Seismic Inversion: Theory and Applications

Table of Contents

Cover

Dedication

Title Page

Copyright

Preface

Chapter 1: Basics of seismic inversion

1.1 The linear inverse problem

1.2 Data, model and mapping

1.3 General solutions

1.4 Regularisation

Chapter 2: Linear systems for inversion

2.1 The governing equation and its solution

2.2 Seismic scattering

2.3 Seismic imaging

2.4 Seismic downward continuation

2.5 Seismic data processing

Chapter 3: Least-squares solutions

3.1 Determinant and rank

3.2 The inverse of a square matrix

3.3 LU decomposition and Cholesky factorisation

3.4 Least-squares solutions of linear systems

3.5 Least-squares solution for a nonlinear system

3.6 Least-squares solution by QR decomposition

Chapter 4: Singular value analysis

4.1 Eigenvalues and eigenvectors

4.2 Singular value concept

4.3 Generalised inverse solution by SVD

4.4 SVD applications

Chapter 5: Gradient-based methods

5.1 The step length

5.2 The steepest descent method

5.3 Conjugate gradient method

5.4 Biconjugate gradient method

5.5 The subspace gradient method

Chapter 6: Regularisation

6.1 Regularisation versus conditional probability

6.2 The Lp norm constraint

6.3 The maximum entropy constraint

6.4 The Cauchy constraint

6.5 Comparison of various regularisations

Chapter 7: Localised average solutions

7.1 The average solution

7.2 The deltaness

7.3 The spread criterion

7.4 The Backus-Gilbert stable solution

Chapter 8: Seismic wavelet estimation

8.1 Wavelet extraction from seismic-to-well correlation

8.2 Generalised wavelet constructed from power spectrum

8.3 Kurtosis matching for a constant-phase wavelet

8.4 Cumulant matching for a mixed-phase wavelet

Chapter 9: Seismic reflectivity inversion

9.1 The least-squares problem with a Gaussian constraint

9.2 Reflectivity inversion with an Lp norm constraint

9.3 Reflectivity inversion with the Cauchy constraint

9.4 Multichannel reflectivity inversion

9.5 Multichannel conjugate gradient method

Chapter 10: Seismic ray-impedance inversion

10.1 Acoustic and elastic impedances

10.2 Ray impedance

10.3 Workflow of ray-impedance inversion

10.4 Reflectivity inversion in the ray-parameter domain

10.5 Ray-impedance inversion with a model constraint

Chapter 11: Seismic tomography based on ray theory

11.1 Seismic tomography

11.2 Velocity-depth ambiguity in reflection tomography

11.3 Ray tracing by a path bending method

11.4 Geometrical spreading of curved interfaces

11.5 Joint inversion of traveltime and amplitude data

Chapter 12: Waveform tomography for the velocity model

12.1 Inversion theory for waveform tomography

12.2 The optimal step length

12.3 Strategy for reflection seismic tomography

12.4 Multiple attenuation and partial compensation

12.5 Reflection waveform tomography

Chapter 13: Waveform tomography with irregular topography

13.1 Body-fitted grids for finite-difference modelling

13.2 Modification of boundary points

13.3 Pseudo-orthogonality and smoothness

13.4 Wave equation and absorbing boundary condition

13.5 Waveform tomography with irregular topography

Chapter 14: Waveform tomography for seismic impedance

14.1 Wave equation and model parameterisation

14.2 The impedance inversion method

14.3 Inversion strategies and the inversion flow

14.4 Application to field seismic data

14.5 Conclusions

Appendices

Appendix A: Householder transform for QR decomposition

Appendix B: Singular value decomposition algorithm

Appendix C: Biconjugate gradient method for complex systems

Appendix D: Gradient calculation in waveform tomography

Exercises and solutions

References

Author index

Subject index

End User License Agreement

List of Illustrations

Chapter 1: Basics of seismic inversion

Figure 1.1 The dependence of model perturbation equation Δm equation on the data errors equation ε equation . There are three types of dependence: linear (solid curves), power (dotted curves) and logarithmic (dashed curves). The three panels (left to right) are cases with α = 0.3, 0.6, 0.9, respectively.

Figure 1.2 A function f(r), that is not differentiable, convolved with a function h(r) produces a differentiable function c01-math-031 . The latter is differentiable without singularities, and the difference c01-math-032 is sufficiently small.

