Institute of Geophysics, University of TehranJournal of the Earth and Space Physics2538-371X39320131023Local gravity field modeling using radial basis functions, case study: coastal area of the Persian GulfLocal gravity field modeling using radial basis functions, case study: coastal area of the Persian Gulf33483559710.22059/jesphys.2013.35597FAAbdol-RezaSafariAssociate Professor, Department of Surveying and Geomatics Engineering, University College of Engineering, University of Tehran, Iran0000-0001-5938-5468MohammadaliSharifiAssistant Professor, Department of Surveying and Geomatics Engineering, University College of Engineering, University of Tehran, Iran0000-0003-0745-4147IsmaeilForoughiM.Sc. Student of Geodesy, Department of Surveying and Geomatics Engineering, University College of Engineering, University of Tehran, IranJournal Article20121225<sup>*</sup>نگارنده رابط: تلفن: 66752215-021 دورنگار: 66752214-021 E-mail:e.foroughi@ut.ac.ir
The determination of the earth Gravity field has various applications in geodesy and geophysics. Measuring the earth gravity field can be divided into satellite, airborne and terrestrial methods. Traditional method for gravity field modeling using these data is approximation by spherical harmonics expansion. Although spherical harmonic is one of the most popular methods to approximate gravity filed, based on their global characteristic, a small regional variation make big changes in whole spherical harmonic coefficients:
Where is a point with spherical coordinate, and are normalized Legendre functions up to degree and order and are spherical harmonic coefficients.
To deal with this problem, different groups of regional basis systems were introduced, as in case we can refer to gravity field modeling using radial basis functions:
Where are the expansion coefficients (scale coefficients) and Bjerhammar is a sphere with radius which is entirely inside the topographic masses of the earth, are the set of radial basis functions with following representation:
Where are points inside and outside of the Bjerhammar sphere respectively, is the Legendre polynomial of degree and are the Legendre coefficients, the point <em>y </em>is called the centre of the RBF. If locations and depths and coefficients as Radial basis function’s parameters are chosen properly we will have a good representation of potential anomaly and related earth functions. In this paper we used Levenberg Marquardt algorithm (LM) to find optimal RBF parameters, LM is a iterative regularization method, can find the best answer with following equation:
Where is the Hessian matrix evaluated at , this update rule is used as follows, if the error goes down following an update, it implies that our quadratic assumption on is working well and we reduce (usually by a factor of 10) and vice versa.
In this paper, we used combination of gravity data (gravity anomaly) and gravity potential data (gravity potential anomaly) derived from satellite altimetry. Significant points in this algorithm are: removing global effect of gravity anomaly by spherical harmonic up to degree and order 360 (EGM2008) and centrifugal force from gravity anomaly data, using potential anomaly in test area those are calculated on Bruns formula from satellite altimetry data, removing global effect of potential anomaly by spherical harmonic up to degree and order 360 (EGM2008) and centrifugal force from the previous step data, forming the observation equations with radial multipole of order 1 by residual gravity anomaly and residual potential anomaly data. Levenberg Marquardt algorithm is then used to choose optimal number and, location and depth of the radial basis functions. We also used some of potential anomaly observations as control point s in the region of coastal Persian Gulf for appraisal this algorithm and then present the gravity field in this area.<sup>*</sup>نگارنده رابط: تلفن: 66752215-021 دورنگار: 66752214-021 E-mail:e.foroughi@ut.ac.ir
The determination of the earth Gravity field has various applications in geodesy and geophysics. Measuring the earth gravity field can be divided into satellite, airborne and terrestrial methods. Traditional method for gravity field modeling using these data is approximation by spherical harmonics expansion. Although spherical harmonic is one of the most popular methods to approximate gravity filed, based on their global characteristic, a small regional variation make big changes in whole spherical harmonic coefficients:
Where is a point with spherical coordinate, and are normalized Legendre functions up to degree and order and are spherical harmonic coefficients.
To deal with this problem, different groups of regional basis systems were introduced, as in case we can refer to gravity field modeling using radial basis functions:
Where are the expansion coefficients (scale coefficients) and Bjerhammar is a sphere with radius which is entirely inside the topographic masses of the earth, are the set of radial basis functions with following representation:
Where are points inside and outside of the Bjerhammar sphere respectively, is the Legendre polynomial of degree and are the Legendre coefficients, the point <em>y </em>is called the centre of the RBF. If locations and depths and coefficients as Radial basis function’s parameters are chosen properly we will have a good representation of potential anomaly and related earth functions. In this paper we used Levenberg Marquardt algorithm (LM) to find optimal RBF parameters, LM is a iterative regularization method, can find the best answer with following equation:
Where is the Hessian matrix evaluated at , this update rule is used as follows, if the error goes down following an update, it implies that our quadratic assumption on is working well and we reduce (usually by a factor of 10) and vice versa.
In this paper, we used combination of gravity data (gravity anomaly) and gravity potential data (gravity potential anomaly) derived from satellite altimetry. Significant points in this algorithm are: removing global effect of gravity anomaly by spherical harmonic up to degree and order 360 (EGM2008) and centrifugal force from gravity anomaly data, using potential anomaly in test area those are calculated on Bruns formula from satellite altimetry data, removing global effect of potential anomaly by spherical harmonic up to degree and order 360 (EGM2008) and centrifugal force from the previous step data, forming the observation equations with radial multipole of order 1 by residual gravity anomaly and residual potential anomaly data. Levenberg Marquardt algorithm is then used to choose optimal number and, location and depth of the radial basis functions. We also used some of potential anomaly observations as control point s in the region of coastal Persian Gulf for appraisal this algorithm and then present the gravity field in this area.https://jesphys.ut.ac.ir/article_35597_26ab908c1977b25326b5f19b75c97cc9.pdf