Nonlinear amplitude versus angle inversion using hybrid quantum ant colony optimization and the exact Zoeppritz equation

The Zoeppritz equations describe the relationship between seismic properties that are useful for the interpretation of lithology and fluid properties. Various approximations to the Zoeppritz equations are often used in linear amplitude variation with offset and amplitude variation with angle inversions, assuming low-contrast boundaries. These approximations restrict the inversion method to not-too-large-angle cases and reduce the inversion accuracy. Therefore, it is necessary to develop joint PP and PS prestack inversion methods using the exact Zoeppritz equations. We have developed a method for nonlinear joint prestack inversion using the exact Zoeppritz equations. We established an objective function to combine PP and PS information based on a least-squares approach, and we used the Taylor expansion method to derive a model updating formula for the inversion. We also used a mean shift method to improve the accuracy of inversion results. We validated our inversion using synthetic and field seismic data. The model for calculating synthetic data contained high-contrast interfaces to match the reservoir layers, which included, for example, coal seams and unconsolidated sandstone. The outputs of the inversion were the elastic parameters (P- and S-wave velocities, as well as density), rather than the changes in elastic parameters. We found that the nonlinear joint prestack inversion achieved more accurate results than the linear joint prestack inversion, regardless of incidence angle size. Furthermore, if the prestack data were of high enough quality, it was possible to identify thin layers from the inversion result.

Young’s modulus and Poisson ratio are essential parameters for rock brittleness evaluation. At present, prestack inversions of these parameters mostly adopt linear approximation, whereas the nonlinear inversion methods that are based on the exact Zoeppritz equation are seldomly conducted. Nevertheless, in the case of large incident angles and high-impedance contrast, the nonlinear equation has higher accuracy than the linear approximation. It is more in line with the nature of geophysical inversion problems, and it describes the variation of formations’ elastic parameters better than the linear approximation does. Therefore, based on the exact solution of the Zoeppritz equation, we have parameterized and derived a new nonlinear reflection coefficient equation using nonlinear Young’s modulus, Poisson ratio, and density (YPD) as variables. Subsequently, by combining the inverse operator algorithm with the YPD equation, we develop a new nonlinear amplitude-variation-with-offset inversion scheme. This approach can robustly extract Young’s modulus and Poisson ratio from seismic data and identify shale-gas sweet spot locations within the reservoir, as well as predict reservoir fluid properties. We have tested the accuracy and applicability of this approach with synthetic and real data examples.

Improving seismic interpretation: a high-contrast approximation to the reflection coefficient of a plane longitudinal wave

This paper presents a new method for amplitude versus offset(AVO) nonlinear inversion that solved by support vector machine(SVM).First the main principle of SVM is described,then the SVM model are established for AVO inversion.Finally,both real data and synthetic model are inverted based on SVM,the results show that the proposed method can resolve limitations associated with conventional inversion methods without sacrificing inversion effects;it can extract elastic parameters from synthetic seismogram directly;meanwhile the method is characterized by high inversion velocity and good stability.

The conventional Markov chain Monte Carlo (MCMC) method is limited to the selected shape and size of proposal distribution and is not easy to start when the initial proposal distribution is far away from the target distribution. To overcome these drawbacks of the conventional MCMC method, two useful improvements in MCMC method, adaptive Metropolis (AM) algorithm and delayed rejection (DR) algorithm, are attempted to be combined. The AM algorithm aims at adapting the proposal distribution by using the generated estimators, and the DR algorithm aims at enhancing the efficiency of the improved MCMC method. Based on the improved MCMC method, a Bayesian amplitude versus offset (AVO) inversion method on the basis of the exact Zoeppritz equation has been developed, with which the P- and S-wave velocities and the density can be obtained directly, and the uncertainty of AVO inversion results has been estimated as well. The study based on the logging data and the seismic data demonstrates the feasibility and robustness of the method and shows that all three parameters are well retrieved. So the exact Zoeppritz-based nonlinear inversion method by using the improved MCMC is not only suitable for reservoirs with strong-contrast interfaces and long-offset ranges but also it is more stable, accurate and anti-noise.

