Geophysical Inverse Problems Research Articles

Coupling strategies in multiparameter geophysical joint inversion

SUMMARYA big potential lies in the quantitative integration of multiple geophysical measurements for what concerns more unique and robust inversion results, complementary sensitivity to geological features and enhanced resolution. The mechanisms that enable such integration typically rely on statistics where the multiphysics parameter values are related to each other through regression functions (rock physics) or through the shape of the parameter spatial distributions (structure). Such coupling operators are jointly minimized with the data misfit to obtain coupled parameter distributions. We explore the application of various coupling mechanisms to synthetic and real data comprising seismic and electromagnetic measurements acquired in complex geological conditions such as salt geology and complex near surface in desert environment. Two types of structure operators, consisting of the standard cross-gradient and a newly introduced summative gradient, together with rock-physics operators are tested and combined for velocity model reconstruction of salt overburden in a marine environment and in a complex near surface case. Results of seismic-EM joint minimization suggest that the summative gradient operator with the introduction of the sign of the gradient correlation provides a strong coupling mechanism that can become advantageous in the presence of noise-affected seismic data. The rock-physics coupling mechanism is extremely strong and its effectiveness depends primarily on the reliability of the rock-physics relation. A Bayesian approach for the rock-physics operator is introduced to balance the related uncertainties and it is successfully tested in a joint minimization scheme. The combined use of rock-physics and structure coupling operators provides the best results in synthetic and real data applications. The use and combination of various regularization operators, as described, provide a formidable toolbox for solving a wide variety of ill-posed and non-unique geophysical inverse problems.

An improved grey wolf optimizer algorithm for the inversion of geoelectrical data

The grey wolf optimizer (GWO) is a novel bionics algorithm inspired by the social rank and prey-seeking behaviors of grey wolves. The GWO algorithm is easy to implement because of its basic concept, simple formula, and small number of parameters. This paper develops a GWO algorithm with a nonlinear convergence factor and an adaptive location updating strategy and applies this improved grey wolf optimizer (improved grey wolf optimizer, IGWO) algorithm to geophysical inversion problems using magnetotelluric (MT), DC resistivity and induced polarization (IP) methods. Numerical tests in MATLAB 2010b for the forward modeling data and the observed data show that the IGWO algorithm can find the global minimum and rarely sinks to the local minima. For further study, inverted results using the IGWO are contrasted with particle swarm optimization (PSO) and the simulated annealing (SA) algorithm. The outcomes of the comparison reveal that the IGWO and PSO similarly perform better in counterpoising exploration and exploitation with a given number of iterations than the SA.

Practical aspects of using a neural network to solve inverse geophysical problems

In this paper, an approach to solve an inverse problem of geophysics, such as determining the position of an object (cavity or cavern) and its geometrical parameters according to the propagation picture of a wave field, is proposed. At present there are no fast and accurate methods for determining such parameters. In this paper, a method based on neural networks (NNs) is proposed and a possible architecture of the NN is presented. The results of experiments on implementing and training the NN are also presented. The model obtained shows the presence of an "understanding" of the input data, demonstrating answers that are similar to the original data. In the NN answers, one can identify a relationship between the quality of the network response and the number of waves that have passed through the medium’s object being investigated.

Trans-dimensional Bayesian inversion of airborne electromagnetic data for 2D conductivity profiles

This paper presents the application of a novel trans-dimensional sampling approach to a time domain airborne electromagnetic (AEM) inverse problem to solve for plausible conductivities of the subsurface. Geophysical inverse field problems, such as time domain AEM, are well known to have a large degree of non-uniqueness. Common least-squares optimisation approaches fail to take this into account and provide a single solution with linearised estimates of uncertainty that can result in overly optimistic appraisal of the conductivity of the subsurface. In this new non-linear approach, the spatial complexity of a 2D profile is controlled directly by the data. By examining an ensemble of proposed conductivity profiles it accommodates non-uniqueness and provides more robust estimates of uncertainties.