Chapter 2: Linear systems for inversion

Figure 2.1 A step of the interval velocity model is defined by a Boltzman function, bk(t), which consists of four parameters {vk, vk+1, tk, ck}. Within the preset time segment (between two dashed lines), inversion produces time tk for each step. Hence, it has a flexible time interval for defining the interval velocity.

Chapter 3: Least-squares solutions

Figure 3.1 Two row vectors r2 and r3 form a parallelogram, where b is the height of the parallelogram, and i, j and k are the basis vectors.

Figure 3.2 Three row vectors of a 3 × 3 matrix form a parallelepiped. The cross product r2 × r3 represents the parallelogram area of the parallelepiped base and has the direction perpendicular to either vectors. The inner product projects vector r1 to the vector r2 × r3. This projection is the height h. The volume of this parallelepiped is the absolute value of the determinant of the 3 × 3 matrix.

Figure 3.3 Fitting a straight line to three available measurements is a least-squares inverse problem. In this case, the number of data samples is more than the number of unknown variables (a0,a1).

Chapter 5: Gradient-based methods

Figure 5.1 Given a gradient vector γ(k) at x(k), search for the local minimum position x(k+1) along the negative gradient direction, −γ(k). The step length αk is originated from the trial solution x(k).

Figure 5.2 The steepest descent method: ϕ(x(k)) and ϕ(x(k) + 2αke(k)) are on the same contour of the error function, and the steepest descent algorithm moves to x(k) + αke(k) the midpoint in between.

Figure 5.3 Illustration of the steepest descent method: Start from the trial solution x(1), which is updated along the steepest descent direction e(1), then c05-math-081 , superposed on top of the contours of the error function ϕ(x).

Figure 5.4 Solid lines illustrate solution updates in the conjugate gradient method, whereas dashed lines are the solution update vectors in the steepest descent method.

Chapter 6: Regularisation

Figure 6.1 The Cauchy distribution describes the distribution of a random angle θ with respect to the vertical axis. The height is fixed to be λ, that is, the rotation point of a line segment is fixed, and the tilted line segment cuts the horizontal axis at distance x.

Figure 6.2 The regularisation function R(λ) versus the Cauchy parameter λ. The estimated λ value corresponding to the minimum R value is λ = 0.0033.

Figure 6.3 Schematic comparison between an exponential probability density function (solid curve) and a Gaussian probability density function (dotted curve). The exponential probability density function is longer-tailed than the Gaussian function.

Figure 6.4 Schematic comparison between the maximum entropy probability function (solid curve) and a Gaussian probability function (dotted curve). The maximum entropy constraint suppresses the random variables with low absolute values and pushes random values away from the mean zero.

Figure 6.5 Schematic comparison between a Cauchy probability density function (solid curve) and a Gaussian probability density function (dotted curve).

Chapter 7: Localised average solutions

Figure 7.1 Two spatial series, R(r), where r is the space coordinate, have the same variance, but the bottom case is better localised at r0. Localisation can be measured quantitatively by the spread length, with respect to r0.

Figure 7.2 A trade-off curve for appraising solution estimates. The horizontal axis is the spread (or the resolution width) and the vertical axis is the variance of the solution estimates. The best trade-off occurs at the most closest point to the origin in the curve whose corresponding value of θ = θ0 is the best trade-off parameter.

Chapter 8: Seismic wavelet estimation

Figure 8.1 Two window functions. A Gaussian function g(τ) (the solid curve) with the standard deviation σ = T/π, where T is a half of the expected wavelet length, and the Parzen window function p(τ) (the dashed curve).

Figure 8.2 A seismic trace (top) at a well location, a reflectivity series (the middle) calculated based on well-log information, and the wavelet (the bottom) extracted from well-seismic correlation.

Figure 8.3 The amplitude spectrum of a seismic trace (the solid curve) and the amplitude spectrum of a wavelet (the dotted curve). Both the seismic trace and the estimated wavelet were shown in Figure 8.2.

Figure 8.4 (a) A field seismic profile, consisting of 600 traces. The vertical straight line indicates a vertical borehole, and the black curve is the acoustic impedance calculated from logging information. (b) The amplitude spectrum of a seismic profile (the fluctuated curve) and the spectrum of the generalised wavelet (the smooth curve). (c) The wavelet extracted using the generalised wavelet method (the solid curve) and a constant-phase wavelet (the dashed curve) extracted from the well-seismic correlation. (d) The spectra of the generalised wavelet (the solid curve) and the constant-phase wavelet (the dashed curve).