Prestack AVA inversion of exact Zoeppritz equations based on modified Trivariate Cauchy distribution

Seismic pre-stack AVA inversion using the Zoeppritz equation and its approximations as a forward engine yields P- and S-wave velocities and density. Due to the presence of seismic noise and other factors, the solution to seismic inversion is generally ill-posed and it is necessary to add constraints to regularize the algorithm. Moreover, since pre-stack inversion is a nonlinear problem, linearized optimization algorithms may fall into false local minima. The simulated annealing (SA) algorithm, on the other hand, is capable of finding the global optimal solution regardless of the initial model. However, when applied to multi-parameter pre-stack inversion, standard SA suffers from instability. Thus, a nonlinear pre-stack inversion method is proposed based on lithology constraints. Specifically, correlations among the elastic parameters are introduced to establish constraints based on a Bayesian framework, with special intention of mitigating the ill-posedness of the inversion problem as well as addressing the lithological characteristics of the formations. In particular, to improve the stability, a multivariate Gaussian distribution of elastic parameters is incorporated into the model updating the SA algorithm. We apply the algorithm to synthetic and field seismic data, indicating that the proposed method has a good resolution and stability performance.

Crosshole seismic tomography often is applied to image the velocity structure of an interwell medium. If the rocks are anisotropic, the tomographic technique must be adapted to the complex situation; otherwise, it leads to a false interpretation. We propose a nonlinear kinematic inversion method for crosshole seismic tomography in composite transversely isotropic media with known dipping symmetry axes. This method is based on a new version of the first-order traveltime perturbation equation. It directly uses the derivative of the phase velocity rather than the eigenvectors of the body-wave modes to overcome the singularity problem for application to the two quasi-shear waves. We applied an iterative nonlinear solver incorporating our kinematic ray-tracing scheme and directly compute the Jacobian matrix in an arbitrary reference medium. This reconstructs the five elastic moduli or Thomsen parameters from the first-arrival traveltimes of the three seismic body waves (qP, qSV, qSH) in strongly and weakly anisotropic media. We conducted three synthetic experiments that involve determining anisotropic parameters for a homogeneous rock, reconstructing a fault embedded in a strongly anisotropic background, and imaging a complicated four-layer model containing a small channel and a buried dipping interface. We compared results of our nonlinear inversion method with isotropic tomography and the traditional linear anisotropic inversion scheme, which showed the capability and superiority of the new scheme for crosshole tomographic imaging.

Seismic Amplitude Inversion in Reflection Tomography

Rock-physics inversion has been widely used in reservoir prediction. However, the prediction and evaluation of source rocks still predominantly rely on data-driven inversion methods. To directly incorporate the rock-physics relationship into the seismic inversion of source rocks, we first introduce a linearization approximation of the rock-physics model through the first-order approximation of the Taylor series. Model examples and well-logging tests substantiate the reliability of this linear approximation in establishing the connection between the elastic moduli and essential physical parameters (EPPs) of source rocks. Subsequently, we develop a novel equation for the P-wave reflection coefficient by combining the linear rock-physics relationship and Gray approximation. The newly developed reflectivity equation is a linear expression of differences in EPPs of source rocks. Model-simulating results demonstrate a favorable agreement between the novel P-wave reflectivity equation and the exact Zoeppritz equation until the incident angle reaches 60°. Furthermore, we explore the amplitude-variation-with-angle (AVA) effects of EPPs to validate their contributions meet the requirement of seismic inversion. Under the Bayesian scheme, we further develop an AVA inversion method, based on the novel P-wave reflectivity equation, to predict the EPPs of source rocks. Our AVA inversion method obtains satisfactory results under noise testing and demonstrates promising application potential in field examples. The predicted EPPs indicate a strong agreement with the actual well loggings. Comparative analysis against data-driven inversion results reveals that the EPPs predicted by our AVA inversion method possess enhanced rock-physics support and significance. Specifically, this study focuses on isotropic clay-rich source rocks with low maturity.