Geostatistical regularization operators for geophysical inverse problems on irregular meshes

Summary Irregular meshes allow to include complicated subsurface structures into geophysical modelling and inverse problems. The non-uniqueness of these inverse problems requires appropriate regularization that can incorporate a priori information. However, defining regularization operators for irregular discretizations is not trivial. Different schemes for calculating smoothness operators on irregular meshes have been proposed. In contrast to classical regularization constraints that are only defined using the nearest neighbours of a cell, geostatistical operators include a larger neighbourhood around a particular cell. A correlation model defines the extent of the neighbourhood and allows to incorporate information about geological structures. We propose an approach to calculate geostatistical operators for inverse problems on irregular meshes by eigendecomposition of a covariance matrix that contains the a priori geological information. Using our approach, the calculation of the operator matrix becomes tractable for 3-D inverse problems on irregular meshes. We tested the performance of the geostatistical regularization operators and compared them against the results of anisotropic smoothing in inversions of 2-D surface synthetic electrical resistivity tomography (ERT) data as well as in the inversion of a realistic 3-D cross-well synthetic ERT scenario. The inversions of 2-D ERT and seismic traveltime field data with geostatistical regularization provide results that are in good accordance with the expected geology and thus facilitate their interpretation. In particular, for layered structures the geostatistical regularization provides geologically more plausible results compared to the anisotropic smoothness constraints.

Diagnostics of the shape and orientation of a rock failure zone based on electrical measurements as an inverse problem of geophysics

Exploration geophysics is concerned with the development of methods and techniques for remote search for mineral deposits, solution of various engineering and geological problems, mining monitoring, and diagnostics of zones of rock bursts and natural and technogenic tectonic earthquakes. In this paper, we consider the diagnostic problem of exploration geophysics related to the determination of the shape and inclination angle of a plane source of natural geoelectric field, which, under certain conditions, simulates a rock failure zone, e.g., during the preparation of rock bursts at the sites of developed mineral deposits. This approach may also be useful in determining the shape and size of ore shoots by electrical measurements on the ground surface. Evaluation of the parameters of a rock failure zone is required in the case of accumulation of multiple fractures having charges of the same sign before a catastrophic failure. This problem is formulated as a Fredholm-Urysohn integral equation of the first kind. The solution of the integral equation is sought using the Tikhonov regularization method of the second order.

EIT in a layered anisotropic medium

We consider the inverse problem in geophysics of imaging the subsurface of the Earth in cases where a region below the surface is known to be formed by strata of different materials and the depths and thicknesses of the strata and the (possibly anisotropic) conductivity of each of them need to be identified simultaneously. This problem is treated as a special case of the inverse problem of determining a family of nested inclusions in a medium \begin{document}$Ω\subset\mathbb{R}^n$\end{document} , \begin{document}$n ≥ 3$\end{document} .

Combined method of F-, S-, and R-approximations in solving the problems of geophysics and geomorphology

The interrelation between different modifications of the method of linear integral representation is studied. Combined approximations of the topography and geopotential fields enable more refined tuning of the method in solving inverse problems of geophysics and geomorphology and provide a more complete allowance for the a priori information about the surface elevation data and elements of anomalous fields. A technique for finding the numerical solution for the inverse problem for determining the mass distributions equivalent in terms of the external field is presented. The results of the mathematical experiment are discussed.

Parameter estimation by Differential Search Algorithm from horizontal loop electromagnetic (HLEM) data

We present an efficient inversion tool for parameter estimation from horizontal loop electromagnetic (HLEM) data using Differential Search Algorithm (DSA) which is a swarm-intelligence-based metaheuristic proposed recently. The depth, dip, and origin of a thin subsurface conductor causing the anomaly are the parameters estimated by the HLEM method commonly known as Slingram. The applicability of the developed scheme was firstly tested on two synthetically generated anomalies with and without noise content. Two control parameters affecting the convergence characteristic to the solution of the algorithm were tuned for the so-called anomalies including one and two conductive bodies, respectively. Tuned control parameters yielded more successful statistical results compared to widely used parameter couples in DSA applications. Two field anomalies measured over a dipping graphitic shale from Northern Australia were then considered, and the algorithm provided the depth estimations being in good agreement with those of previous studies and drilling information. Furthermore, the efficiency and reliability of the results obtained were investigated via probability density function. Considering the results obtained, we can conclude that DSA characterized by the simple algorithmic structure is an efficient and promising metaheuristic for the other relatively low-dimensional geophysical inverse problems. Finally, the researchers after being familiar with the content of developed scheme displaying an easy to use and flexible characteristic can easily modify and expand it for their scientific optimization problems.

