Discrete Smooth Interpolation Research Articles

Deterministic modelling for driving factors of mineralization in Shanggong gold deposit (China)

1. Three-dimensional (3D) deterministic modeling is crucial for analyzing the controlling factors of mineralization. The Shanggong gold deposit, situated in the southern margin of North China Craton, represents a magmatic hydrothermal deposit. To construct the 3D deterministic model, boreholes, cross-sections, and geochemistry assays are integrated based on the geological characteristics of the Shanggong gold deposit. This paper summarizes its contents as follows: 1) GoCAD software can be utilized to build a 3D geology model encompassing faults and ore bodies; 2) Geostatistics and discrete smooth interpolation (DSI) can be employed to create a 3D attribute model involving grade and Au/Pb ratio. The findings reveal that: 1) High-value zones are associated with NE and NNE trending cross faults; 2) Deterministic modeling, such as grade model and Au/Pb ratio model, effectively elucidates the metallogenic process mechanism.

3D mineral exploration targeting with multi-dimensional geoscience datasets, Tongling Cu(-Au) District, China

The Tongling Cu(-Au) district is the most important magma-skarn type Cu(-Au) polymetallic districts in the Middle-Lower Yangtze metallogenic belt of China. The previous studies in the Tongling Cu(Au) district compiled available digital geological, geochemical, geophysical, and remote sensing datasets. As the discovery of near-surface Cu(-Au) deposits becomes more and more difficult in the Tongling Cu(-Au) district, we face the challenge of exploring for blind mineral deposits in the subsurface. Three-dimensional (3D) mineral potential mapping can provide effective exploration targeting by integrating multi-source and multi-dimension geoscience datasets. In this paper, firstly, a 3D district-scale geological model was constructed from multiple geoscience datasets. Secondly, multi-level (0 m, −300 m, −600 m, −900 m, −1200 m) exploration criteria were extracted by spatial analyses. Thirdly, the weights-of-evidence method, discrete smooth interpolation, and ordinary kriging interpolation were used to integrate multi-level exploration criteria including two-dimensional (2D) surface geochemical and remote sensing data into a 3D posterior-probability model. Finally, the concentration-area and concentration-volume fractal methods were used to define posterior-probability thresholds to outline 2D and 3D targets, respectively. The results indicated that the integrating performance of ordinary kriging interpolation is better than that of discrete smooth interpolation in 3D posterior-probability modeling. The whole procedure yielded a series of high-potential Cu(-Au) targets. The methodology discussed here can be used in other magma-skarn districts in the world.

From 2D to 3D Modeling of Mineral Prospectivity Using Multi-source Geoscience Datasets, Wulong Gold District, China

The Wulong gold district (WGD) is located at the northeastern margin of the North China Carton (NCC) on the Liaodong Peninsula, China. The results of early mineral exploration, mining, and metallogenic modeling research show that gold deposition was associated with lithospheric thinning of the NCC and the emplacement of voluminous Early Cretaceous intrusions. In this study, multi-source information on the ore formation and geology (including geophysical, geochemical, and remote sensing data) was used to construct two-dimensional (2D) prospectivity maps at multiple levels (surface (0 m), − 500 m, − 1000 m, − 1500 m, − 2000 m, − 2500 m, and − 3000 m) for subsurface mineral exploration in the WGD. The methodology employed consists of four steps. Firstly, three-dimensional (3D) geological and geophysical models were constructed using UBC-GIF and SKUA-GOCAD software. 3D geophysical modeling included 3D gravity and magnetic inversions, and 3D resistivity models were based on 2D magnetotelluric (MT) inversion. Secondly, multi-source, ore-forming, and geoscientific data were integrated using radial basis function link net modeling technology within ArcGIS-SDM. Thirdly, multiple-level 2D prospectivity maps were integrated using the discrete smooth interpolation method within the SKUA-GOCAD software to produce 3D prospecting models. Finally, posterior probabilities were classified to delineate 3D exploration targets using the concentration–volume fractal method (GeoCube software, v. 2.0). The results suggest that: (1) the Early Cretaceous intrusions are key ore-forming and ore-controlling objects within the WGD, and a potential target is the western part of the Sanguliu intrusion at depths from 1000 to 3000 m, in the zone where the Wulong and Yangjia gold deposits cross; (2) there are potential targets within 2500 m of the Wulong gold deposit; however, (3) there are no potential targets from 1000 to 3000 m in the Sigdaogou gold deposit.