Figure 8.5 (a) A seismic reflectivity profile. Because of the spikiness, the kurtosis of this reflectivity profile is very high c08-math-051 . (b) A seismic section generated by convolution of the reflectivity profile and a wavelet (in the corner). After the convolution, the kurtosis is reduced drastically c08-math-052 .

Figure 8.6 Constant-phase wavelets with different phase angles, and the kurtosis values of corresponding seismic profiles. The seismic profile with a zero-phased wavelet has the largest kurtosis value.

Figure 8.7 The kurtosis c08-math-070 versus the scanning rotation angle θ, applied to the seismic profile. The maximum kurtosis occurs at c08-math-071 , which is the negative value of the wavelet phase angle θ0.

Figure 8.8 (a) A slice of the fourth-order cumulant, c08-math-099 . (b) A slice of the 3-D Gaussian window a(τ1,τ2,τ3 = 0). (c) A slice of the approximate fourth-order moment of the wavelet, c08-math-100 , which is a windowed fourth-order cumulant of the seismic trace. (d) A slice of the fourth-order moment of the wavelet c08-math-101 .

Figure 8.9 (a) A group of five seismic traces (zero mean). (b) Estimated zero-phase wavelet. (c) Estimated constant-phase wavelet. (d) Estimated mixed-phase wavelet using the cumulant matching inversion scheme.

Chapter 9: Seismic reflectivity inversion

Figure 9.1 (a) A seismic section. (b) Reflectivity solution of the L2 norm method. (c) Reflectivity solution of the L1 norm method. (d) Reflectivity solution of the Lp norm method with p = 0.1. This inversion solution is sparse and spiky enough to be the reflectivity series.

Figure 9.2 Seismic reflectivity inversion with Cauchy constraint. (a) A seismic profile. (b) The first iteration result, which is a least-squares solution. (c) The inversion results after the second iteration. (d) The inversion results after the eighth iteration.

Figure 9.3 The amplitude spectra of inverted reflectivity series. The curve labelled by ‘Iteration = 1’ is the spectrum of the result with no sparseness constraint and others are sparse solutions during different iterations, which gradually flatten the spectrum with more high-frequency components in reflectivity series.

Figure 9.4 The statistical information of seismic reflectivity after a different number of iterations. (a) Reflectivity distribution after the first iteration, which is in Gaussian distribution. (b) Cauchy fitting of the reflectivity distribution after the second iteration (the first Cauchy constrained inversion). (c) After the eighth iteration, the reflectivity distribution changes towards Gaussian (solid line) rather than Cauchy (dashed line).

Figure 9.5 (a) A synthetic seismic profile. (b) The reflectivity profile obtained by the standard single channel deconvolution. (c) The reflectivity profile obtained by the multichannel inversion algorithm.

Figure 9.6 (a) A field seismic profile. (b) The reflectivity profile obtained by the multichannel reflectivity inversion method.

Figure 9.7 (a) A zoomed-in field seismic profile. (b) The reflectivity profile obtained by the single channel inversion, followed by spatial prediction. (c) The reflectivity profile obtained by multichannel reflectivity inversion (the first three steps), followed by a spatial prediction (the fourth and final step).

Chapter 10: Seismic ray-impedance inversion

Figure 10.1 Difference between two formulae for the acoustic impedance ratio.

Figure 10.2 Cross-plots of the ray impedance at p = 150 ms/km and that at p = 0, for (a) revealing the shale content (%) within a target reservoir at depth 2200–2300 m, and (b) the porosity (%) of different sand bodies.

Figure 10.3 The PP- and PS-wave reflections with a constant ray-parameter (p) share the same reflection point. Therefore, the PP-wave and PS-wave data can be used jointly for the elastic parameters.

Figure 10.4 A typical workflow for the ray-impedance inversion.

Figure 10.5 A common-reflection-point (CRP) gather in the offset domain and in the ray-parameter domain.

Figure 10.6 CRP gathers, presented in the ray-parameter (p) domain, are sorted into constant-p sections.

Figure 10.7 (a) A seismic section across two boreholes A and B. (b) Crosshole seismic velocity image obtained from waveform tomography.

Figure 10.8 (a) A constant ray-parameter section (with p = 150 ms/km). (b) Inverted reflectivity section, corresponding to the constant p value. (c) The result of ray-impedance inversion, in which a crosshole seismic constraint is used.

