Basis Pursuit Inversion Research Articles

A depth-variant wavelet extraction method based on the orthogonal matching pursuit for direct inversion

The results of pre-stack depth migration reflect subsurface structures directly. However, it is an urgent geophysical problem that how to deal with this non-stationarity exhibited by depth-domain seismic data and achieve high precision reservoir prediction. To address this issue, we develop a depth-variant wavelet extraction method based on the orthogonal matching pursuit and attain direct inversion of the depth-domain acoustic impedance using the depth-variant wavelets. First, the parameters of source wavelets estimated by complex seismic trace analysis are utilized to construct the overcomplete dictionary. Secondly, the overcomplete dictionary is used to decompose the depth-domain seismic traces by orthogonal matching pursuit. Thirdly, we extract the depth-variant Ricker wavelets from the decomposed optimal atoms. Test of examples demonstrates that the proposed depth-variant wavelet method has high accuracy. The synthetic seismogram obtained by the proposed method has high Pearson correlation coefficients of 99%, which verifies that our method is feasible and accurate. Next, a depth-domain impedance trend constraint is introduced into the objective function of the conventional basis pursuit inversion to enhance the lateral continuity. Finally, the objective function with impedance trend constraint is solved by basis pursuit for depth-domain acoustic impedance. We test the direct depth-domain acoustic impedance inversion method on the synthetic and field data, which indicates that our method has high accuracy and robustness. The mean relative error of its inversion result is only 0.9% for the noise-free example and about 5% for the strong noisy example. Therefore, our method can be effectively applied to the seismic-well calibration and seismic impedance inversion in the depth domain. And it has powerful potential in reservoir characterization, fluid prediction and attribute extraction in the depth domain. Additionally, the energy distribution features of the Ricker wavelet are derived and analyzed, which can guide the application of the Ricker wavelet in seismic signal analysis.

A depth-variant seismic wavelet extraction method for basis pursuit inversion with an impedance trend constraint

The seismic images produced by prestack depth migration indicate more accurate subsurface structures than time images, resulting in a growing need for depth-domain inversion. However, due to the strong nonstationarity exhibited by depth-domain seismic data, time-domain inversion methods based on the convolutional model cannot be directly applied in the depth domain. To address this issue, we develop a method for extracting a depth-variant seismic wavelet, which is then combined with a nonstationary convolutional model to enable direct inversion of the depth-domain acoustic impedance (AI). First, we extend the Morlet wavelet to the depth domain and develop an orthogonal matching pursuit spectral decomposition method using the depth-domain Morlet wavelet. We then investigate the waveforms and wavenumber spectra similarities between the depth-domain Morlet wavelet and depth-domain Ricker wavelet and extract depth-variant Ricker wavelets from the depth-wavenumber spectrum. We add a depth-domain impedance trend constraint to the conventional basis pursuit inversion to enhance the lateral continuity of the inversion results. Then, we attain direct inversion of the depth-domain AI. Tests of synthetic and field data demonstrate that our method achieves high-accuracy inversion results while maintaining high computational efficiency, highlighting our approach’s effectiveness and strong reservoir characterization potential.

The ℓ1−2-norm-regularized basis pursuit seismic inversion based on the exact Zoeppritz equation

Sparsity constraints have been widely adopted in the regularization of ill-posed problems to obtain subsurface properties with sparseness features. However, the target parameters are generally not sparsely distributed, and sparsity constraints lead to results that are missing information. Smooth constraints (e.g., [Formula: see text] norm) lead to the insufficient resolution of the inversion results. To overcome this issue, an effective solution is to convert the target parameters to a sparse representation, which can then be solved with sparsity constraints. For the estimation of elastic parameters, a high-resolution and reliable seismic basis pursuit inversion is developed based on the exact Zoeppritz equation. Furthermore, the [Formula: see text] norm is developed as a constraint, wherein a regularized function is minimized with the alternating direction method of the multipliers algorithm. Numerical examples and real data applications demonstrate that our method can not only improve the accuracy of the inversion results, especially the S-wave velocity and density information but also increase the resolution of the inversion results. Furthermore, the [Formula: see text]-norm constraint has better noise suppression demonstrating great potential in practical applications.

Depth-domain angle and depth variant seismic wavelets extraction for prestack seismic inversion

Many prestack depth migration methods have been developed and widely used to generate depth-domain seismic images, resulting in a need for depth-domain prestack seismic inversion of the subsurface elastic properties for reservoir characterization. Time-domain inversion techniques often are used after the depth-domain data set is transformed to the time domain. We provide a new technique for directly inverting depth-domain prestack seismic data for subsurface elastic properties: P-impedance ([Formula: see text]), S-impedance ([Formula: see text]), and density ([Formula: see text]) in the depth domain. The proposed depth-domain workflow eliminates the need for time-depth/depth-time conversion, making it efficient and effective. Using a depth-wavenumber decomposition approach, the suggested workflow first extracts a collection of depth and angle varying wavelets to handle the potential nonstationarity of the depth-domain prestack seismic data. The extracted depth and angle variant wavelets are used in a basis pursuit inversion, which enhances the resolution of the inversion results. The workflow is tested on the Wenan 3D field data set, demonstrating its viability for practical applications.

