Computational and asymptotic aspects of velocity inversion

Geophysics ◽  
1985 ◽  
Vol 50 (8) ◽  
pp. 1253-1265 ◽  
Author(s):  
Norman Bleistein ◽  
Jack K. Cohen ◽  
Frank G. Hagin
Keyword(s):  
Asymptotic Analysis ◽  
Synthetic Data ◽  
Small Variation ◽  
Computer Time ◽  
Velocity Variation ◽  
Inversion Method ◽  
Singular Function ◽  
Velocity Inversion ◽  
Dirac Delta ◽  
Peak Value

We discuss computational and asymptotic aspects of the Born inversion method and show how asymptotic analysis is exploited to reduce the number of integrations in an f-k like solution formula for the velocity variation. The output of this alternative algorithm produces the reflectivity function of the surface. This is an array of singular functions—Dirac delta functions which peak on the reflecting surfaces—each scaled by the normal reflection strength at the surface. Thus, imaging of a reflector is achieved by construction of its singular function and estimation of the reflection strength is deduced from the peak value of that function. By asymptotic analysis of the application of the algorithm to the Kirchhoff representation of the backscattered field, we show that the peak value of the output estimates the reflection strength even when the condition of small variation in velocity (an assumption of the original derivation) is violated. Furthermore, this analysis demonstrates that the method provides a migration algorithm when the amplitude has not been preserved in the data. The design of the computer algorithm is discussed, including such aspects as constraints due to causality and spatial aliasing. We also provide O‐estimates of computer time. This algorithm has been successfully implemented on both synthetic data and common‐midpoint stacked field data.

Refinements to the linear velocity inversion theory

Geophysics ◽  
1984 ◽  
Vol 49 (2) ◽  
pp. 112-118 ◽  
Author(s):  
Frank G. Hagin ◽  
Jack K. Cohen
Keyword(s):  
Synthetic Data ◽  
Scattered Wave ◽  
Scattering Data ◽  
Inversion Method ◽  
Impedance Change ◽  
Inversion Algorithm ◽  
Linear Inversion ◽  
Velocity Inversion ◽  
Linear Algorithm ◽  
Inversion Theory

The linear inversion method presented by Cohen and Bleistein in 1979 gives seriously degraded results when large reflectors are encountered. Obviously there is an irrecoverable loss of information when such a linear algorithm is applied to a nonlinear world. However, in many cases, excellent results can be achieved by suitable postprocessing of the output of the basic linear inversion algorithm. Although a certain degree of helpful postprocessing can be and has been performed by straightforward consideration of the linearization process, we present here a substantially improved postprocessing algorithm. The basis for these improvements is a more accurate scattering model due to Lahlou et al where, among other things, a WKB analysis of the wave equation led to a much more accurate accounting of the geometric spreading of the scattered wave. These notions plus an effective use of traveltime are used in the new algorithm to improve both the estimate of the reflector locations and the estimate of amplitude (velocity or acoustic impedance) change across the reflectors. The basic idea is to insert this idealized scattering data into the original linear algorithm, and then use the result of this computation as a guide in the interpretation of the numerical output of the algorithm. We demonstrate the result of computer implementation of this algorithm on synthetic data, with and without noise, and verify that the postprocessing algorithm produces dramatically improved reflector locations and speed estimates. Moreover, the new algorithm adds only very modest cost to the basic processing, which is, in turn, very competitive in cost to other multidimensional algorithms.

REAL-TIME 2.5D INVERSION OF LWD RESISTIVITY MEASUREMENTS USING DEEP LEARNING FOR GEOSTEERING APPLICATIONS ACROSS FAULTED FORMATIONS

2021 ◽  
Author(s):  
Kyubo Noh ◽  
◽  
Carlos Torres-Verdín ◽  
David Pardo ◽  
◽  
...  
Keyword(s):  
Deep Learning ◽  
Real Time ◽  
Fault Plane ◽  
Synthetic Data ◽  
Field Measurements ◽  
Geological Structure ◽  
Inversion Method ◽  
Adaptive Finite Element ◽  
Resistivity Measurements ◽  
Lwd Resistivity