Iteratively re‐weighted and refined least squares algorithm for robust inversion of geophysical data

ABSTRACTA robust metric of data misfit such as the ℓ1‐norm is required for geophysical parameter estimation when the data are contaminated by erratic noise. Recently, the iteratively re‐weighted and refined least‐squares algorithm was introduced for efficient solution of geophysical inverse problems in the presence of additive Gaussian noise in the data. We extend the algorithm in two practically important directions to make it applicable to data with non‐Gaussian noise and to make its regularisation parameter tuning more efficient and automatic. The regularisation parameter in iteratively reweighted and refined least‐squares algorithm varies with iteration, allowing the efficient solution of constrained problems. A technique is proposed based on the secant method for root finding to concentrate on finding a solution that satisfies the constraint, either fitting to a target misfit (if a bound on the noise is available) or having a target size (if a bound on the solution is available). This technique leads to an automatic update of the regularisation parameter at each and every iteration. We further propose a simple and efficient scheme that tunes the regularisation parameter without requiring target bounds. This is of great importance for the field data inversion where there is no information about the size of the noise and the solution. Numerical examples from non‐stationary seismic deconvolution and velocity‐stack inversion show that the proposed algorithm is efficient, stable, and robust and outperforms the conventional and state‐of‐the‐art methods.

Nonlinear Rayleigh wave inversion based on the shuffled frog-leaping algorithm

At present, near-surface shear wave velocities are mainly calculated through Rayleigh wave dispersion-curve inversions in engineering surface investigations, but the required calculations pose a highly nonlinear global optimization problem. In order to alleviate the risk of falling into a local optimal solution, this paper introduces a new global optimization method, the shuffle frog-leaping algorithm (SFLA), into the Rayleigh wave dispersion-curve inversion process. SFLA is a swarm-intelligence-based algorithm that simulates a group of frogs searching for food. It uses a few parameters, achieves rapid convergence, and is capability of effective global searching. In order to test the reliability and calculation performance of SFLA, noise-free and noisy synthetic datasets were inverted. We conducted a comparative analysis with other established algorithms using the noise-free dataset, and then tested the ability of SFLA to cope with data noise. Finally, we inverted a real-world example to examine the applicability of SFLA. Results from both synthetic and field data demonstrated the effectiveness of SFLA in the interpretation of Rayleigh wave dispersion curves. We found that SFLA is superior to the established methods in terms of both reliability and computational efficiency, so it offers great potential to improve our ability to solve geophysical inversion problems.

On the optimization of electromagnetic geophysical data: Application of the PSO algorithm

Particle Swarm optimization (PSO) algorithm resolves constrained multi-parameter problems and is suitable for simultaneous optimization of linear and nonlinear problems, with the assumption that forward modeling is based on good understanding of ill-posed problem for geophysical inversion.We apply PSO for solving the geophysical inverse problem to infer an Earth model, i.e. the electrical resistivity at depth, consistent with the observed geophysical data.The method doesn't require an initial model and can be easily constrained, according to external information for each single sounding.The optimization process to estimate the model parameters from the electromagnetic soundings focuses on the discussion of the objective function to be minimized. We discuss the possibility to introduce in the objective function vertical and lateral constraints, with an Occam-like regularization. A sensitivity analysis allowed us to check the performance of the algorithm. The reliability of the approach is tested on synthetic, real Audio-Magnetotelluric (AMT) and Long Period MT data. The method appears able to solve complex problems and allows us to estimate the a posteriori distribution of the model parameters.

Accounting for model error in Bayesian solutions to hydrogeophysical inverse problems using a local basis approach

Abstract Bayesian solutions to geophysical and hydrological inverse problems are dependent upon a forward model linking subsurface physical properties to measured data, which is typically assumed to be perfectly known in the inversion procedure. However, to make the stochastic solution of the inverse problem computationally tractable using methods such as Markov-chain-Monte-Carlo (MCMC), fast approximations of the forward model are commonly employed. This gives rise to model error, which has the potential to significantly bias posterior statistics if not properly accounted for. Here, we present a new methodology for dealing with the model error arising from the use of approximate forward solvers in Bayesian solutions to hydrogeophysical inverse problems. Our approach is geared towards the common case where this error cannot be (i) effectively characterized through some parametric statistical distribution; or (ii) estimated by interpolating between a small number of computed model-error realizations. To this end, we focus on identification and removal of the model-error component of the residual during MCMC using a projection-based approach, whereby the orthogonal basis employed for the projection is derived in each iteration from the K-nearest-neighboring entries in a model-error dictionary. The latter is constructed during the inversion and grows at a specified rate as the iterations proceed. We demonstrate the performance of our technique on the inversion of synthetic crosshole ground-penetrating radar travel-time data considering three different subsurface parameterizations of varying complexity. Synthetic data are generated using the eikonal equation, whereas a straight-ray forward model is assumed for their inversion. In each case, our developed approach enables us to remove posterior bias and obtain a more realistic characterization of uncertainty.

