{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T01:11:42Z","timestamp":1760145102547,"version":"build-2065373602"},"reference-count":86,"publisher":"MDPI AG","issue":"6","license":[{"start":{"date-parts":[[2024,6,15]],"date-time":"2024-06-15T00:00:00Z","timestamp":1718409600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"National Natural Science Foundation of China","award":["41704138","41974148","2017JJ3069","E51651","E21722"],"award-info":[{"award-number":["41704138","41974148","2017JJ3069","E51651","E21722"]}]},{"name":"Hunan Provincial Science &amp; Technology Department of China","award":["41704138","41974148","2017JJ3069","E51651","E21722"],"award-info":[{"award-number":["41704138","41974148","2017JJ3069","E51651","E21722"]}]},{"name":"Project of the Doctoral Foundation of Hunan University of Science and Technology","award":["41704138","41974148","2017JJ3069","E51651","E21722"],"award-info":[{"award-number":["41704138","41974148","2017JJ3069","E51651","E21722"]}]},{"name":"Hunan Provincial Key Laboratory of Share Gas Resource Exploitation","award":["41704138","41974148","2017JJ3069","E51651","E21722"],"award-info":[{"award-number":["41704138","41974148","2017JJ3069","E51651","E21722"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>When using traditional Euler deconvolution optimization strategies, it is difficult to distinguish between anomalies and their corresponding Euler tails (those solutions are often distributed outside the anomaly source, forming \u201ctail\u201d-shaped spurious solutions, i.e., misplaced Euler solutions, which must be removed or marked) with only the structural index. The nonparametric estimation method based on the normalized B-spline probability density (BSS) is used to separate the Euler solution clusters and mark different anomaly sources according to the similarity and density characteristics of the Euler solutions. For display purposes, the BSS needs to map the samples onto the estimation grid at the points where density will be estimated in order to obtain the probability density distribution. However, if the size of the samples or the estimation grid is too large, this process can lead to high levels of memory consumption and excessive computation times. To address this issue, a fast linear binning approximation algorithm is introduced in the BSS to speed up the computation process and save time. Subsequently, the sample data are quickly projected onto the estimation grid to facilitate the discrete convolution between the grid and the density function using a fast Fourier transform. A method involving multivariate B-spline probability density estimation based on the FFT (BSSFFT), in conjunction with fast linear binning appropriation, is proposed in this paper. The results of two random normal distributions show the correctness of the BSS and BSSFFT algorithms, which is verified via a comparison with the true probability density function (pdf) and Gaussian kernel smoothing estimation algorithms. Then, the Euler solutions of the two synthetic models are analyzed using the BSS and BSSFFT algorithms. The results are consistent with their theoretical values, which verify their correctness regarding Euler solutions. Finally, the BSSFFT is applied to Bishop 5X data, and the numerical results show that the comprehensive analysis of the 3D probability density distributions using the BSSFFT algorithm, derived from the Euler solution subset of\u00a0x0,y0,z0, can effectively separate and locate adjacent anomaly sources, demonstrating strong adaptability.<\/jats:p>","DOI":"10.3390\/e26060517","type":"journal-article","created":{"date-parts":[[2024,6,17]],"date-time":"2024-06-17T08:48:31Z","timestamp":1718614111000},"page":"517","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["FFT-Based Probability Density Imaging of Euler Solutions"],"prefix":"10.3390","volume":"26","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-7033-6077","authenticated-orcid":false,"given":"Shujin","family":"Cao","sequence":"first","affiliation":[{"name":"School of Earth Sciences and Spatial Information Engineering, Hunan University of Science and Technology, Xiangtan 411201, China"},{"name":"School of Geosciences and Info-Physics, Central South University, Changsha 410083, China"},{"name":"Institute of Geophysics & Geomatics, China University of Geosciences, Wuhan 430074, China"}]},{"given":"Peng","family":"Chen","sequence":"additional","affiliation":[{"name":"School of Earth Sciences and Spatial Information Engineering, Hunan University of Science and Technology, Xiangtan 411201, China"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-8014-819X","authenticated-orcid":false,"given":"Guangyin","family":"Lu","sequence":"additional","affiliation":[{"name":"School of Geosciences and Info-Physics, Central South University, Changsha 410083, China"}]},{"given":"Zhiyuan","family":"Ma","sequence":"additional","affiliation":[{"name":"School of Earth Sciences and Spatial Information Engineering, Hunan University of Science and Technology, Xiangtan 411201, China"}]},{"given":"Bo","family":"Yang","sequence":"additional","affiliation":[{"name":"School of Earth Sciences and Spatial Information Engineering, Hunan University of Science and Technology, Xiangtan 411201, China"}]},{"given":"Xinyue","family":"Chen","sequence":"additional","affiliation":[{"name":"School of Earth Sciences and Spatial Information Engineering, Hunan University of Science and Technology, Xiangtan 411201, China"}]}],"member":"1968","published-online":{"date-parts":[[2024,6,15]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"1021","DOI":"10.1190\/1.1438294","article-title":"Elementary approximations in aeromagnetic interpretation","volume":"21","author":"Smellie","year":"1956","journal-title":"Geophysics"},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"891","DOI":"10.1190\/1.1439666","article-title":"Gradient measurements in aeromagnetic surveying","volume":"30","author":"Hood","year":"1965","journal-title":"Geophysics"},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"181","DOI":"10.1190\/1.1440246","article-title":"The Direct Approach to Magnetic Interpretation and Its Practical Application; discussion and reply","volume":"37","author":"Choudhury","year":"1972","journal-title":"Geophysics"},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"31","DOI":"10.1190\/1.1441278","article-title":"EULDPH: A new technique for making computer-assisted depth estimates from magnetic data","volume":"47","author":"Thompson","year":"1982","journal-title":"Geophysics"},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"80","DOI":"10.1190\/1.1442774","article-title":"Magnetic interpretation in three dimensions using Euler deconvolution","volume":"55","author":"Reid","year":"1990","journal-title":"Geophysics"},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"949","DOI":"10.1016\/S0098-3004(03)00101-8","article-title":"Automatic interpretation of magnetic data based on Euler deconvolution with unprescribed structural index","volume":"29","author":"Gerovska","year":"2003","journal-title":"Comput. 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