Blocky Inversion Research Articles

Fuzzy Constrained Inversion of Magnetotelluric Data Using Guided Fuzzy C-Means Clustering

Magnetotelluric (MT) inversion algorithms are an essential tool in exploration geophysics because they provide us with resistivity models of the subsurface. Inverse problems are ill-posed and non-unique. Resistivity models produced by geologically unconstrained MT inversions are typically smooth, which means that the geological interpretations obtained based on the resistivity models will be ambiguous and uncertain. To solve the above problems, geophysicists have made many efforts over the years to introduce any prior information (such as geological structure information and physical property data) into models, which could help to better constrain the inverted model with the inversion algorithms. In the last few years, there has been renewed interest in machine learning techniques. Various machine learning methods have been applied to inversions. Using the fuzzy c-means (FCM) clustering method, geophysicists proposed new inversion algorithms that are capable of building petrophysical information into the inversion. Although the FCM has been used and advanced in previous work, geophysicists concluded that a priori information of the correct number and value of cluster centers is very important for a successful FCM inversion. In fact, it is difficult to obtain the appropriate clustering information in some geophysical survey areas. In this study, we present an effective way to build the petrophysical information for the MT inversion based on FCM clustering algorithm. When the actual petrophysical information is insufficient, considering the characteristics of MT data, we perform a one-dimensional blocky inversion that can clearly identify the interface with distinct electrical property contrast; then, we obtain the number and value of cluster centers from 1D blocky inversion for the further MT inversion with FCM clustering. The algorithm uses guided FCM clustering to improve the model within the iterative minimization during MT inversion. In our MT inversion method, we integrate the geophysical inversion and geological differentiation into a unified scheme, which interact and enhance each other. The resistivity models obtained from the inversion respect the geophysical and petrophysical data and are easy to geologically interpret. We tested the algorithm using two synthetic examples and a field data example.

Post‐stack impedance blocky inversion based on analytic solution of viscous acoustic wave equation

ABSTRACTThe conventional impedance inversion method ignores the attenuation effect, transmission loss and inter‐layer multiple waves; the smooth‐like regularization approach makes the corresponding impedance solution excessively smooth. Both fundamentally limit the resolution of impedance result and lead to the inadequate ability of boundary characterization. Therefore, a post‐stack impedance blocky inversion method based on the analytic solution of viscous acoustic equation is proposed. Based on the derived recursive formula of reflections, the 1D viscous acoustic wave equation is solved analytically to obtain zero‐offset full‐wave field response. Applying chain rule, the analytical expression of the Fréchet derivative is derived for gradient‐descent non‐linear inversion. Combined with smooth constraints, the blocky constraints can be introduced into the Bayesian inference framework to obtain stable and well‐defined inversion results. According to the above theory, we firstly use model data to analyse the influence of incompleteness of forward method on seismic response, and further verify the effectiveness of the proposed method. Then the Q‐value sensitivity analysis of seismic trace is carried out to reduce the difficulty of Q‐value estimation. Finally, the real data from Lower Congo Basin in West Africa indicate that the proposed approach provide the high‐resolution and well‐defined impedance result. As a supplement and development of linear impedance inversion method, the non‐linear viscous inversion could recover more realistic and reliable impedance profiles.