Prestack seismic inversion using a Rytov-WKBJ approximation

Summary Approaches to seismic modeling using integral methods, notably Born-WKBJ (Wentzel-Kramers-Brillouin-Jeffreys), have seen an increase in applications in geophysical prospecting and near-surface exploration. Moreover, due to its linearity characteristic, which can fast and efficiently simulate full-waveform information, the Born-WKBJ based method performs well in seismic inversion. However, the Born approximation is the linearized wavefield simulation by Taylor expansion of the wavefield and omitting the high-order term, which cannot simulate the wavefiled accurately, including the amplitude, phase, and waveform information. For the seismic inversion, especially the amplitude variation with angle/offset (AVA/AVO) inversion, the amplitude is the most important element for estimating the parameters. In this paper, a Rytov-WKBJ approximation-based method, which can simulate the seismic amplitude information more accurately, is introduced to prestack seismic inversion. Besides, in order to improve the resolution of the inversion results, the ℓ1 − 2-norm regularized basis pursuit inversion (BPI) is introduced to the inversion algorithm. Then, we demonstrate the superiority of the proposed method with the zero-offset and angle-dependent seismic data simulation. The model tests show that the proposed method performs better than the conventional method significantly both on amplitude and phase of the seismic data. Finally, the inversion tests of synthetic and field seismic data indicate that the proposed method can obtain more accurate results with a higher resolution.

Seismic reflectivity inversion using an L1-norm basis-pursuit method and GPU parallelisation

Abstract Seismic reflectivity inversion problem can be formulated using a basis-pursuit method, aiming to generate a sparse reflectivity series of the subsurface media. In the basis-pursuit method, the reflectivity series is composed by large amounts of even and odd dipoles, thus the size of the seismic response matrix is huge and the matrix operations involved in seismic inversion are very time-consuming. In order to accelerate the matrix computation, a basis-pursuit method-based seismic inversion algorithm is implemented on Graphics Processing Unit (GPU). In the basis-persuit inversion algorithm, the problem is imposed with a L1-norm model constraint for sparsity, and this L1-norm basis-pursuit inversion problem is reformulated using a linear programming method. The core problems in the inversion are large-scale linear systems, which are resolved by a parallelised conjugate gradient method. The performance of this fully parallelised implementation is evaluated and compared to the conventional serial coding. Specifically, the investigation using several field seismic data sets with different sizes indicates that GPU-based parallelisation can significantly reduce the computational time with an overall factor up to 145. This efficiency improvement demonstrates a great potential of the basis-pursuit inversion method in practical application to large-scale seismic reflectivity inversion problems.

Pre-stack basis pursuit seismic inversion for brittleness of shale

Brittleness of rock plays a significant role in exploration and development of shale gas reservoirs. Young’s modulus and Poisson’s ratio are the key parameters for evaluating the rock brittleness in shale gas exploration because their combination relationship can quantitatively characterize the rock brittleness. The high-value anomaly of Young’s modulus and the low-value anomaly of Poisson’s ratio represent high brittleness of shale. The technique of pre-stack amplitude variation with angle inversion allows geoscientists to estimate Young’s modulus and Poisson’s ratio from seismic data. A model constrained basis pursuit inversion method is proposed for stably estimating Young’s modulus and Poisson’s ratio. Test results of synthetic gather data show that Young’s modulus and Poisson’s ratio can be estimated reasonably. With the novel method, the inverted Young’s modulus and Poisson’s ratio of real field data focus the layer boundaries better, which is helpful for us to evaluate the brittleness of shale gas reservoirs. The results of brittleness evaluation show a good agreement with the results of well interpretation.

