In this study, the presented algorithm is tested with reconstructing three-dimensional chondrite granules, in which a SEM image (Fig. 3) of the Heyetang meteorite is used as TI (Shen et al., 2013). The Heyetang meteorite, which fell in October 1988, is the most primitive meteorite among non-Antarctic ordinary chondrites in China. It retains the information that is of great significance to study the origin and evolution of the solar system. The selected image is one of backscatter electron images of Heyetang Meteorite. In this image, the chondrites have clear structures that are spherical or sub-spherical. It is composed of various chondrites, opaque minerals, and matrix filling between these granules.
The SEM image of the chondrite section has been manually interpreted as a two-dimensional TI. The interpreted image (Fig. 3b) is a grayscale BMP image with a size of 270×400. The actual size of one pixel is 2.5 μm×2.5 μm. In the manually interpreted image, different attribute is represented with different gray values. The primary purpose of this modeling is to simulate the olivine granules in the center of the image. So the attributes of olivine granules in the SEM image are divided into central olivine granules and peripheral olivine granules. The gray value 33 is the olivine granules in the center, and 56, 64, 74 are the olivine granules in the periphery. The gray value 135 is FeO, 150 is FeS, 194 is Fe-Ni, and 115 is the matrix. The algorithm is implemented based on a python environment, using a desktop workstation with a 20-cores CPU and 128 G of memory for calculations. A multi-process parallel acceleration is used in the GOSIM-based iterative step. Two scales are used in the simulation. The finest scale of the SG is 270×400×400. We tested the presented algorithm under different simulation parameters as shown in Table 1. The time for calculating one realization is no more than 10 h. The visualization of the results is based on Paraview Software.
Pattern size for initialization Pattern size for iteration Using hierarchical strategy Using vertical TIs Para 1 7×7×7 5×5×5 √ √ Para 2 7×7×7 5×5×5 √ - Para 3 7×7×7 5×5×5 - - Para 4 7×7×7 5×5×5 - √ Para 5 5×5×5 5×5×5 √ √ Para 6 9×9×9 5×5×5 √ √ Para 7 7×7×7 7×7×7 √ √ Para 8 7×7×7 9×9×9 √ √ - means that the parameter is not used in the simulation.
Table 1. The simulation parameters in testing the presented algorithm
The size of the initial SG is 135×200×200. Assuming that the samples have certain transverse isotropy in space, two TIs will be perpendicular to each other and introduced into the SG, as shown in Fig. 5a. The copy of the original TI is imported as soft data, and provides constraints perpendicular to the original TI. In the sequential simulation, the pattern size is 7×7×7. The initial model generated by sequential simulation based on the hierarchical strategy is shown in Fig. 6a. The pattern size used in the iterative step is 5×5×5, and the number of executions of the propagation and random search processes is three times in each iteration. Compared with the initial model, artifacts are reduced in the realization after the iterative process as shown in Fig. 6b.
Figure 6. Simulation results using parameters of Para 1. (a) The initial results; (b) the final realization; (c) the olivine granules in the center; (d) the olivine granules distributed on the periphery; (e) FeO, FeS, Fe-Ni granules.
From the visual perception, the central olivine spheroids in the results are ellipsoidal, which meets the expectations under the assumptions. The olivine and other granules in the periphery are also reasonably distributed (Figs. 6b-6e), but the shape of some granules is different from the granules in the TI.
As shown in Fig. 7, the realization reveals reasonable geometry and topology of chondrite granules. When the copy TI as the soft data is removed in the simulation with Para 2, only one TI that extended from the image of Fig. 3b is imported in the SG (Fig. 5b). The size of granules in the initial results in Fig. 7a is larger than that in Fig. 6a. Also, the geometry is stretched along the y-axis. Compared with the simulation results in Figs. 7b-7e, the distribution and morphology of the outer olivine granules and other granules in the results are similar. But, the central olivine spheroids as shown in Fig. 7c are flatter in the direction perpendicular to the hard data, though the ductility of the direction is preserved along the x-axis.
