The inversion of gravity data into three-dimensional polyhedral models (original) (raw)
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2017
A constrained nonlinear optimization method based on nonlinear programming techniques has been applied to map geometry of bedrock of sedimentary basins by inversion of gravity anomaly data. In the inversion, the applying model is a 2-D model that is composed of a set of juxtaposed prisms whose lower depths have been considered as unknown model parameters. The applied inversion method is a nonlinear one, which minimizes the objective functions by definition of different objective functions and an initial simple model to improve the initial model parameters. In this study, for different cases, sufficient objective functions are defined based on the condition which is encountered in the inverse problem. To control the under- determinacy part of the inverse problem and to prevent unreasonable instability in the resultant model, damping terms are added to the objective function. The act of synthetic inversion for different cases of parameterization has been examined and the results are a...
Next Generation Three-Dimensional Geologic Modeling and Inversion
Existing three-dimensional (3-D) geologic systems are well adapted to high data-density environments, such as at the mine scale where abundant drill core exists, or in basins where 3-D seismic provides stratigraphic constraints but are poorly adapted to regional geologic problems. There are three areas where improvements in the 3-D workflow need to be made: (1) the handling of uncertainty, (2) the model-building algorithms themselves, and (3) the interface with geophysical inversion.
A method for 2-dimensional inversion of gravity data
Applying 2D algorithms for inverting the potential field data is more useful and efficient than their 3D counterparts, whenever the geologic situation permits. This is because the computation time is less and modeling the subsurface is easier. In this paper we present a 2D inversion algorithm for interpreting gravity data by employing a set of constraints including minimum distance, smoothness, and compactness. Using different combination of these constraints provide either smooth images of the underground geological structures or models with sharp geological boundaries. We model the study area by a large number of infinitely long horizontal prisms with square cross-sections and unknown densities. The final density distribution is obtained by minimizing an objective function that is composed of the model objective function and equality constraints, which are combined using a Lagrangian multipliers. Each block's weight depends on depth, a priori information on density and the allowed density ranges for the specified area. A MATLAB code has been developed and tested on a synthetic model consists of vertical and dipping dikes. The algorithm is applied with different combinations of constraints and the practical aspects are discussed. Results indicate that when a combination of constraints is used, the geometry and density distribution of both structures can be reconstructed. The method is applied on Zereshlu Mining Camp in Zanjan-Iran, which is well known for the Manganese ores. Result represents a high density distribution with the horizontal extension of about 30 m, and the vertical extension shows a trend in the E-W direction with a depth interval between 7 to 22 m in the east and 15 to 35 m in the west.
Tectonophysics, 2005
The sensitivity of gravity and magnetic data to deep structures and the broad availability of regional data sets and surveys of high resolution make them suitable for determining detailed three-dimensional (3D) models of the subsurface. However, the sole consideration of gravity and magnetic information cannot properly resolve heterogeneous 3D environments. Advocated to solve this problem, we present an automated refinement technique for three-dimensional multilayer models as conditioned by gravity and magnetic data and by meaningful geometrical and physical constraints. We construct our model by an aggregate of rectangular prisms and aim to estimate their bottom depths, which define the geological layers. We summarize mathematically our concept of refinement in an objective function that includes the misfit to the data, the similitude to an a priori geologicalgeophysical model, and the smoothness of the relief of the layers. Importantly, our objective function also includes inequality constraints that prevent the superposition of layers and integrate the surface and borehole geology with the multilayer deep model. The objective function is solved using quadratic programming in a stable iterative scheme. The resulting algorithm is tested on synthetic data and applied to crustal and sedimentary basin environments from southern Baja California, Mexico. The assimilation of the geological and geometrical constraints to the inversion process produces models that correlate with the surface geology and reveal the three-dimensional features of the subsurface. D
Constrained Two-Dimensional Inversion of Gravity Data
The non-uniqueness in the solution of gravity inversion poses a major problem in the interpretation of gravity data. To overcome this ambiguity, “a priori” information is introduced by minimizing a functional that describes the geometrical or physical properties of the solution. This paper presents a 2D gravity inversion technique incorporating axes of anomalous mass concentration as constraints. The inverse problem is formulated as a minimization of the moment of inertia of the causative body with respect to the axes of the mass concentration. The proposed method is particularly applicable to homogeneous, linear mass distributions, such as mineralization along faults and intruded sills or dikes. Inversions of synthetic and field data illustrate the versatility of the implemented algorithm.