We develop a Deep Learning (DL) inversion method for the interpretation of 2.5-dimensional (2.5D) borehole resistivity measurements that requires negligible online computational costs. The method is successfully verified with the inversion of triaxial LWD resistivity measurements acquired across faulted and anisotropic formations. Our DL inversion workflow employs four independent DL architectures. The first one identifies the type of geological structure among several predefined types. Subsequently, the second, third, and fourth architectures estimate the corresponding spatial resistivity distributions that are parameterized (1) without the crossings of bed boundaries or fault plane, (2) with the crossing of a bed boundary but without the crossing of a fault plane, and (3) with the crossing of the fault plane, respectively. Each DL architecture employs convolutional layers and is trained with synthetic data obtained from an accurate high-order, mesh-adaptive finite-element forward numerical simulator. Numerical results confirm the importance of using multi-component resistivity measurements -specifically cross-coupling resistivity components- for the successful reconstruction of 2.5D resistivity distributions adjacent to the well trajectory. The feasibility and effectiveness of the developed inversion workflow is assessed with two synthetic examples inspired by actual field measurements. Results confirm that the proposed DL method successfully reconstructs 2.5D resistivity distributions, location and dip angles of bed boundaries, and the location of the fault plane, and is therefore reliable for real-time well geosteering applications.

Simultaneous inversion of Q and reflectivity using dictionary learning

Geophysics ◽  
2021 ◽  
pp. 1-69
Author(s):  
Jie Shao ◽  
Yibo Wang
Keyword(s):  
Conventional Method ◽  
Dictionary Learning ◽  
Computational Efficiency ◽  
Synthetic Data ◽  
Inversion Method ◽  
Learning Method ◽  
Q Value ◽  
Value Set ◽  
Simultaneous Inversion ◽  
Convolution Model

Quality factor ( Q) and reflectivity are two important subsurface properties in seismic data processing and interpretation. They can be calculated simultaneously from a seismic trace corresponding to an anelastic layered model by a simultaneous inversion method based on the nonstationary convolution model. However, the conventional simultaneous inversion method calculates the optimum Q and reflectivity based on the minimum of the reflectivity sparsity by sweeping each Q value within a predefined range. As a result, the accuracy and computational efficiency of the conventional method depend heavily on the predefined Q value set. To improve the performance of the conventional simultaneous inversion method, we have developed a dictionary learning-based simultaneous inversion of Q and reflectivity. The parametric dictionary learning method is used to update the initial predefined Q value set automatically. The optimum Q and reflectivity are calculated from the updated Q value set based on minimizing not only the sparsity of the reflectivity but also the data residual. Synthetic data and two field data sets were used to test the effectiveness of our method. The results demonstrated that our method can effectively improve the accuracy of these two parameters compared to the conventional simultaneous inversion method. In addition, the dictionary learning method can improve computational efficiency up to approximately seven times when compared to the conventional method with a large predefined dictionary.

scholarly journals Seismic AVOA Inversion for Weak Anisotropy Parameters and Fracture Density in a Monoclinic Medium

2020 ◽  
Vol 10 (15) ◽  
pp. 5136 ◽  
Author(s):  
Zijian Ge ◽  
Shulin Pan ◽  
Jingye Li
Keyword(s):  
Fractured Rock ◽  
Synthetic Data ◽  
Transversely Isotropic ◽  
P Wave ◽  
Inversion Method ◽  
Fracture Density ◽  
Weak Anisotropy ◽  
Amplitude Versus Offset ◽  
Porosity And Permeability ◽  
Transverse Isotropic

In shale gas development, fracture density is an important lithologic parameter to properly characterize reservoir reconstruction, establish a fracturing scheme, and calculate porosity and permeability. The traditional methods usually assume that the fracture reservoir is one set of aligned vertical fractures, embedded in an isotropic background, and estimate some alternative parameters associated with fracture density. Thus, the low accuracy caused by this simplified model, and the intrinsic errors caused by the indirect substitution, affect the estimation of fracture density. In this paper, the fractured rock of monoclinic symmetry assumes two non-orthogonal vertical fracture sets, embedded in a transversely isotropic background. Firstly, assuming that the fracture radius, width, and orientation are known, a new form of P-wave reflection coefficient, in terms of weak anisotropy (WA) parameters and fracture density, was obtained by substituting the stiffness coefficients of vertical transverse isotropic (VTI) background, normal, and tangential fracture compliances. Then, a linear amplitude versus offset and azimuth (AVOA) inversion method, of WA parameters and fracture density, was constructed by using Bayesian theory. Tests on synthetic data showed that WA parameters, and fracture density, are stably estimated in the case of seismic data containing a moderate noise, which can provide a reliable tool in fracture prediction.

