation. It is also used to describe the attenuation property of seismic wave in a fluid‐saturated medium. The attenuation of diffusive–viscous wave is mainly characterised by the effective attenuation parameters in the equation. Thus, it is essential to obtain those parameters and further characterise the features of the diffusive–viscous wave. In this work, we use inversion method to obtain the effective attenuation parameters through quality factor to investigate the characteristics of diffusive–viscous wave by comparing with those of the viscoacoustic wave. Then, the reflection/transmission coefficients in a dip plane‐layered medium are studied through coordinate transform and plane‐wave theory. Consequently, the reflectivity method is extended to compute seismograms of diffusive–viscous wave in a dip plane multi‐layered medium. Finally, we present two models to simulate the propagation of diffusive–viscous wave in a dip plane multi‐layered medium by comparing the results with those in a viscoacoustic medium. The numerical results demonstrate the validity of our extension of reflectivity method to the diffusive–viscous medium. The numerical examples in both time domain and time–frequency domain show that the reflections from a dip plane interface have significant phase shift and amplitude change compared with the results of horizontal plane interface due to the differences in reflection/transmission coefficients. Moreover, the modelling results show strong attenuation and phase shift in the diffusive–viscous wave compared to those of the viscoacoustic wave.     

Parsimonious wave‐equation travel‐time inversion for refraction waves

We present a parsimonious wave‐equation travel‐time inversion technique for refraction waves. A dense virtual refraction dataset can be generated from just two reciprocal shot gathers for the sources at the endpoints of the survey line, with N geophones evenly deployed along the line. These two reciprocal shots contain approximately 2N refraction travel times, which can be spawned into refraction travel times by an interferometric transformation. Then, these virtual refraction travel times are used with a source wavelet to create N virtual refraction shot gathers, which are the input data for wave‐equation travel‐time inversion. Numerical results show that the parsimonious wave‐equation travel‐time tomogram has about the same accuracy as the tomogram computed by standard wave‐equation travel‐time inversion. The most significant benefit is that a reciprocal survey is far less time consuming than the standard refraction survey where a source is excited at each geophone location.     

Surface nuclear magnetic resonance signals recovery by integration of a non‐linear decomposition method with statistical analysis

Presence of noise in the acquisition of surface nuclear magnetic resonance data is inevitable. There are various types of noise, including Gaussian noise, spiky events, and harmonic noise that affect the signal quality of surface nuclear magnetic resonance measurements. In this paper, we describe an application of a two‐step noise suppression approach based on a non‐linear adaptive decomposition technique called complete ensemble empirical mode decomposition in conjunction with a statistical optimization process for enhancing the signal‐to‐noise ratio of the surface nuclear magnetic resonance signal. The filtering procedure starts with applying the complete ensemble empirical mode decomposition method to decompose the noisy surface nuclear magnetic resonance signal into a finite number of intrinsic mode functions. Afterwards, a threshold region based on de‐trended fluctuation analysis is defined to identify the noisy intrinsic mode functions, and then the no‐noise intrinsic mode functions are used to recover the partially de‐noised signal. In the second stage, we applied a statistical method based on the variance criterion to the signal obtained from the initial phase to mitigate the remaining noise. To demonstrate the functionality of the proposed strategy, the method was evaluated on an added‐noise synthetic surface nuclear magnetic resonance signal and on field data. The results show that the proposed procedure allows us to improve the signal‐to‐noise ratio significantly and, consequently, extract the signal parameters (i.e., and V₀) from noisy surface nuclear magnetic resonance data efficiently.