By Majid Naeimi, Jakob Flury
This booklet presents a legitimate theoretical foundation for the the several gravity box restoration equipment and the numerics of satellite-to-satellite monitoring facts. It represents lectures given on the ‘Wilhelm and Else Heraeus Autumn tuition’ in undesirable Honnef, Germany, October 4-9, 2015. The emphasis of the varsity used to be on delivering a valid theoretical foundation for the various gravity box restoration tools and the numerics of knowledge research. The methods coated listed here are the variational equations (classical approach), the acceleration method and the power stability strategy, all of that are used for international gravity box restoration at the foundation of satellite tv for pc observations. the idea of parameter estimation in satellite tv for pc gravimetry and ideas for orbit selection also are integrated. The ebook courses readers via a large variety of themes in satellite tv for pc gravimetry, supplemented by means of the mandatory theoretical history and numerical examples. whereas it presents a finished assessment for these readers who're already accustomed to satellite tv for pc gravity info processing, it additionally deals a vital reference advisor for graduate and undergraduate scholars drawn to this field.
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Additional info for Global Gravity Field Modeling from Satellite-to-Satellite Tracking Data
It is the model which has been considered when we discussed partitioning into global and local parameters with ⎡ (l) ⎤ ⎡ (l) (c) ⎤ x1 A1 0 . . ⎥ E(y) = E ⎣. ⎦ = ⎣ . . . . . . ⎦ ⎢ (l) ⎦ ⎣ x k yk 0 . . 32) ⎡ ⎤ ⎡ 2 −1 ⎤ y1 σ1 P1 ⎦. ··· D(y) = D ⎣. ⎦ = ⎣ −1 2 yk σk Pk In other words, next to x we seek σ = (σ12 , . . , σk2 )T . It is clear that this is a difficult problem. For example, the estimates for σ have to be strictly positive. It is further not obvious under which circumstances the problem has full rank.
MATLAB functions: function [Pnm] = calculatePnm(theta, nmax ) Calculation of normalized Legendre Functions using stable recursion formulas. Input • theta: co-latitude at which the Legendre Functions are computed. The dimension is 1 × 1. [rad] • nmax : maximum degree up to which the Legendre Functions are computed. The dimension is 1 × 1. Output • Pnm: n max × n max matrix containing the normalized Legendre functions. function [cnm , snm ] = sortCoefficients(x, nmax ) Divides the parameter vector x into two triangular matrix for cosine spherical harmonic coefficients cnm and sine spherical harmonic coefficients snm .
Output • 2D-plot of the normal equation matrix. function showParameters(x,nmax ,titleString) Visualization of the estimated parameters in a triangular plot. , sn max ,n max . The dimension n is the number of the parameters. • nmax : maximum degree up to which the parameters are estimated. The dimension is 1 × 1. Output • 2D-plot of the spherical harmonic coefficients. function showSqrtDegreeVariances(x,nmax ,titleString) Computation and visualization of the square root of the degree variances from degree 3 to degree n max .