“Unsmooth” 1D inversion of frequency domain marine controlled source EM data

The goal of geophysical inversion of electromagnetic (EM) data is to recover a model of the geoelectrical properties of the sub-surface. The standard practice for 1D inversion of marine controlled source EM (CSEM) data is to generate smooth conductivity models using least squares (L2-norm) methods. However, sedimentary geology is stratified and piece-wise continuous. As such, smooth resistivity models cannot represent this character. In response to this inconsistency, a means was sought to generate more geologically plausible, piece-wise continuous models. The common approach in the literature when generating piece-wise continuous inversion models is to regularize L2-norm methods in such a manner as to induce blocky behaviour. Although effective, these techniques are self-conflictive; forcing non-smooth behaviour from an implicitly smooth algorithm. In contrast, L1-norm inversion inherently produces piece-wise continuous models. To investigate the possible utility of this approach, a L1-norm inversion algorithm has been developed and tested on synthetic and real datasets. The L1-norm results were compared with those generated using an industry standard L2-norm algorithm. The synthetic inversions focused on previously published examples. The real data inversions focused on electric and magnetic field measurements recorded over the main reservoir sand of the Pluto gas field in block WA350-P, North West Shelf, WA. The L1-norm inversions recovered, to within the resolution limits of the CSEM method, the depth, thickness and resistivity of the synthetic geological models and the Pluto-1 resistivity well log, whilst fitting the input data to within noise. When compared against the L2-norm profiles, the L1-norm inversion more closely represented the stratified character of the sedimentary sequence. It was therefore concluded that L1-norm inversion is an attractive alternative to smooth L2-norm methods when blocky inversion models are desired.

Analysis of prior models for a blocky inversion of seismic AVA data

Resolving thinner layers and focusing layer boundaries better in inverted seismic sections are important challenges in exploration and production seismology to better identify a potential drilling target. Many seismic inversion methods are based on a least-squares optimization approach that can intrinsically lead to unfocused transitions between adjacent layers. A Bayesian seismic amplitude variation with angle (AVA) inversion algorithm forms sharper boundaries between layers when enforcing sparseness in the vertical gradients of the inversion results. The underlying principle is similar to high-resolution processing algorithms and has been adapted from digital-image-sharpening algorithms. We have investigated the Cauchy and Laplace statistical distributions for their potential to improve contrasts betweenlayers. An inversion algorithm is derived statistically from Bayes’ theorem and results in a nonlinear problem that requires an iterative solution approach. Bayesian inversions require knowledge of certain statistical properties of the model we want to invert for. The blocky inversion method requires an additional parameter besides the usual properties for a multivariate covariance matrix, which we can estimate from borehole data. Tests on synthetic and field data show that the blocky inversion algorithm can detect and enhance layer boundaries in seismic inversions by effectively suppressing side lobes. The analysis of the synthetic data suggests that the Laplace constraint performs more reliably, whereas the Cauchy constraint may not find the optimum solution by converging to a local minimum of the cost function and thereby introducing some numerical artifacts.

Regularized and Blocky 3D Controlled Source Electromagnetic Inversion

Since the interpretation of controlled source electromagnetic (CSEM) data proves challenging in complex geological settings, 3D CSEM anisotropic resistivity imaging problem is formulated as an inverse problem. A least-squares misflt functional is minimized with a quasi- Newton algorithm to cope with the large number of unknowns. Furthermore, model and data weights are applied to speed up the convergence of the non-linear inversion. A-priori information obtained for instance from seismic interpretation can be included either in a blocky inversion, i.e., where the resistivity cube is parameterized with a small number of parameters or with a regularized inversion. Since earth resistivity contrasts can be high and spatially well deflned, minimum norm support regularization terms are implemented. Although the blocky inversion proves quite powerful, we show that results can be misleading in complex geological settings. A combination of the blocky and regularized inversions may provide a more robust approach to interpret CSEM imaging results. We illustrate this with a deep-water example.