3D geological modelling and uncertainty analysis for 3D targeting in Shanggong gold deposit (China)

Three dimensional (3D) deposit-scale modelling was used to identify exploration targets at depth and evaluation of mineral resources including 3D geometry and attribute models. Constructing 3D deposit-scale model need to combine exploration boreholes, cross-sections, and geochemistry assays additional mining dataset. The Shanggong gold deposit is a magmatic hydrothermal deposit with an orogenic setting, and it is located in the south margin of North China Craton. On basis of the geological features of the Shanggong gold deposit, the 3D geology model and attribute modelling aims at potential targets identification and mineral resource estimation adopting different mining and exploration datasets and uncertainty analysis in 3D space. The methodology is summarized as follows: 1) the 3D geological bodies, such as stratum, fault, ore bodies model, constructed using GoCAD software, an initial range of 3D model is 15 × 13 × 3 km3; 2) Ordinary Kriging and discrete smooth interpolation (DSI) methods were used to analyze and identify the trend of the mineralization (Au) features in 3D space combined boreholes and the primary geochemical haloes dataset, and the Au/Pb ratio model is constructed using primary geochemical haloes data with DSI interpolation method; 3) the grade simulation modelling and assessing the uncertainty of the geometry and feature models using the geostatistical method. The results show as follows: 1) the high gold grade value zones are associated with intrusion and NE-trending and NNE-trending cross fault structures; 2) the high gold grade value blocks of Au/Pb ratio can be used to depict mineralization zones; 3) Ordinary Kriging and DSI methods can show the trend of the attribute model; 4) the simulation results can characterize the uncertainty of the 3D model through uncertainty analysis.

3D geochemical modeling for subsurface targets of Dashui Au deposit in Western Qinling (China)

Three dimensional (3D) geochemical modeling was used to identify deposit scale exploration targets combining the geological (110 boreholes and 6 tunnels with 709 assay borehole samples) dataset and 374 litho-geochemical samples in the Dashui Au deposit (China). The Dashui Au deposit, located in the Western Qinling orogenic belt, is a magmatic-hydrothermal type large Au deposit discovered in recent years. The methodologies are summarized as follows: (1) the 3D geological bodies (ore bodies, granitic stocks/dikes, fault fractures, and alteration zones) modeling using GOCAD software. (2) 3D deposit-scale geological modeling in mining zones combination the deposit-scale plane maps, cross sections, longitudinal sections, tunnel planes, borehole datasets, and the range of 3D model is 2.3 × 0.55 × 0.6 km3. (3) The ordinary kriging and discrete smooth interpolation (DSI) interpolation methods were used to analyze the distribution trend of Au grade features combined with borehole and the primary litho-geochemical halos dataset, and the Cu/Pb ratio model is analyzed and constructed by metallogenic model/rule using primary geochemical halos data with DSI interpolation method. (4) The concentration-volume (C–V) fractal method is used to analyze the anomaly classification of grade and Cu/Pb ratio in 3D space using GeoCube software. The results show as follows: (1) the high Au grade value zones are associated with ore-bearing intrusion and the NW- and NE-trending fault structures; (2) the high Au value blocks of Cu/Pb ratio can be used to delineate ore-bearing magmatic fluids channel and mineralization zones; (3) the integrated exploration criteria are Cu/Pb ratio, the NE- and NW-trending faults, and intrusions in the Dashui deposits, the potential targets at depth are located in the southwest and southeast zones under the mining levels of No. 2, No. 170, No. 20, and No. 35 Au orebodies, and the Au resource is >172 t (the depth is <1000 m) in the study area.

Three-dimensional analyses of the sliding surface distribution in the Huangtupo No. 1 riverside sliding mass in the Three Gorges Reservoir area of China

The Huangtupo landslide is one of the largest and most destructive landslides still deforming in the Three Gorges area of China. In recent years, most studies on this landslide have been based on the data obtained from an investigation conducted in 2001. To further elucidate the geological structure and evolution of the landslide, we began building a field test site in the area of the sliding mass with the fastest deformation velocity in 2009. A group of tunnels with a total length of 1.1 km has been excavated, and nine boreholes with depths between 76.8 and 127.1 m have been drilled into the sliding body. Additionally, relative monitoring devices, such as borehole inclinometers and crack meters, have been installed. Based on the findings of the previous investigation and the latest tunneling, drilling, and monitoring data, a spatial distribution model of the sliding surfaces of the Huangtupo No. 1 riverside sliding mass has been established using the discrete smooth interpolation (DSI) method. Significant differences are revealed between the previous proposed sliding surface and the latest monitoring data. We propose that the Huangtupo No. 1 sliding mass has two sliding surfaces. Thus, the sliding mass can be further divided into two secondary sliding bodies, which are named the No. 1-1 (east) and No. 1-2 (west) sliding masses. The No. 1-1 sliding mass slid first, and the material along the western boundary slid later, producing the No. 1-2 sliding body, which has a smaller volume and shallower depth. The areas, volumes, and thicknesses of each sliding body have been calculated using a digital 3D model.