Chapter 11: Seismic tomography based on ray theory

Figure 11.1 (a) A layered Earth structure with interpolation between interfaces. A reflected ray trajectory intersects all the interfaces and interpolated levels. (b) The geometry of seismic reflection rays within a variable velocity medium, reflected from a curved interface.

Figure 11.2 Geometry of incidence and reflection (or refraction), ϕ and ϕ′ are the angles of incidence and reflection (or refraction) for a ray of take-off angle ψ, ϕm and c011-math-018 represent modified angles where the ray take-off angle is ψ + Δψ. N0 represents the source point, c011-math-019 is the virtual image of the source, N1 is the incident point, and N2 is the initial observation point. The distance between N0 and N1 is ℓ1, between c011-math-020 and N1 is c011-math-021 , and between N1 and N2 is ℓ2.

Figure 11.3 A 2-D stratified structure model considered in the joint inversion of traveltime and amplitude data, inverting simultaneously for the interface geometry and the elastic parameters along the interfaces.

Figure 11.4 Reflection seismic profile from the North Sea. Traveltimes and amplitudes, extracted from migrated CRP gathers, are used in the inversion for interface geometry and elastic parameters along reflectors. Four reflections c011-math-043 are considered in the inversion. Labels A, B, C and D indicate phase changes of reflection events.

Figure 11.5 Reflection times picked from the reference profile in Figure 11.3. The reflection time is used as a reference time in a cross-correlation procedure for picking the relative reflection traveltimes and amplitudes on individual traces of CRP gathers.

Figure 11.6 Simultaneous inversion of reflections 3, 4 and 5. In each case, the inversion model is consisted of a constant interval velocity, the interface geometry z and three elastic parameters (Δα/α, Δβ/β, β/α). The interface geometry (solid line) is compared with inversion result from traveltime inversion (dotted line).

Figure 11.7 Lithology log data and seismic traces. The first trace is the impedance log. The second trace is the broadband synthetic computed from well logs. The third trace is the estimated seismic wavelet. The fourth trace is the filtered synthetic, filtered by estimated seismic wavelet. The fifth trace is the field seismic segment. The synthetic and field seismic traces match accurately.

Chapter 12: Waveform tomography for the velocity model

Figure 12.1 (a) The Marmousi velocity model. (b) Waveform inversion image, in which only the top portion of the model is well-reconstructed. (c) Final waveform tomography image, with scaled model updates during the iterative inversion.

Figure 12.2 (a) A sample shot record with 120 traces. (b) The shot record after multiple attenuation. (c) The same shot record after partial compensation.

Figure 12.3 (a) A marine seismic section. (b) The section after pre-stack free-surface multiple attenuation. (c) The same section with amplitude and phase compensations applied to the shot records before stacking. The compensations to the shot records make the original point sources become the equivalent line sources, before they are used in waveform tomography.

Figure 12.4 Comparison between a seismic trace from a point-source (solid curve) and the trace after partial compensation (dashed curve) and their amplitude spectra. Wavelets in a trace from a line source (i.e. after partial compensation) are broader than those from a point-source.

Figure 12.5 Seismic waveform tomography. (a) The initial velocity model, built from traveltime tomography. (b–d) Three velocity models reconstructed by waveform tomography, using frequencies in the ranges of 6.9–7.5, 6.9–13.8 and 6.9–30 Hz, respectively.

Chapter 13: Waveform tomography with irregular topography

Figure 13.1 (a) Body-fitted grids (without boundary point modification). (b) Body-fitted grids with boundary-point modification. (c) Zoomed-in meshes between x = [100, 160] m, and z = [−10, −50] m, and between x = [180, 240] m and z = [100, 160] m. Because of boundary-point modification, both boundary and internal grids have a good orthogonal performance.

Figure 13.2 (a) Seismic wave simulation in a homogeneous fan area: the orthogonal meshes to partition the study area, plotted by each 5 grids, and a snapshot of the wavefield at a time 80 ms. (b) Wavefield simulation in a homogeneous area with a skewness: the meshes to partition the area, and a snapshot at 50 ms, showing a twisted wavefront because of the skewness of meshes.

Figure 13.3 (a) Velocity model with staircase boundary, caused by quadrate grids partition at a dip subsurface boundary, and the corresponding snapshot (at 190 ms), showing that the staircase boundary could cause a strong