Image processing of 2D resistivity data for imaging faults

A methodology to locate automatically limits or boundaries between different geological bodies in 2D electrical tomography is proposed, using a crest line extraction process in gradient images. This method is applied on several synthetic models and on field data set acquired on three experimental sites during the European project PALEOSIS where trenches were dug. The results presented in this work are valid for electrical tomographies data collected with a Wenner-alpha array and computed with an l 1 norm (blocky inversion) as optimization method. For the synthetic cases, three geometric contexts are modelled: a vertical and a dipping fault juxtaposing two different geological formations and a step-like structure. A superficial layer can cover each geological structure. In these three situations, the method locates the synthetic faults and layer boundaries, and determines fault displacement but with several limitations. The estimated fault positions correlate exactly with the synthetic ones if a conductive (or no superficial) layer overlies the studied structure. When a resistive layer with a thickness of 6 m covers the model, faults are positioned with a maximum error of 1 m. Moreover, when a resistive and/or a thick top layer is present, the resolution significantly decreases for the fault displacement estimation (error up to 150%). The tests with the synthetic models for surveys using the Wenner-alpha array indicate that the proposed methodology is best suited to vertical and horizontal contacts. Application of the methodology to real data sets shows that a lateral resistivity contrast of 1:5–1:10 leads to exact faults location. A fault contact with a resistivity contrast of 1:0.75 and overlaid by a resistive layer with a thickness of 1 m gives an error location ranging from 1 to 3 m. Moreover, no result is obtained for a contact with very low contrasts (∼1:0.85) overlaid by a resistive soil. The method shows poor results when vertical gradients are greater than horizontal ones. This kind of image processing technique should be systematically used for improving the objectiveness of tomography interpretation when looking for limits between geological objects.

A comparison of smooth and blocky inversion methods in 2D electrical imaging surveys

Two-dimensional electrical imaging surveys are now widely used in engineering and environmental surveys to map moderately complex structures. In order to adequately resolve such structures with arbitrary resistivity distributions, the regularised least-squares optimisation method with a cell-based model is frequently used in the inversion of the electrical imaging data. The L norm based least-squares optimisation method that attempts to minimise the sum of squares of the spatial changes in the model resistivity is often used. The resulting inversion model has a smooth variation in the resistivity values. In cases where the true subsurface resistivity consists of several regions that are approximately homogenous internally and separated by sharp boundaries, the result obtained by the smooth inversion method is not optimal. It tends to smear out the boundaries and give resistivity values that are too low or too high. The blocky or L norm optimisation method can be used for such situations. This method attempts to minimise the sum of the absolute values of the spatial changes in the model resistivity. It tends to produce models with regions that are piecewise constant and separated by sharp boundaries. Results from tests of the smooth and blocky inversion methods with several synthetic and field data sets highlight the strengths and weaknesses of both methods. The smooth inversion method gives better results for areas where the subsurface resistivity changes in a gradual manner, while the blocky inversion method gives significantly better results where there are sharp boundaries. While fast computers and software have made the task of interpreting data from electrical imaging surveys much easier, it remains the responsibility of the interpreter to choose the appropriate tool for the task based on the available geological information.

A comparison of smooth and blocky inversion methods in 2-D electrical imaging surveys

Two-dimensional electrical imaging surveys are now widely used in engineering and environmental surveys to map moderately complex structures. In order to adequately resolve such structures with arbitrary resistivity distributions, the regularised least-squares optimisation method with a cell-based model is frequently used in the inversion of the electrical imaging data. The 12 norm or smoothness-constrained optimisation method that attempts to minimise the sum of squares of the spatial changes in the model resistivity is frequently used. The resulting inversion model has a smooth variation in the resistivity values. In cases where the true subsurface resistivity consists of several regions that are approximately homogenous internally and separated by sharp boundaries, the result obtained by the smooth inversion method is not optimal. It tends to smear out the boundaries and give resistivity values that are too low or too high. The li norm or blocky optimisation method can be used for such situations. This method attempts to minimise the sum of the absolute values of the spatial changes in the model resistivity. It tends to produce models that are piecewise constant. Results from tests with the smooth and blocky inversion methods with several synthetic and field data sets are given to highlight the strengths and weaknesses of both methods. The smooth inversion method gives better results for areas where the subsurface resistivity changes in a gradual manner, while the blocky inversion method gives significantly better results where there are sharp boundaries. While fast computers and software has made the task of interpreting data from electrical imaging surveys much easier, it still remains the responsibility of the interpreter to choose the appropriate tool for the task based on the available geological information.