Towards incorporating uncertainty of structural data in 3D geological inversion
2010
All geological models are subject to several kinds of uncertainty. These can be classified into three different types: data imprecision and quality, inherent randomness and incomplete knowledge. With our approach, we address uncertainties introduced by input data quality in complex three-dimensional (3D) models of subsurface structures (geological models). As input data, we consider parameters of geological structures, i.e. formation and fault boundary points and orientation measurements. Our method consists of five steps: construction of an initial geological model with an implicit potential-field method, assignment of probability distributions to data positions and orientation measurements, simulation of several input data sets, construction of several model realisations based on these simulated data sets and finally the visualisation and analysis of the uncertainties. We test our approach in two generic models, a simple graben setting and a complex dome structure. The first model shows that our approach can evaluate uncertainties from different structures and their interaction. Furthermore, it indicates that the final uncertainty of the model is not simply the sum of all input data uncertainties but complex interactions exist. The second example demonstrates that our approach can handle full three-dimensional settings with overturned surfaces. Results of the uncertainty simulation can be visualised and analysed in several ways, ranging from borehole histograms to uncertainty maps. For complex 3D visualisation and analysis, we use indicator functions. When we apply these, we can treat the visualisation of uncertainties in complex settings in a self-consistent manner. With our simulation method, it is possible to analyse and visualise the uncertainties directly introduced by imprecision in the input data. Our approach is intuitive and straight-forward and suitable in both simple and complex geological settings. It enables detailed insights into the model quality, even for the non-expert. In cases where a geological model is the basis for geophysical simulation, it opens-up the way to geological data-driven ensemble modelling and inversion.
First International Meeting for Applied Geoscience & Energy Expanded Abstracts, 2021
Geophysical inversion has been recognized as a useful tool to interpret measurements in many geoscientific applications. However, analyzing uncertainty for inversions is still an open question and has been challenging so far, especially for regional scale 3D inversions. The goal of our work is to develop an empirical method of quantifying uncertainties of 3D inversions in the deterministic framework. We performed inversions using a mixed Lp-norm regularization where various combinations of Lp (0 <= p <= 2) norm can be applied to different components of the regularization term. We randomly sampled two user-specified parameters in multiple times and generated a large set of density models that span a wide spectrum of the possible model characteristics. Based on prior density information from drillholes or lab measurements, we have developed an acceptance-rejection strategy to determine which models to keep for subsequent uncertainty analysis. The accepted density models allow us to quantify the uncertainties of the recovered density values through statistical calculations and 3D visualizations. We applied our method to a set of airborne gravity gradient data over the Decorah area. We were able to quantify the uncertainties of the density values, as well as volume and mass estimates.
Constraints in 3D gravity inversion
Geophysical Prospecting, 2001
A B S T R A C T A three-dimensional (3D) inversion program is developed to interpret gravity data using a selection of constraints. This selection includes minimum distance, flatness, smoothness and compactness constraints, which can be combined using a Lagrangian formulation. A multigrid technique is also implemented to resolve separately large and short gravity wavelengths. The subsurface in the survey area is divided into rectangular prismatic blocks and the problem is solved by calculating the model parameters, i.e. the densities of each block. Weights are given to each block depending on depth, a priori information on density and the density range allowed for the region under investigation. The present computer code is tested on modelled data for a dipping dike and multiple bodies. Results combining different constraints and a weight depending on depth are shown for the dipping dike. The advantages and behaviour of each method are compared in the 3D reconstruction. Recovery of geometry (depth, size) and density distribution of the original model is dependent on the set of constraints used. From experimentation, the best combination of constraints for multiple bodies seems to be flatness and a minimum volume for the multiple bodies. The inversion method is tested on real gravity data from the Rouyn-Noranda (Quebec) mining camp. The 3D inversion model for the first 10 km is in agreement with the known major lithological contacts at the surface; it enables the determination of the geometry of plutons and intrusive rocks at depth.
Inversion of gravity‐field inclination to map the basement relief of sedimentary basins
GEOPHYSICS, 2004
This paper presents a method to map the basement relief of homogeneous sedimentary basins that does not require the knowledge of the basin density contrast. To reach this task, the proposed method relies on the invariance of the inclination of the anomalous gravity field with the density contrast caused by models constituted by two homogeneous media. This invariance occurs because the density contrast appears as a constant factor in both vertical and horizontal gravity components, therefore being canceled out when these components are divided during the evaluation of the field inclination. For such media, the field inclination is independent of the density contrast, thus allowing the source geometry reconstruction even when the density contrast is unknown. As the inclination is rarely measured, the gravity anomaly (i.e., the field vertical component) is initially used to compute the horizontal component of the gravity field by applying a suitable linear transform. The field inclination is estimated from both components and then used to invert the source geometry by fitting the inclination values under the geologic constraints attributed to the causative sources. In this process, the density contrast is not required nor introduced as an unknown parameter in the formulated inverse problem. Moreover, it can be estimated later by solving a new inverse problem where the source geometry determined from the inverted inclination is fixed and the constant density contrast is determined by fitting the gravity anomaly. This paper applies such ideas to map the basement relief of a sedimentary basin and to estimate its density contrast. The inversion is implemented by a random search procedure that excludes extreme models, and imposes constraints that the unknown interface is smooth everywhere and assumes known depth values at isolated points investigated by wells. The proposed technique is tested with synthetic noisy data from homogeneous and heterogeneous basin models and is applied to invert a gravity profile from the Recôncavo Basin, Brazil. The results from the real data application are compared with well data and previously published results.