This paper presents an efficient three-dimensional nonlinear electromagnetic inversion method in a multilayered medium for geophysical exploration. The stabilized biconjugate-gradient fast Fourier transform algorithm is used for forward modeling, whereas the distorted Born iterative method is used in the inversion. The conjugate-gradient method is applied at each inversion iteration to minimize the cost function. The usage of an iterative solver based on the FFT algorithm and the developed recursive matrix method combined with an interpolation technique to evaluate the layered medium Green's functions rapidly, make this method highly efficient, with the requirement of very low time and memory, O (N log N ) and O (N) respectively. Numerical experiments about the 3D forward modeling with application in Borehole to Surface Electromagnetic Method will be presented in this paper; in addition, we will show the examples about 3D inversion applied in geophysical exploration. All these results will demonstrate the high efficiency and accuracy of the proposed method.

First-order approximate analytical expressions of oblique incident elastic wave at an interface of porous media saturated with a non-viscous fluid

Magnetotelluric is a geophysical method which uses natural time variations of the Earth’s magnetic and electric fields. One-dimensional magnetotelluric inversion was carried out to estimate the electrical resistivity distribution of rocks which varied with depth. Non-linear relationships between model parameters and observation data lead to difficulties in the inversion process. This problem can be solved with a non-linear inversion method with a global search technique. One of the global search technique that gaining interest in the inverse problem is Particle Swarm Optimization (PSO). PSO can find acceptable solutions from a very broad set of initial parameters. This study also carried out a non-linear local search technique, namely Levenberg-Marquardt (LM) which has been used globally for the geophysical problem. Then the two inversion algorithm was compared in determining the best model solution. There are two data used in this study which are generated from a three-layer model, namely data without noise and also data that has been added by Gaussian noise with 10% standard deviation. It was found that the results of the two inversions were quite good at determining the actual model. LM algorithm is an algorithm that is truly influenced by the initial value of the damping factor ε2 while PSO algorithm depends on several parameters, namely number of particles n, inertia weight w, cognitive parameter c1 and social parameter c2. Trial experiments suggested that the global best solution could be achieved with controlling parameters wmin = 0.4, wmax = 0.9, c1 = 1.3 and c2 = 1.5 without noise data and wmin = 0.7, wmax = 0.9, c2 = 0.8 and c2 = 1.4 for data with 10% noise.

During my PhD, starting from the original formulations proposed by Bertrand et al., 2000 and Emolo & Zollo 2005, I developed inversion methods and applied then at different earthquakes. In particular large efforts have been devoted to the study of the model resolution and to the estimation of the model parameter errors. To study the source kinematic characteristics of the Christchurch earthquake we performed a joint inversion of strong-motion, GPS and InSAR data using a non-linear inversion method. Considering the complexity highlighted by superficial deformation data, we adopted a fault model consisting of two partially overlapping segments, with dimensions 15x11 and 7x7 km2, having different faulting styles. This two-fault model allows to better reconstruct the complex shape of the superficial deformation data. The total seismic moment resulting from the joint inversion is 3.0x1025 dyne.cm (Mw = 6.2) with an average rupture velocity of 2.0 km/s. Errors associated with the kinematic model have been estimated of around 20-30 %. The 2009 Aquila sequence was characterized by an intense aftershocks sequence that lasted several months. In this study we applied an inversion method that assumes as data the apparent Source Time Functions (aSTFs), to a Mw 4.0 aftershock of the Aquila sequence. The estimation of aSTFs was obtained using the deconvolution method proposed by Vallee et al., 2004. The inversion results show a heterogeneous slip distribution, characterized by two main slip patches located NW of the hypocenter, and a variable rupture velocity distribution (mean value of 2.5 km/s), showing a rupture front acceleration in between the two high slip zones. Errors of about 20% characterize the final estimated parameters.

The statistical independence between the distributions of different chromophores in tissue has previously been used for linear unmixing with independent component analysis (ICA). In this study, we propose exploiting this statistical property in a nonlinear model-based inversion method. The aim is to reduce the sensitivity of the inversion scheme to errors in the modelling of the fluence, and hence provide more accurate quantification of the concentration of independent chromophores. A gradient-based optimisation algorithm is used to minimise the error functional, which includes a term representing the mutual information between the chromophores in addition to the standard least-squares data error. Both numerical simulations and an experimental phantom study are conducted to demonstrate that, in the presence of experimental errors in the fluence model, the proposed inversion method results in more accurate estimation of the concentrations of independent chromophores compared to the standard model-based inversion.