Introduction to the special issue of unconventional hydrocarbon research

Unconventional hydrocarbon research has developed rapidly in the recent years. In order to review the advances in unconventional hydrocarbon research in the areas of reservoir geology, geophysics, engineering, and economic evaluation, the Unconventional Natural Gas Institute (UNGI) of China University of Petroleum (Beijing) was invited to organize this special issue. Papers in this special issue are mainly about tight oil and gas, shale oil and gas, and coalbed methane, which are the main types of unconventional hydrocarbon. In the area of geology and accumulation mechanisms, Song et al. divided unconventional oil and gas into different types based on relationships between source rock evolution and different unconventional reservoirs and further noted that the fundamental difference between unconventional and conventional hydrocarbons is the driving force. Xiao et al. discussed the controlling factors and enrichment area evaluation of shale gas of the Lower Paleozoic marine strata in south China, and established geological and loss models of shale gas for uplifted/folded and faulted/folded areas. Jiang et al. identified three types of tight sandstone gas reservoirs according to the timing of coupling between the two phases of gas charging and reservoir tightening, and clarified the controlling factors, accumulation mechanisms and modes of different types of tight gas reservoirs. Zou et al. optimized the evaluation parameters and standards of the tight oil ‘‘sweet spots’’ which are significant for the exploration of tight oil in China. In the area of geophysical exploration, Yin et al. proposed a constrained basis pursuit inversion (BPI) method for stably estimating the Young’s modulus and Poisson’s ratio, and effectively predicting the brittleness of unconventional reservoirs. In the area of hydraulic fracturing, Zeng et al. used the finite element method to quantify the stress distribution and investigated the controlling mechanism of the stress distribution of multi-stage hydraulic fracturing and fracture expansion. Ge et al. discussed the interaction between hydraulic fracturing fluids and shale matrix, clarified the mechanism of low flowback efficiency of fracturing fluids, and established its relationship to gas productivity. In the area of development and economic evaluation, Hu et al. discussed the pore structure of typical American and Chinese shales and investigated the effect of pore structure on the decline of shale gas well production. Wang et al. discussed the influence of gas transport mechanisms on the productivity of multi-stage fractured horizontal wells in shale gas reservoirs. Zhang et al. proposed a model of hydraulic fracture in tight oil reservoirs by using a combined finite and discrete element method to direct the hydraulic fracturing of tight reservoirs. Tang et al. quantitatively divided the coalbed methane well-production process into four stages and quantitatively characterized the coal reservoir permeability change in different stages. Zhang et al. simulated the absolute adsorption and excess adsorption capacity of medium rank coal to CO2, CH4, and C2H6 gas using molecular dynamics, and provided a theoretical support for improving coalbed methane recovery using CO2 injection. Luo et al. applied the discounted cash & Yan Song sya@petrochina.com.cn

Rock-physics-based double-difference inversion for CO2 saturation and porosity at the Cranfield CO2 injection site

Large-scale subsurface injection of [Formula: see text] has the potential to reduce emissions of atmospheric [Formula: see text] and improve oil recovery. Studying the effects of injected [Formula: see text] on the elastic properties of the saturated reservoir rock can help to improve long-term monitoring effectiveness and accuracy at locations undergoing [Formula: see text] injection. We used two vintages of existing 3D surface seismic data and well logs to probabilistically invert for the [Formula: see text] saturation and porosity at the Cranfield reservoir using a double-difference approach. The first step of this work was to calibrate the rock-physics model to the well-log data. Next, the baseline and time-lapse seismic data sets were inverted for acoustic impedance [Formula: see text] using a high-resolution basis pursuit inversion technique. The reservoir porosity was derived statistically from the rock-physics model based on the [Formula: see text] estimates inverted from the baseline survey. The porosity estimates were used in the double-difference routine as the fixed initial model from which [Formula: see text] saturation was then estimated from the time-lapse [Formula: see text] data. Porosity was assumed to remain constant between survey vintages; therefore, the changes between the baseline and time-lapse [Formula: see text] data may be inverted for [Formula: see text] saturation from the injection activities using the calibrated rock-physics model. Comparisons of inverted and measured porosity from well logs indicated quite accurate results. Estimates of [Formula: see text] saturation found less accuracy than the porosity estimates.

Time-lapse seismic data registration and inversion for CO2 sequestration study at Cranfield

The time-lapse seismic survey for [Formula: see text] sequestration study at Cranfield can be problematic because of misalignments between time-lapse data sets. Such misalignments can be caused by the seismic data processing workflow and may result in the wrong interpretation of time-lapse seismic amplitude differences. We propose an efficient local-correlation-based warping method of registering the time-lapse poststack data sets, which can align these data sets without changing original amplitudes. Application of our registration method to Cranfield time-lapse data demonstrates its effectiveness in separating time-shift character from seismic amplitude signature. After registration, time-lapse differences show an improved consistency in vertical cross sections and a more localized distribution of difference amplitudes along the horizon, allowing us to apply a high-resolution basis pursuit inversion (BPI) for acoustic impedances. Inversion results show that decreases in acoustic impedances occur mostly at the top of the injection interval, which can be used as an indicator of rock properties to detect a subsurface [Formula: see text] plume.

