{"title":"Spherical radial basis functions model: approximating an integral functional of an isotropic Gaussian random field","authors":"Guobin Chang, Xun Zhang, Haipeng Yu","doi":"10.1007/s00190-024-01910-w","DOIUrl":null,"url":null,"abstract":"<p>The spherical radial basis function (SRBF) approach, widely used in gravity modeling, is theoretically surveyed from a viewpoint of random field theory. Let the gravity potential be a random field which is represented as an integral functional of another random field, namely an isotropic Gaussian random field (IGRF) on a sphere inside the Bjerhammar sphere with the SRBF as the integral kernel. When the integration is approximated by a discrete sum within a local region, one gets the widely applicable SRBF model. With this theoretical study, the following two findings are made. First, the IGRF implies a Gaussian prior on the spherical harmonic coefficients (SHCs) of the gravity potential; for this prior the SHCs are independent with each other and their variances are degree-only dependent. This should be reminiscent of two well-known priors, namely the power-law Kaula’s rule and the asymptotic power-law Tscherning-Rapp model. Second, the IGRF-SRBF representation is non-unique. Benefiting from this redundant representation, one can employ a simple IGRF, e.g., the simplest white field, and then design the SRBF accordingly to represent a potential with desired prior statistical properties. This can simplify the corresponding SRBF modeling significantly; to be more specific, the regularization matrix in parameter estimation of the SRBF modeling can be chosen to be a diagonal matrix, or even the naïve identity matrix.</p>","PeriodicalId":54822,"journal":{"name":"Journal of Geodesy","volume":"247 1","pages":""},"PeriodicalIF":3.9000,"publicationDate":"2024-11-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Journal of Geodesy","FirstCategoryId":"89","ListUrlMain":"https://doi.org/10.1007/s00190-024-01910-w","RegionNum":2,"RegionCategory":"地球科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"GEOCHEMISTRY & GEOPHYSICS","Score":null,"Total":0}
引用次数: 0
Abstract
The spherical radial basis function (SRBF) approach, widely used in gravity modeling, is theoretically surveyed from a viewpoint of random field theory. Let the gravity potential be a random field which is represented as an integral functional of another random field, namely an isotropic Gaussian random field (IGRF) on a sphere inside the Bjerhammar sphere with the SRBF as the integral kernel. When the integration is approximated by a discrete sum within a local region, one gets the widely applicable SRBF model. With this theoretical study, the following two findings are made. First, the IGRF implies a Gaussian prior on the spherical harmonic coefficients (SHCs) of the gravity potential; for this prior the SHCs are independent with each other and their variances are degree-only dependent. This should be reminiscent of two well-known priors, namely the power-law Kaula’s rule and the asymptotic power-law Tscherning-Rapp model. Second, the IGRF-SRBF representation is non-unique. Benefiting from this redundant representation, one can employ a simple IGRF, e.g., the simplest white field, and then design the SRBF accordingly to represent a potential with desired prior statistical properties. This can simplify the corresponding SRBF modeling significantly; to be more specific, the regularization matrix in parameter estimation of the SRBF modeling can be chosen to be a diagonal matrix, or even the naïve identity matrix.
期刊介绍:
The Journal of Geodesy is an international journal concerned with the study of scientific problems of geodesy and related interdisciplinary sciences. Peer-reviewed papers are published on theoretical or modeling studies, and on results of experiments and interpretations. Besides original research papers, the journal includes commissioned review papers on topical subjects and special issues arising from chosen scientific symposia or workshops. The journal covers the whole range of geodetic science and reports on theoretical and applied studies in research areas such as:
-Positioning
-Reference frame
-Geodetic networks
-Modeling and quality control
-Space geodesy
-Remote sensing
-Gravity fields
-Geodynamics