When the simulation is implemented without any auxiliary three-dimensional TIs and hierarchical strategy, as the Para 3 in Table 1, the realization will be fully decided by the pattern size and stochastic reconstructing process. The construction of the initial model did not use the hierarchical strategy and directly used all the patterns extracted from the three-dimensional TI. Granules in all results including initial (Fig. 8a) and final results (Figs. 8b-8e) are distorted along the y-axis direction because of lacking hierarchical strategy. The geometry of granules along the x-axis direction parallel to the TI direction is more reasonable, as shown in Figs. 8a-8e. When two TIs (Fig. 5a) are used as the Para 4 in Table 1, also, the geometry of granules in realizations (Figs. 9a-9e) is stretched. Note that the central olivine granules distribute outside of the central area in the TIs in Fig. 8c and Fig. 9c. It means that the hierarchical strategy plays a major role in simulating the reasonable granule geometry, although more TIs provide more constraints in the simulation.
Figure 8. Simulation results using parameters of Para 3. (a) The initial results; (b) the final realization; (c) the olivine granules in the center; (d) the olivine granules distributed on the periphery; (e) FeO, FeS, Fe-Ni granules.
The pattern size in the algorithm is related to the size of the SG. When a multiple-scale strategy is used in the implementation, the changing of the SG size will ensure that the multiple-scale structure can be captured (Yang et al, 2016). In the presented algorithm, the pattern size can be different in the initialization and the iterative process. When changing the pattern size for initialization and other parameters are the same as Para 1 (Para 5 in Table 1), the size of granules in the initial realization is much larger than that in Fig. 6a. In the final realization, the central olivine granules (Fig. 10c) become more concentrated. Other granules in Figs. 10d-10e) are larger than those in Figs. 6d-6e and show a stronger continuity. When the pattern size increases to 9×9×9 as the Para 5 and other parameters are the same as the Para 1, granules size of the initial and final realization (in Figs. 11a-11e) become lager, and numbers of granules decrease. When the pattern size for the iterative process increases (e.g., Para 7 and Para 8) and other parameters is the same as Para 1, the volume of the central granules realization becomes more concentrated (Figs. 12c and 13c) and granule size increases as the pattern size increases as shown in Figs. 12d, 12e and Figs. 13d, 13e.
Figure 10. Simulation results using parameters of Para 5. (a) The initial results; (b) the final realization; (c) the olivine granules in the center; (d) the olivine granules distributed on the periphery; (e) FeO, FeS, Fe-Ni granules.
Figure 11. Simulation results using parameters of Para 6. (a) The initial results; (b) the final realization; (c) the olivine granules in the center; (d) the olivine granules distributed on the periphery; (e) FeO, FeS, Fe-Ni granules.
Figure 12. Simulation results using parameters of Para 7. (a) The initial results; (b) the final realization; (c) the olivine granules in the center; (d) the olivine granules distributed on the periphery; (e) FeO, FeS, Fe-Ni granules.
2.1. Data Description
2.2. Simulation Using Hierarchical Strategy Based on a Set of Vertical TIs
2.3. Simulation Using Hierarchical Strategy Based on One TI
2.4. Simulation Without Hierarchical Strategy
2.5. Simulation with Different Pattern Sizes
As a statistical feature of micro-structure, connectivity analysis is of great significance for analyzing the physical properties, geometric characteristics, and distribution of geological structures (Western et al., 2001). In this study, the two-point connectivity probability function is calculated for the quantitative analysis of the geometric characteristics of granules in TI and realizations. The shapes of the two-point connectivity probability functions of realizations and TI appear similar (in Fig. 14). In the realizations without hierarchical strategy (Fig. 8 and Fig. 9), the granules, especially the central olivine, exhibited abnormal ductility (Para 3 and Para 4 in Fig. 14). The volume proportion of the granules in the realizations with different parameters is different as shown in Table 2.