Comparing the performances of four stochastic optimisation methods using analytic objective functions, 1D elastic full‐waveform inversion, and residual static computation

ABSTRACTWe compare the performances of four stochastic optimisation methods using four analytic objective functions and two highly non‐linear geophysical optimisation problems: one‐dimensional elastic full‐waveform inversion and residual static computation. The four methods we consider, namely, adaptive simulated annealing, genetic algorithm, neighbourhood algorithm, and particle swarm optimisation, are frequently employed for solving geophysical inverse problems. Because geophysical optimisations typically involve many unknown model parameters, we are particularly interested in comparing the performances of these stochastic methods as the number of unknown parameters increases. The four analytic functions we choose simulate common types of objective functions encountered in solving geophysical optimisations: a convex function, two multi‐minima functions that differ in the distribution of minima, and a nearly flat function. Similar to the analytic tests, the two seismic optimisation problems we analyse are characterised by very different objective functions. The first problem is a one‐dimensional elastic full‐waveform inversion, which is strongly ill‐conditioned and exhibits a nearly flat objective function, with a valley of minima extended along the density direction. The second problem is the residual static computation, which is characterised by a multi‐minima objective function produced by the so‐called cycle‐skipping phenomenon. According to the tests on the analytic functions and on the seismic data, genetic algorithm generally displays the best scaling with the number of parameters. It encounters problems only in the case of irregular distribution of minima, that is, when the global minimum is at the border of the search space and a number of important local minima are distant from the global minimum. The adaptive simulated annealing method is often the best‐performing method for low‐dimensional model spaces, but its performance worsens as the number of unknowns increases. The particle swarm optimisation is effective in finding the global minimum in the case of low‐dimensional model spaces with few local minima or in the case of a narrow flat valley. Finally, the neighbourhood algorithm method is competitive with the other methods only for low‐dimensional model spaces; its performance sensibly worsens in the case of multi‐minima objective functions.

Paleoclimate's factors of reconstruction of thermal history of petroleum bazhenov and togur suites southeastern West Siberia

The regions of the West Siberian oil and gas province have unique paleoclimatic features: 1) Mesozoic-Cenozoic secular course of temperature, individual for regional paleoclimatic zones; 2) different scale processes of formation and degradation of Neopleistocene permafrost sections; 3) zonal and periodically forming late Quaternary ice sheets. The purpose of research is to assess the influence of factors paleoclimate (secular temperature course on the earth's surface and Neopleistocene permafrost strata) on the calculated geothermal regime of parent oil-source suites that determines the calculation of the density of hydrocarbon resources by three-dimensional-genetic method. The object of research are the Bazhenov and Togur deposits of Mesozoic and Cenozoic section, stripped by deep wells in the south-east of Western Siberia (North Festival and Luginetskoye hydrocarbon deposits of Tomsk region). Research is based on the method paleotemperature simulation — solving the inverse and direct problems of unsteady geothermy under conditions of sedimentation on multivariance of paleotectonic and paleotemperature reconstructions, on conformity assessment of results reconstructions to the criteria of optimal solution of the inverse problems of geophysics and the degree of harmonization of rated centers of generation of oils with established oil and gas presence in the interior. The necessity to take into account Neopleistocene permafrost of about 300 meters thick and of the use of local (regional) secular progress of temperatures on the earth's surface for adequately recover of thermal history of parent oil source deposits on the lands of southeast Western Siberia has been established. In case of permafrost strata are neglected estimated hydrocarbon resources HR, determined by the volume-genetic method, can be underestimated up to 25—30 %. In the case of neglecting paleoclimatic temperatures running calculated HR can be underestimated from 50 % to 4 times lower.