Typomorphic characteristics of pyrite: Criteria for 3D exploration targeting in the xishan gold deposit, China

This paper describes 3D modeling of exploration criteria derived from typomorphic characteristics of pyrite in the Xishan quartz vein-type gold deposit (China). The methodology consists of five steps: (1) modeling of orebody thickness and grade using ordinary kriging in longitudinal section; (2) analysis of major/trace element content of Au-bearing pyrite from each ore paragenetic stage; (3) analysis of thermoelectric parameters of Au-bearing pyrites and estimation of ore-forming temperatures and comparison with homogenization temperatures from fluid inclusion analysis; (4) 3D modeling of orebodies using surface geological mapping, mining tunnels in different levels, and a borehole dataset; and (5) 3D modeling of thermoelectricity coefficients and estimated temperatures from Au-bearing pyrites for exploration targeting via discrete smooth interpolation and concentration-volume fractal modeling. The results indicate that: (1) Au-bearing pyrites from four ore paragenetic stages record gradually decreasing temperatures from the earliest to the latest stages, and the frequencies of occurrence of pyrite crystal combination forms and element components are closely correlated with P-type values of pyrite; (2) orebodies Nos. 108–1 and 107 are continuous at depth and potential exploration targets of their continuations extend more than 700m downward from their present mining levels; whereas orebodies Nos. 55 and 108–2 discontinue at depth.

An improved buffer analysis technique for model-based 3D mineral potential mapping and its application

Buffer zones in bo\ h two-dimensional (2D) and three-dimensional (3D) spaces are commonly used in prospectivity mapping. The method completes a modelling that starts with a real example and progresses to the development of a virtual model. This includes the consideration of lithological or structural contacts at depth, which is a theoretical concept based on extrapolation of data collected in the field, rather than an empirical observation of the feature based on physical samples. This contribution documents an improved buffer analysis method for the study of 3D-space that is implicit (rapid), precise (smooth) and based on triangulated characteristics, which can be used to construct influence domains of geological models. As traditional 2D GIS-based mineral potential mapping is gradually becoming limited with time, mineral potential mapping in three dimensions (3D) is increasingly becoming an important tool in finding concealed economic mineralization. This contribution documents an improved methodology of buffer analysis for prospectivity mapping processing mineralized favourable models rather than describing an advance in the geometry of surface rendering of “geological complexity”. Measures used in this buffer analysis include the: (1) voxelization of geological objects (i.e. assigning numerical values of features on a regular cube in 3D-space); (2) revision of the 3D Euclidean distance transform and the calculation of signed distance field; (3) extracting surfaces from the field; and (4) construction of a buffer-surface based on a “discrete smooth interpolation” (DSI) algorithm. Furthermore, this contribution constructs 3D models using a buffer analysis algorithm and prospectivity mapping introduced here, which is based on real data from the Jiama Cu-polymetallic deposit in Tibet and Daye Fe deposit in the Hubei Province, China. This contribution also presents a comparison between voxel and irregular triangle models, which illustrate that irregular triangle mesh buffer analysis (ITB) can improve modelling techniques for GIS-based 3D mineral potential mapping. The outcome is an increase in the accuracy of prospectivity mapping.

Comprehensive Geophysical Data Integration and Stratigraphic Contacts Delineation in a Regional Hydrogeological Characterization Study

Accurate inference of the interfaces between geological units showing different hydraulic properties is a key step for a reliable hydrogeological characterization of regional aquifers. In this study, we developed a workflow that combines multiple geological and geophysical data sets having different intrinsic resolution to map a stratigraphic interface of the regional aquifer located in Montérégie, Quebec, Canada. One of the principal goals was to optimally assimilate all the data at all stages in the workflow. Firstly, the experimental variogram showed two structures of different ranges: one coming from highly sampled geophysical data and the other from conventional borehole geological markers. Secondly, a secondary variable is constructed with all secondary data (e.g., geological interpretations of low-resolution electromagnetic surveys), each having its own accuracy and resolution. To account for the variable secondary data, different reliability indexes were assigned as weights in a discrete smooth interpolation (DSI). Thirdly, a classic kriging with an external drift (KED) operator was used to interpolate the more reliable well data on the entire region. The approach was tested on the estimation of the bedrock interface elevation in a regional hydrogeological characterization study. The resulting map shows bedrock elevations coherent with geological structure of the region, representing main features such as outcrops and valleys. A cross section is presented to illustrate the philosophy behind the tools employed to achieve the estimation process. It also shows an example of visual quality control undertaken to validate the workflow.