Proportion of component (%) Para 1 Para 2 Para 3 Para 4 Para 5 Para 6 Para 7 Para 8 Central olivine granules 8.62 7.84 13.62 10.57 8.77 8.67 8.55 8.60 Peripheral olivine granules with high iron 15.07 21.53 16.79 11.39 10.70 15.63 13.46 13.41 Peripheral olivine granules with medium iron 5.65 4.38 6.16 5.97 6.52 5.45 3.93 3.05 Peripheral olivine granules with low iron 0.70 0.50 2.02 1.03 0.53 0.55 0.53 0.48 FeO granules 0.17 0.08 0.81 0.63 0.11 0.32 0.12 0.09 FeS granules 0.87 0.75 1.42 1.26 0.55 1.05 0.75 0.79 Fe-Ni granules 0.40 0.24 1.16 0.35 0.50 0.28 0.24 0.21 Matrix 68.52 64.68 58.02 68.80 72.32 68.05 72.42 73.37
Table 2. Granules proportion in realizations with different parameters
Unlike the macro-structures, the micro-structures have more singularities. As a quantitative index, the fractal dimension can describe the complexity and irregularity of the micro-structure. Here, Minkowski-Bouligand dimension is used to describe the geometric characteristics of the central olivine granules. Table 3 shows the fractal dimension of the central olivine granules in the TI, the results and two-dimensional slices of which the azimuth angle are 45° and 135° in each result. The fractal dimension of central olivine granules in the three-dimensional model with hierarchical strategy is between 2.434-2.507. And the fractal dimension of the central olivine granules in the two-dimensional slices of each result is slightly smaller than the fractal dimension in the TI. Note that the fractal dimension of central olivine granules in the results without hierarchical strategy (Para 3 and Para 4) is slightly smaller than in the results with hierarchical strategy.
TI Para 1 Para 2 Para 3 Para 4 Para 5 Para 6 Para 7 Para 8 Realizations 2.495 2.459 2.323 2.383 2.498 2.434 2.485 2.507 Slice in 45° 1.709 1.650 1.660 1.557 1.620 1.677 1.662 1.675 1.669 Slice in 135° 1.709 1.679 1.659 1.610 1.651 1.689 1.694 1.682 1.695
Table 3. Central olivine granules' fractal dimension in realizations with different parameters
In essence, the pattern used for the GOSIM-based iterative process only extracts local information. For the ductile structures that are larger than the pattern size, the optimization process does not perform well. It illustrates that the hierarchical strategy in this paper is effective in simulating non-stationary targets.
Simulation results show that the shape of the object is affected by the pattern size used in sequential simulation and iterative process. When the pattern size for sequential simulation is too large, it is not conducive to reconstruct the spatial structures with a small scope. If the pattern size is too small, the extension characteristics of the spatial structure will be exaggerated. Therefore, the pattern size should be carefully set in a proper value range.
The proportion of the central olivine granules in TI is 25.1%. When the central granule of the TI is an ideal circle, the volume ratio of the sphere with the same diameter in the two-dimensional image is about 7.75%. Therefore, in the simulation results of the example using hierarchical strategy, the volume proportion of the central olivine granules is about 7.55% to 8.77%. This proportion is basically in line with expectations. According to the proportion of different components in realizations as shown in Table 2, pattern size has a limited impact on the proportions in the final results.
The volume proportion of the simulation granules except the central olivine granules, such as FeO, FeS, and Fe-Ni granules, shows relatively large fluctuations. In the simulation results, the maximum volume proportion of FeO granules is four times the minimum volume proportion. Note that the distance calculation for pattern matching using Hamming distance has the same weight for all attributes. As a result, the attribute values with small volume proportion in the patterns are likely to be ignored during the calculation, and uncertainty of these attribute values in the reconstruction increases.
Some of the olivine granules in the periphery of the simulation results with the hierarchical strategy (Fig. 15) are still unnatural in shape and show excessive continuity. Essentially, the three-dimensional TI used in this study is a simple extension of the two-dimensional image, and it does not guarantee that sufficient morphological information is provided in the direction of expansion. To solve this problem, it is necessary to add constraints on the morphology of peripheral granules in the sequential simulation process, or to adjust the two-dimensional TI extension method based on the morphological characteristics of the peripheral olivine granules.
Figure 15. The peripheral olivine granules with abnormal geometry in the simulation results with parameters of Para 1.
Labeling attribute of each grid is one of the problems in the presented method. The presented method, in essence, is implemented on the gray value of the interpreted image. Without interpretation, structure boundaries cannot be simulated and revealed effectively with the proposed algorithm. Because the original image is not strict in discriminating attributes based on gray values, the same category will show different attribute values in different positions. In the original SEM image, the olivine granules in the center of the image have no special gray value. Therefore, it is still costly to classify granules and matrix by gray value.