Warping least-squares inversion for 4D velocity change: Gulf of Mexico case study

2019 ◽  
Vol 7 (2) ◽  
pp. SB23-SB31
Author(s):  
Chang Li ◽  
Mark Meadows ◽  
Todd Dygert
Keyword(s):  
Gulf Of Mexico ◽  
Least Squares ◽  
Synthetic Data ◽  
Inversion Method ◽  
Velocity Change ◽  
Data Set ◽  
Phase Constraints ◽  
Free Case ◽  
Velocity Changes

We have developed a new trace-based, warping least-squares inversion method to quantify 4D velocity changes. There are two steps to solve for these velocity changes: (1) dynamic warping with phase constraints to align the baseline and monitor traces and (2) least-squares inversion for 4D velocity changes incorporating the time shifts and 4D amplitude differences (computed after trace alignment by warping). We have demonstrated this new inversion workflow using simple synthetic layered models. For the noise-free case, phase-constrained warping is superior to standard, amplitude-based warping by improving trace alignment, resulting in more accurate inverted velocity changes (less than 1% error). For synthetic data with 6% rms noise, inverted velocity changes are reasonably accurate (less than 10% error). Additional inversion tests with migrated finite-difference data shot over a realistic anticline model result in less than 10% error. The inverted velocity changes on a 4D field data set from the Gulf of Mexico are more interpretable and consistent with the dynamic reservoir model than those estimated from the conventional time-strain method.

scholarly journals Relative moment tensors and deep Yakutat seismicity

2019 ◽  
Vol 219 (2) ◽  
pp. 1447-1462 ◽  
Author(s):  
Alexandre P Plourde ◽  
Michael G Bostock
Keyword(s):  
Principal Components ◽  
Moment Tensor ◽  
Synthetic Data ◽  
Principal Component ◽  
P Wave ◽  
Inversion Method ◽  
Low Velocity ◽  
Stress Regime ◽  
Moment Tensors ◽  
S Waves

SUMMARY We introduce a new relative moment tensor (MT) inversion method for clusters of nearby earthquakes. The method extends previous work by introducing constraints from S-waves that do not require modal decomposition and by employing principal component analysis to produce robust estimates of excitation. At each receiver, P and S waves from each event are independently aligned and decomposed into principal components. P-wave constraints on MTs are obtained from a ratio of coefficients corresponding to the first principal component, equivalent to a relative amplitude. For S waves we produce constraints on MTs involving three events, where one event is described as a linear combination of the other two, and coefficients are derived from the first two principal components. Nonlinear optimization is applied to efficiently find best-fitting tensile-earthquake and double-couple solutions for relative MT systems. Using synthetic data, we demonstrate the effectiveness of the P and S constraints both individually and in combination. We then apply the relative MT inversion to a set of 16 earthquakes from southern Alaska, at ∼125 km depth within the subducted Yakutat terrane. Most events are compatible with a stress tensor dominated by downdip tension, however, we observe several pairs of earthquakes with nearly antiparallel slip implying that the stress regime is heterogeneous and/or faults are extremely weak. The location of these events near the abrupt downdip termination of seismicity and the low-velocity zone suggest that they are caused by weakening via grain-size and volume reduction associated with eclogitization of the lower crustal gabbro layer.

