Variance component estimation of heterogeneous boundary values in downward continuation step for geoid determination involved fix- free two-boundary value problem


1 M.Sc. in Survey, Department of Survey, Exploration Directorate, National Iranian Oil Company, Tehran, Iran

2 Associate Professor, Department of Geomatics, Faculty of Engineering, University of Tehran, Iran



*نگارنده رابط:         تلفن: 82702604-021         دورنگار: 88604550-021                           E-mail:


Geoid computation with boundary value problem is an inverse and Ill-posed problem in potential theory that needs regulation.
Ardalan and Grafarend (2001) proposed Free-Fixed Two Boundary Value Problem (F.F.T. B.V.P) for geoid computation that enable to combination of heterogeneousness gravity observation. In this B.V.P., any observation that has gravity information could be used as Boundary value; for example gravity acceleration from gravimetry, gravity potential from leveling, geoid from GPS/Leveling and geoid from Satellite Altimetry, astronomical observation and etc. Gravity Field of the earth has an influence on the geodetic observations. So all of geodetic observations have gravity information and used in geoid determination as boundary value in B.V.P (Safari, 2004).
In precise geoid computation and gravity field modeling, combination of gravity observation with different type makes high degree of free down and precision. In this paper, geoid from GPS/Leveling () and satellite altimetry () is used as boundary value beside the gravity acceleration observation () for gravity field modeling. The steps of geoid computation are given by Jomegi, 2006.
The fix- free two boundary value problem is a Laplace-Poisson partial differential equation (Ardalan, 1999). The summary of the Over determine Nonlinear Fix- Free Two Boundary Value Problem is given in table 1.
In table 1, the  is gravity potential,  is angular of velocity of the earth,  is mass density of the earth,  is Newton gravitational constant,  is mathematical expectation operator,  is the mathematical expectation of the norm of gravity acceleration vector,  is the mathematical expectation of geoid from satellite altimetry,  is the mathematical expectation of geoid from GPS/Leveling, is geoid's potential,  position vector of point on the physical surface of the earth or geoid and  is position vector of point on the reference ellipsoid.
For solving this B.V.P., the external space of the reference ellipsoid should be harmonic, this means where there are no masses. For this problem and linearization of F.F.T.B.V.P, the reference effect involve ellipsoidal harmonic expansion of the earth up to degree/ order 360/360 and residual topography, is removed from quantities in table 1. Therefore, the B.V.P. for gravity observations converts to B.V.P. for incremental quantities. The summary of the Over determine linear Fix- Free Two Boundary Value Problem for incremental quantities is given in table 3.
As we see in table 3, after the removal step, the field differential equation becomes a Laplace equation for out of the reference ellipsoid .
The ellipsoidal Abel-Poisson integral is the solution of field differential equation on the reference ellipsoid with boundary data type of gravity acceleration. For more study about Abel-Poisson integral, refer to (Ardalan, 1999) (Safari, 2004) (Jomegi, 2006).
The ellipsoidal Abel-Poisson integral observation equation is shown in table 4. The inverse Ellipsoidal Able- Poisson integral equation converts incremental gravity acceleration on the observation point, located on the earth to incremental gravity potential at the reference ellipsoid. This means downward continuation.
In table 4,  are three components of the reference gravity acceleration vector () in Jacobi Ellipsoidal coordinates ,  is weight function and  is the modified Abel-Poisson kernel.
The aim of gravity field modeling and geoid determination is computation of gravity potential on the reference ellipsoid. So, in this paper, the ellipsoidal Abel-Poisson integral is used as an observation equation for gravity acceleration beside the equation of geoid from Satellite altimetry and GPS/Leveling to compute of incremental potential on the reference ellipsoid is shown in equation 1.
This equation system is over determine and Ill-Posed problem (Jomegi, 2006). So the regularization should be used for computation of incremental potential on the reference ellipsoid.
Regularization has been applied by implicitly assuming that the weight matrix of measurements is known. If measurements are assumed heteroscedastic with different unknown variance components, not all regularization techniques may be proper to apply, unless techniques of variance component estimation are directly implemented. Although variance component estimation techniques have been proposed by R.Koch, to simultaneously estimate the variance components and provide the regularization together. The steps of R.Koch method is shown in figure 1.
Figure1. Technique of variance component estimation By R.Koch.
In this research for the first time, a new method for variance estimation by R.Koch (Koch K-R, Kusche J, 2002) is used for estimation of variance component and regularization of boundary value in type of (1) gravity acceleration vector, (2) geoid from satellite altimetry and (3) geoid from GPS/Leveling involved in over determine, linear Free-Fixed Two Boundary Value Problem. With this method, all of gravity observation could be used for gravity field modeling and geoid computation together involved in FFTBVP.
The verity of this method is tested in Coastal region of Pars. The result shows the ability of R.Koch method to use in gravity field modeling by heterogeneous gravity observations.