The neural network approximation method for solving multidimensional nonlinear inverse problems of geophysics

The iterative approximation neural network method for solving conditionally well-posed nonlinear inverse problems of geophysics is presented. The method is based on the neural network approximation of the inverse operator. The inverse problem is solved in the class of grid (block) models of the medium on a regularized parameterization grid. The construction principle of this grid relies on using the calculated values of the continuity modulus of the inverse operator and its modifications determining the degree of ambiguity of the solutions. The method provides approximate solutions of inverse problems with the maximal degree of detail given the specified degree of ambiguity with the total number of the sought parameters ~n × 103 of the medium. The a priori and a posteriori estimates of the degree of ambiguity of the approximated solutions are calculated. The work of the method is illustrated by the example of the three-dimensional (3D) inversion of the synthesized 2D areal geoelectrical (audio magnetotelluric sounding, AMTS) data corresponding to the schematic model of a kimberlite pipe.

Polynomial chaos expansion for nonlinear geophysical inverse problems

There are lots of geophysical problems that include computationally expensive functions (forward models). Polynomial chaos (PC) expansion aims to approximate such an expensive equation or system with a polynomial expansion on the basis of orthogonal polynomials. Evaluation of this expansion is extremely fast because it is a polynomial function. This property of the PC expansion is of great importance for stochastic problems, in which an expensive function needs to be evaluated thousands of times. We have developed PC expansion as a novel technique to solve nonlinear geophysical problems. To better evaluate the methodology, we use PC expansion for automating the velocity analysis. For this purpose, we define the optimally picked velocity model as an optimizer of a variational integral in a semblance field. However, because computation of a variational integral with respect to a given velocity model is rather expensive, it is impossible to use stochastic methods to search for the optimal velocity model. Thus, we replace the variational integral with its PC expansion, in which computation of the new function is extremely faster than the original one. This makes it possible to perturb thousands of velocity models in a matter of seconds. We use particle swarm optimization as the stochastic optimization method to find the optimum velocity model. The methodology is tested on synthetic and field data, and in both cases, reasonable results are achieved in a rather short time.

Differentiability of the objective in a class of coefficient inverse problems

In this paper we study the Fréchet differentiability of the objective functional for a quite general class of coefficient inverse problems. We present sufficient conditions for the existence of arbitrary-order differentials as well as formulae for the calculation of first and second order derivatives. The general case theorems are then applied to two real-world geophysical inverse problems.

Particle Swarm Optimization (PSO) for Magnetotelluric (MT) 1D Inversion Modeling

Particle Swarm Optimization (PSO) is one of nature-inspired optimization algorithms that adopts swarm (insects, school of fish, flock of birds etc.) behaviour in search for food or common target in a collaborative manner. The particles (or agents) in the swarm learn from their neighbours as well as themselves regarding the promising area in the search space. The information is then used to update their position in order to reach the target. The search algorithm of a particle is dictated by the best position of that particle during the process (individual learning term) and the best particle in its surroundings (social learning term) at a particular iteration. In terms of optimization, the particles are models defined by their parameters, while the promising area in the model space is characterized by a low misfit associated with optimum models. Being a global search approach, PSO is suitable for nonlinear inverse problem resolution. The algorithm was applied to a simple minimization problem for illustration purpose. The application of PSO in geophysical inverse problem is demonstrated by inversion of synthetic magnetotelluric (MT) data associated with simple 1D models with satisfactory results in terms of model recovery as well as data misfit.

Bayesian Gaussian Mixture Linear Inversion for Geophysical Inverse Problems

A Bayesian linear inversion methodology based on Gaussian mixture models and its application to geophysical inverse problems are presented in this paper. The proposed inverse method is based on a Bayesian approach under the assumptions of a Gaussian mixture random field for the prior model and a Gaussian linear likelihood function. The model for the latent discrete variable is defined to be a stationary first-order Markov chain. In this approach, a recursive exact solution to an approximation of the posterior distribution of the inverse problem is proposed. A Markov chain Monte Carlo algorithm can be used to efficiently simulate realizations from the correct posterior model. Two inversion studies based on real well log data are presented, and the main results are the posterior distributions of the reservoir properties of interest, the corresponding predictions and prediction intervals, and a set of conditional realizations. The first application is a seismic inversion study for the prediction of lithological facies, P- and S-impedance, where an improvement of 30% in the root-mean-square error of the predictions compared to the traditional Gaussian inversion is obtained. The second application is a rock physics inversion study for the prediction of lithological facies, porosity, and clay volume, where predictions slightly improve compared to the Gaussian inversion approach.