Plumes: Response of time migration to lateral velocity variation

Geophysics ◽  
1995 ◽  
Vol 60 (4) ◽  
pp. 1118-1127 ◽  
Author(s):  
Dimitri Bevc ◽  
James L. Black ◽  
Gopal Palacharla
Keyword(s):  
Impulse Response ◽  
Synthetic Data ◽  
Migration Velocity ◽  
Velocity Variation ◽  
Velocity Dependence ◽  
Geometrical Construction ◽  
Lateral Velocity ◽  
Time Migration ◽  
Offset Time ◽  
Zero Offset

We analyze how time migration mispositions events in the presence of lateral velocity variation by examining the impulse response of depth modeling followed by time migration. By examining this impulse response, we lay the groundwork for the development of a remedial migration operator that links time and depth migration. A simple theory by Black and Brzostowski predicted that the response of zero‐offset time migration to a point diffractor in a v(x, z) medium would be a distinctive, cusp‐shaped curve called a plume. We have constructed these plumes by migrating synthetic data using several time‐migration methods. We have also computed the shape of the plumes by two geometrical construction methods. These two geometrical methods compare well and explain the observed migration results. The plume response is strongly influenced by migration velocity. We have studied this dependency by migrating synthetic data with different velocities. The observed velocity dependence is confirmed by geometrical construction. A simple first‐order theory qualitatively explains the behavior of zero‐offset time migration, but a more complete understanding of migration velocity dependence in a v(x, z) medium requires a higher order finite‐offset theory.

Monte Carlo background velocity inversion method

2014 ◽  
Author(s):  
Yue Ling* ◽  
Huazhong Wang ◽  
Shaoyong Liu
Keyword(s):  
Monte Carlo ◽  
Inversion Method ◽  
Velocity Inversion

Regularized two‐dimensional Fourier gravity inversion method with application to the Silent Canyon caldera, Nevada

Geophysics ◽  
1989 ◽  
Vol 54 (4) ◽  
pp. 486-496 ◽  
Author(s):  
Sharon K. Reamer ◽  
John F. Ferguson
Keyword(s):  
Random Noise ◽  
Empirical Relationship ◽  
Synthetic Data ◽  
Test Site ◽  
Gravity Inversion ◽  
Inversion Method ◽  
Type Model ◽  
Filter Parameter ◽  
Mass Distributions ◽  
Rms Error

A modification of the 2‐D Fourier gravity inversion method includes regularization and a linear density variation with depth. Explicit downward continuation in the Fourier inversion of gravity observations from mass distributions at depth produces instability in the presence of noise and shallow mass distributions. A data‐adaptive regularization filter tapers growth of the exponential continuation function. An empirical relationship between the regularization filter parameter and a parametric model of potential field spectra results in automatic selection of the filter parameter for a given continuation depth. Inversion of synthetic data from a random noise‐contaminated basin type model produces a depth model that agrees with the synthetic structure with an rms error commensurate with the data noise. A model of the Silent Canyon caldera, buried beneath Pahute Mesa at the Nevada Test Site, results in a gravity field that agrees with the observations to within a 4 percent rms error. The caldera gravity model supports the hypothesis of a high‐density half‐space (precaldera lithology) beneath a lower density caldera infill (postcaldera volcanic activity).

Traveltime inversion of offset vertical seismic profiles—A feasibility study

Geophysics ◽  
1984 ◽  
Vol 49 (3) ◽  
pp. 250-264 ◽  
Author(s):  
L. R. Lines ◽  
A. Bourgeois ◽  
J. D. Covey
Keyword(s):  
Inverse Method ◽  
Synthetic Data ◽  
Real Data ◽  
Seismic Profile ◽  
Inversion Method ◽  
Earth Model ◽  
Seismic Profiles ◽  
Traveltime Inversion ◽  
Vertical Seismic Profiles ◽  
Inversion Techniques

Traveltimes from an offset vertical seismic profile (VSP) are used to estimate subsurface two‐dimensional dip by applying an iterative least‐squares inverse method. Tests on synthetic data demonstrate that inversion techniques are capable of estimating dips in the vicinity of a wellbore by using the traveltimes of the direct arrivals and the primary reflections. The inversion method involves a “layer stripping” approach in which the dips of the shallow layers are estimated before proceeding to estimate deeper dips. Examples demonstrate that the primary reflections become essential whenever the ratio of source offset to layer depth becomes small. Traveltime inversion also requires careful estimation of layer velocities and proper statics corrections. Aside from these difficulties and the ubiquitous nonuniqueness problem, the VSP traveltime inversion was able to produce a valid earth model for tests on a real data case.

Sign in / Sign up

Export Citation Format

Share Document