Real 3D geotechnical modeling in engineering geology: a case study from the inner city of Aachen, Germany

A comprehensive three-dimensional geological and geotechnical modeling procedure is proposed that combines many useful technologies in order to reduce the uncertainties associated with geological and geotechnical models of the shallow subsurface in urban areas and facilitate sustainable city development. An approach that couples geostatistical analysis with discrete smooth interpolation to describe a reliable geological boundary was first developed. This can support attribution models with more reliable geological frameworks. Then a more informative geotechnical modeling method was proposed that consisted of three steps: assigning discrete property values to the geological model at real positions, analyzing the spatial variability of the property, and predicting the property across the whole model. These two methodologies were applied, as an example, to the inner city of Aachen, Germany. The shallow subsurface of the modeled area is characterized by a fault system and discontinuous distribution. Three-dimensional soil fraction models were constructed and visualized using the proposed methodology. The resulting models demonstrate the feasibility of the presented methodology and provide a robust way to solve some engineering geology problems.

3D geological grid model based on discrete smooth interpolation

3D geological grid model based on discrete smooth interpolation

A high-precision 3D geological surface modeling method based on discrete smooth interpolation

A high-precision 3D geological surface modeling method based on discrete smooth interpolation

Reservoir flow uncertainty assessment using response surface constrained by secondary information

This paper develops a novel response surface method for characterizing the flow and its uncertainty in hydrocarbon reservoirs. Usually, due to computational requirement, the number of flow simulations that can be carried out is limited, which (1) leads to an under sampling of the uncertain space and (2) does not allow to capture curvatures and non-linearities in the reservoir flow behavior in face of uncertain parameters. The proposed method combines conventional flow simulation results with secondary information provided by fast alternatives such as connectivity measurements, single phase tracer or streamline simulation to overcome this sampling problem. Two response surface interpolation algorithms are investigated: discrete smooth interpolation and kriging with an external drift. Both (1) are independent of any predefined regression model, (2) can model complex response surfaces and (3) are able to integrate a secondary “soft” information. Cross validation is then carried out to evaluate the response surface reliability and the contribution of alternative simulation data. A modified version of the SPE 10th benchmark model is used to compare this methodology with standard ones. Results show: (1) the developed approach generates more accurate response surface than conventional techniques and (2) the cost of the method in time is clearly affordable compared to the gain in accuracy which allows a better appraisal of uncertainty on the reservoir flow response.

3D-reconstruction of complex geological interfaces from irregularly distributed and noisy point data

In this paper we introduce a new, precise and adaptive method for the implicit reconstruction of faulted surfaces with complex geometry from scattered, unorganized points as obtained from seismic data or laser scanners. We embed the point set into a 3d-complex on which a 3d-implicit function is interpolated. The 3d-complex is a set of tetrahedrons and the implicit function represents a surface that lies as close as possible to the input data points. The density of the 3d-complex can be adapted to efficiently control both the precision of the implicit function and the size of triangles of the reconstructed surface. Discontinuities in the topology of the tetrahedral mesh make it possible to reconstruct discontinuous, bounded surfaces and very close parallel patches without introducing unwanted connections (topological “handles”) between these regions. To compute the implicit function we use the discrete smooth interpolation (DSI) method with a set of boundary, off-boundary and smoothness constraints. The interpolation problem does not primarily depend on the number of input data points but on the magnitude of the 3d-complex. This method can be applied to the construction of faulted horizons and salt-top surfaces.

Editing faults within tetrahedral volume models in real time

Abstract Accurately positioning faults in a geological model is a major concern because they are responsible for offsets of geological sequences. In the tetrahedral models studied in this paper, faults are discontinuities: faces of tetrahedra on either side of a fault are disconnected. Building tetrahedral models can require a large amount of time, especially when there are many faults. We present a tool for making small, real-time, modifications of faults in tetrahedral models arising from geometrical changes required either by new subsurface data or by new interpretations of existing subsurface data. Fault editing is achieved by moving control points on the fault in the tetrahedral grid and by computing a distortion property over an editable volume relative to the control point and spreading this distortion to neighbouring points using the Discrete Smooth Interpolation technique. The editable volume in which tetrahedron vertices are allowed to move is defined by a given distance to the fault. This approach provides a means of editing faults and fault-related features, such as branch-lines.