Multi-trace basis pursuit inversion with spatial regularization

Basis pursuit inversion (BPI) was developed to derive spiky reflection coefficients, which was applied on post-stack seismic data trace-by-trace for subsurface reflectivity. Such a trace-by-trace operation has no control on lateral continuity. To address this, we extend the former BPI to a multi-trace procedure with spatial regularization—a spatial derivative operator—to stabilize the lateral variation. The original objective function of BPI is reformed with a ‘Z’ shape spatial derivative imposed, which can balance the lateral continuity and vertical resolution. Synthetic tests show improved lateral continuity by spatial regularized BPI in the case of noise contamination. Real data applications demonstrate its effectiveness to enhance the lateral continuity of inverted spiky reflectivity.

A prestack basis pursuit seismic inversion

Resolving thin layers and clearly delineating layer boundaries in inverted seismic sections are very important goals for exploration and production. Many seismic inversion methods based on a least-squares optimization approach with Tikhonov-type regularization can lead to unfocused transitions between adjacent layers. A basis pursuit inversion (BPI) algorithm based on the [Formula: see text] norm optimization method can, however, resolve sharp boundaries between layers. We have formulated a BPI algorithm for amplitude-versus-angle inversion and investigated its potential to improve contrasts between layers. Like the BPI for poststack case, the sparse layer constraint, rather than the sparse spike constraint, is used to construct the model space as a wedge dictionary. All the elements of the dictionary are bed reflectivities, which include solutions consisting of thin beds as well. With this dictionary, we use an [Formula: see text] norm optimization framework to derive three reflectivities, namely, [Formula: see text], [Formula: see text], and [Formula: see text]. Although BPI does not require a starting model, high-resolution absolute velocities ([Formula: see text], [Formula: see text]) and density ([Formula: see text]) can be obtained by incorporating initial models in the BPI derived reflectivities. Tests on synthetic and field data show that the BPI algorithm can indeed detect and enhance layer boundaries by effectively removing the wavelet interference.

Time-lapse surface seismic inversion with thin bed resolution for monitoring CO2 sequestration: A case study from Cranfield, Mississippi

The feasibility of carbon dioxide sequestration research at Cranfield, Mississippi is studied by injecting millions tonnes of CO2 into the lower Tuscaloosa sandstone Formation over a two year period. Time-lapse surface seismic surveys were recorded at pre-(2007) and post-(2010) injection stages to monitor the subsurface fluid plume. The injection interval, appearing as a thin layer in the well-log data, shows very weak signature of CO2 injection in the time-lapse seismic amplitude data. In order to improve the capability of tracking CO2 plume movement using seismic data, we have applied a basis pursuit inversion (BPI) method to the post-stack seismic datasets. This method of inversion incorporates a priori information as a wedge dictionary and employs a L1-norm optimization for obtaining solutions with improved resolution. The inverted time-lapse acoustic impedances show a strongly decreasing trend mostly at the top of the injection interval, which is in agreement with well-log measurements for CO2 saturation. Improved resolution time-lapse impedance mapping therefore is an effective tool for imaging the displacement of the CO2 plume.

Stochastic and deterministic seismic inversion methods for thin-bed resolution

Results from a typical post-stack or pre-stack seismic inversion are band-limited primarily due to missing low and high frequencies in the wavelet. Thin beds that are of primary interest in reservoir characterization are generally poorly resolved. Inversion methods are designed to incorporate a priori information in several different ways. In this paper, we examine and compare performances of two such methods: a stochastic method that draws a starting broad-band model based on some statistics, derived from well logs, and a deterministic method that incorporates constraints using a model norm, based on the basis pursuit decomposition. We apply a stochastic method employing very fast simulated annealing for optimization, using fractal-based starting models, and a basis pursuit inversion (deterministic method) to a single post-stack dataset to assess the performance of the two algorithms. We find that both methods are able to derive a broad-band acoustic impedance from the band-limited post-stack seismic data. The results show a similar structure and therefore result in possible increasing confidence in the interpretation. The essence of our study is that the calibration of results from two different methods can provide an interpreter with more information in interpreting broad-band inversion results hitherto not derived from standard deterministic algorithms.

Seismic sparse-layer reflectivity inversion using basis pursuit decomposition

A basis pursuit inversion of seismic reflection data for reflection coefficients is introduced as an alternative method of incorporating a priori information in the seismic inversion process. The inversion is accomplished by building a dictionary of functions representing reflectivity patterns and constituting the seismic trace as a superposition of these patterns. Basis pursuit decomposition finds a sparse number of reflection responses that sum to form the seismic trace. When the dictionary of functions is chosen to be a wedge-model of reflection coefficient pairs convolved with the seismic wavelet, the resulting reflectivity inversion is a sparse-layer inversion, rather than a sparse-spike inversion. Synthetic tests suggest that a sparse-layer inversion using basis pursuit can better resolve thin beds than a comparable sparse-spike inversion. Application to field data indicates that sparse-layer inversion results in the potentially improved detectability and resolution of some thin layers and reveals apparent stratigraphic features that are not readily seen on conventional seismic sections.