Research paperForward modeling of gravity anomalies based on cell mergence and parallel computing
Introduction
Gravity surveys find broad application in various areas, including mineral exploration (Zeng, 2005), oil exploration (Aghajani et al., 2011), aquifer monitoring (Davis et al., 2008), engineering exploration (Zeng, 2005), environmental engineering (Butler, 1984), and large-scale geodynamic (Albora et al., 2007; Pamukçu et al., 2007). In gravity field analysis, forward modeling is one of the classical techniques used to interpret observed gravity anomalies. Geophysicists use forward modeling to compute the gravity anomalies of a geological model from their geological and geophysical understanding of the study area (Götze and Lahmeyer, 1988; Blakely, 1995; Paul and Bain, 1998; Pignatelli et al., 2011). The proposed forward modeling approach is geared toward usage within a 3D interactive modeling that computes gravity anomalies. A seismological model or tectonic information are commonly used to define the initial model. After that, we interactively modify the model in small regions until the predicted anomalies fit the observed anomalies. Within the interactive modeling process, faster computation is required because the user needs immediate interactive feedback.
In the previous decades, many efforts have been made to improve the efficiency of the forward modeling approach. Portniaguine and Zhdanov (2002) use a cubic interpolation based compression technique to obtain a sparse representation of the sensitivity matrix used in forward modeling. Li and Oldenburg (2003) employ wavelet transform to compress the sensitivity matrix. In general, these two methods achieve a high compression ratio; however, they suffer from loss of accuracy (Wang, 2017). Yao et al. (2003) consider an equivalent storage strategy to reduce the memory requirements of the sensitivity matrix, involving model cells of the same size and uniformly distributed survey locations. Butler and Sinha (2012) calculate gravity anomalies by directly solving Poisson's equation. By adjusting parameters (domain size, the order of the basis function, etc.), the calculation time and memory usage can be significantly reduced. Zhang et al. (2015) transform block density models into residual node density models to compress the geological model. Different boundary conditions are chosen for different models. Casenave et al. (2016) and Wang (2017) adopt a fast multipole method-based approach to accelerate forward modeling. For a selected accuracy, these studies achieve a speedup ratio up to 7 times higher compared with the classical approach (“summation” approach). In addition, fast Fourier transform-based approaches (Bhattacharyya, 1966, 1967; Caratori et al., 2009; Forsberg, 1985; Parker, 1973; Wu and Tian, 2014) also improve the efficiency of forward modeling. However, they rely on the hypotheses that sources are periodic and survey locations are present on a plane.
Hardware acceleration is another method to improve the efficiency of this algorithm. Moorkamp et al. (2010), Chen et al. (2012a, 2012b), Martin et al. (2013), and Čuma and Zhdanov (2014) adopt parallel computation methods for forward modeling and inversion of scalar and tensor gravimetry data. In general, the use of multiple CPU cores or GPU can achieve a large speedup ratio.
Although all the above mentioned studies present methods to improve efficiency, they involve a trade-off between precision and speed (Portniaguine and Zhdanov, 2002; Li and Oldenburg, 2003; Butler and Sinha, 2012; Casenave et al., 2016; Wang, 2017), or the model, survey locations, and boundary conditions are limited (Yao et al., 2003; Zhang et al., 2015; Bhattacharyya, 1966, 1967; Caratori et al., 2009; Forsberg, 1985; Parker, 1973; Wu and Tian, 2014). In this work, we employ a cell merging technique and parallel computing method to compute gravity anomalies. In the following sections, we show the parameterization of the model in terms of rectangular cells (Li and Chouteau, 1998). Then, a cell merging technique to reduce redundant computations of forward modeling is presented. We adopt parallel computing to speed up the process. The proposed method is implemented on a consumer PC in MATLAB, and its performance is evaluated using a synthetic density model (Zhang et al., 2015) and the SEG/EAGE salt density model (Vaillant, 2002).
Section snippets
Summation approach
In Cartesian coordinate systems, we adopt coordinates (x, y, z) for a survey point and (ξ, η, ζ) for a geological body (Fig. 1). The vertical axes (z and ζ), the north axes (x and ξ), and the east axes (y and η) are arranged in a right-handed system. The gravity effect g caused by a rectangular cell with density ρ (ξ, η, ζ) at survey point (x, y, z) is given by (Li and Chouteau, 1998)where γ is the universal gravitational
Results
In this section, we combine the 1-D cell mergence with parallel computing and provide a practical process for forward modeling (Fig. 9). The experimental platform used for performance evaluation consists of an Intel i7-4710MQ CPU with 2.5 GHz, 8 GB of RAM, and an NVIDIA GTX850M GPU card, which has a total of 2048 MB on-board memory with a bandwidth of 28.8 GB/s. The software development environment is a Windows 10 64-bit operating system and the MATLAB 2016a compiler.
To evaluate the performance
Discussions
All the models presented in this study took a shorter time to compute the gravity anomalies using our proposed approach with a single GPU. In particular, compared with the classical approach with a single CPU core, the proposed approach with a single GPU achieves a speedup ratio of up to 681 for the synthetic model. Furthermore, compared with the classical approach with a single GPU, the proposed approach with a single GPU achieves a speedup ratio of up to 13 for the synthetic model. The
Conclusions
We have proposed cell mergence to reduce the redundant computations in forward modeling of gravity anomalies. First, we execute the cell mergence for model cells to obtain a series of cell groups; then, we use these cell groups and survey locations to calculate the gravity anomalies. The process is accelerated by performing the calculation on a consumer CPU-GPU heterogeneous parallel architecture using MATLAB. The proposed approach has no limitations on the model, survey locations, or boundary
Acknowledgements
We are very grateful to associate editors Dario Grana and Leonardo Azevedo, and three anonymous reviewers for their critique, helpful comments, and valuable suggestions to improve the manuscript significantly. We would like to thank Sheng Zhang for assistance in building the SEG/EAGE salt density model. The code associated with this paper can be found at https://github.com/GeoGoku. This work was co-supported by the National Key R&D Program of China [grant numbers 2017YFC0602204-01 and
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- 1
Tao Chen wrote the code, designed and performed the model tests, analyzed the results of the model tests, prepared figures, wrote the paper, reviewed drafts of the paper.
- 2
Guibin Zhang proposed the idea, conceived and designed the model tests, and reviewed drafts of the paper.