A Construction Method of the Short Operator for Wavefield Extrapolation

AbstractIn the explicit depth extrapolation method of frequency‐space domain, wavefield depth extrapolation is completed with convolution of short explicit difference operators and the wavefields. In this paper, based on the study of the method of creating short explicit difference operators, a method of creating short explicit operators is proposed. Through discrete smooth interpolation with control of the phase‐shift operator, this method, in wavefield‐spread area, uses the phase‐shift operator as a constraint in frequency‐wavenumber domain. Outside the wavefield‐spread area, it uses the zero point at the ends of phase‐shift operator as a constraint to take discrete smooth interpolation so that the resulted operator satisfies two‐order smooth and the length of this operator can be selected shorter in frequency‐space domain. This method has simple construction process and high precision, and can be used in wavefields depth extrapolation.

3‐D Seismic Data Discrete Smooth Interpolation Using the Conjugate Gradient Method

AbstractAimed at the problem of huge computation in the interpolation of three dimensional seismic data, we propose an interpolation method that applies the Laplacian algorithm to do smooth restraint. Using the idea of discrete smooth interpolation (DSI) studied by Mallet for reference, the precondition conjugated gradient method is introduced to compute the values straightly on the grids and avoid to find the function to fit the interpolating equation. In order to compute the converse of the Laplacian operator rapidly, the theory of helical spectral factorization method is applied. Under the helical coordinates, the matrix of the Laplacian algorithm has the Topelitz structure of which the converse can be got using spectral factorization quickly. Based on the two dimensional DSI, a flow of discrete smooth interpointion is presented to do DSI along the time slice by use of the conjugated gradient method combining with NMO. The good results have been obtained through the processing of synthetic data and real three‐dimensional data by using this method.

3D reconstruction of the mouse's mesonephros.

The present work reports on the three-dimensional reconstruction of the segmented mesonephros during the embryonic development of the mouse. With a light microscope and an automatic reconstruction of surfaces, aspects of the mesonephros are described. These surfaces are obtained by using digitized contour lines. A new interpolation method called DSI (Discrete Smooth Interpolation) enables correction of the distortion induced by microtomy in paraffin sections. After a triangulation step, this method uses a smoothing algorithm, which implies a spatial redistribution of the vertices of the triangles to correct the rotational and translational misalignment. The use of this 3D program improves the understanding of the development patterns and helps us to appreciate changes in the rebuilt mesonephros. By 10.5 embryonic days, tubules emerge from the Wolffian body and begin their formation, then between 11.5 and 13.5 embryonic days, tortuous mesonephric tubules bound to the Wolffian duct form small curls, which grow and finally unwind. At the same time, mesonephric tubules unbound to the Wolffian duct appear, and on 13.5 embryonic days, the Müllerian duct is visible. After 14.5 embryonic days, the segmented mesonephros keeps its general aspect but decreases in size. At this time, each gonad is provided with both Wolffian and Müllerian ducts. Later, the Wolffian duct differentiates into the definitive male duct system, whilst the Müllerian duct regresses. Conversely, the paramesonephric duct differentiates into the definitive female duct system, whilst the mesonephric duct in turn degenerates. By this time degeneration has begun in the cranial portion of the mesonephros and this process progresses caudally. The spatial organization of the mesonephric tubules and the precise organization of all connections between these elements and the ducts may be well defined. Such approach can allow for a high definition of the normal pattern of mesonephros differentiation.

Discrete modeling for natural objects

This paper presents a discrete technique specially designed for modeling the geometry and the properties of natural objects as those encountered in biology and geology. Contrary to classical Computer-Aided Design methods based on continuous (polynomial) functions, the proposed approach is based on a discretization of the objects close to the finite-element techniques used for solving differential equations. Each object is modeled as a set of interconnected nodes holding the geometry and the physical properties of the objects and the Discrete Smooth Interpolation method is used for fitting the geometry and the properties to complex data. Data are turned into linear constraints and some constraints related to typical information encountered in geology are presented.

Three-Dimensional Modeling of an Activated Sludge Floc

A new interpolation method called discrete smooth interpolation was used to construct a 3-D model of an activated sludge floc from digitized microtome sections. This method comprises several advantages: rapidity, almost completely automated, precision, high quality of representation, and, most importantly, availability of specific tools to manipulate the 3-D model. Some parameters of the floc such as the size, the surface area, and the volume were directly measured on the 3-D reconstructed object. The contour and surface are fractal with fractal dimensions of 1.4 and 2.4, respectively. Local curvatures were also calculated and show great deformations of the surface due to anisotropic growth within the floc. Discrete smooth interpolation constitutes a powerful tool to improve the understanding of the floc structure complexity.