Eshagh, Sequential Tikhonov Regularization: An Alternative Way for Integral Inversion … Fachbeitrag Sequential Tikhonov Regularization: An Alternative Way for Integral Inversion of Satellite Gradiometric Data Mehdi Eshagh Abstract of this mission will be a set of spherical harmonic coeffi- Numerous regularization methods exist for solving the ill- cients of the Earth’s gravity field and their corresponding posed problem of downward continuation of satellite grav- errors to degree and order 200 corresponding to a spatial ity gradiometry (SGG) data to gravity anomaly at sea level. resolution of 0.9° × 0.9° (100 km × 100 km). This set is Generally, the use of a dense set of data is recommended in expected to deliver the geoid and gravity anomalies with the downward continuation. However, when such dense data accuracies of 1–2 cm and 1 mGal, respectively from joint are used some of the regularization methods are not efficient inversion of SGG and satellite-to-satellite tracking data. and applicable. In this paper, a sequential way of using the GOCE measures the full tensor of gravitation, contain- Tikhonov regularization is developed for solving large systems ing second-order partial derivatives of the geopotential, and compared to methods of direct truncated singular val- in the gradiometer reference frame, second by second, ue decomposition and iterative methods of range restricted during its life. However, it should be stated that the full minimum residual, algebraic reconstruction technique, ν and tensor of gravitation is not measured with the same ac- conjugate gradient for recovering gravity anomaly at sea level curacy and there are highly sensitive and less sensitive from the SGG data. Numerical studies show that the sequen- gradiometer axes. GOCE will provide a very dense set of tial Tikhonov regularization is comparable to the conjugate the SGG data all over the globe, except polar gaps. The gradient and yields similar result. SGG data of GOCE can be used directly to recover gravity anomalies at sea level. Zusammenfassung The problem of determining the gravity anomalies at Zur Stabilisierung des unterbestimmten Problems der har- sea level from SGG data was the issue investigated by monischen Fortsetzung von Satellitengradiometriedaten zu Reed (1973). He used the second-order partial derivatives Schwere anoma lien auf Geoidhöhe gibt es unterschiedliche of the extended Stokes formula to present the integral Regularisierungsmethoden. Im Allgemeinen wird die Verwen- relation between gravity anomaly and the SGG data and dung eines räumlich dichten Datensatzes für die harmonische inverted them to recover the gravity anomaly. Since in- Fortsetzung nach unten empfohlen. Aber für diesen Fall sind version of such integral formulas is an ill-posed problem einige der Regularisierungsverfahren numerisch ineffizient he used some a priori constraints for regularization of und daher nicht gut anwendbar. In diesem Beitrag wird ein the integral equations. Later on, this idea was followed sequenzielles Verfahren der Tikhonov-Regularisierung für die by Xu (1992) who presented a technique to invert the Fortsetzung großer Datensätze vorgestellt und mit direkten integral formulas without a priori information and it was Regularisierungsmethoden, wie der abgeschnittenen/abge- further investigated in Xu (1998) by comparing it with brochenen Singulärwertzerlegung, sowie iterativen Methoden, some direct regularization methods. Koch and Kusche wie der entfernungsbeschränkten Minimierung der Residuen, (2002) developed an iterative method for simultaneous der algebraischen Rekonstruktionsmethode, der ν Methode estimation of variance components and regularization und der Methode der konjugierten Gradienten, verglichen. parameter. Kotsakis (2007) used the integral inversion for Numerische Untersuchungen zeigen, dass die sequenzielle recovering the gravity anomaly at sea level by a cova- Tikhonov-Regularisierung mit der Methode der konjugierten riance-adaptive method. Eshagh (2009) used the integral Gradienten vergleichbar ist und zu gleichwertigen Ergebnis- inversion method for the recovery of the anomalies from sen führt. full gravitational tensor. Xu (2009) presented a method based on generalized cross validation for simultaneous Keywords: bias-correction, integral inversion, iterative estimation of variance components and regularization of regularization, Krylov subspaces, Tikhonov regularization system of equations. Janak et al. (2009) carried out the inversion of full gravitational tensor to gravity anomalies using the truncated singular value decomposition. Inver- sion of stochastically modified integral of second-order radial derivative of the extended Stokes formula was 1 Introduction done by Eshagh (2011a). Regional gravity field recovery from SGG data using least-squares collocation was done The gravity field and steady-state ocean circulation ex- by Tscherning (1988, 1989) and Arabelos and Tscherning plorer (GOCE) (ESA 1999, 2008) is the recent European (1990, 1993, 1995 and 1999). Tscherning et al. (1990) Space Agency satellite mission which uses the satellite studied three different methods of regional gravity field gravity gradiometry (SGG) technique. The main product recovery: least-squares collocation, Fourier Transform 136. Jg. 2/2011 zfv 113 Fachbeitrag Eshagh, Sequential Tikhonov Regularization: An Alternative Way for Integral Inversion … and the integral approach which was further developed where t = R r, D = 1 − 2t cos ψ + t 2 and r is the geo- by Eshagh (2011b). centric distance of P. In each one of the reviewed studies one regularization Eq. (1a) is the Fredholm integral equation of the first method was considered for inversion of integral formu- kind, and inversion of such an integral is an ill-posed las. However, it is obvious that there are more regular- problem. The integral in the right hand side of this equa- ization methods with their own benefits. In this study, tion should be discretized and solved. Let us present the the goal is to investigate some of these regularization discretized integral (1a) in the following matrix form: methods, in the same conditions, to see which one of T 2 them is suitable for recovering gravity anomalies from Ax = L − ε , E{εε } = σ 0Q and E{ε} = 0 , (2) the SGG data. These regularization methods are classi- fied into two main groups of direct and iterative. Here, where A is the n × m coefficient matrix (right hand side direct methods of Tikhonov regularization (TR) (Tikhonov of Eq. (1a)), L is the n × 1 vector of Trr (left hand side of 1963) and the truncated singular value decomposition Eq. (1a)), e is the n × 1 vector of error of L, x is the m × 1 (TSVD) (Hansen 1998) are considered as well as iterative vector of unknown parameters or the gravity anomalies methods of ν (Brakhage 1987), algebraic reconstruction at sea level, E stands for statistical expectation operator, 2 technique (ART) (Kaczmarz 1937), the range restricted σ 0 is the a priori variance factor which is equal to 1 and generalized minimum residual (RRGMRES) (Calvetti et al. Q is the variance-covariance matrix of observations 2000) and conjugate gradient (CG) (Hanke 1995). Also a which is considered an identity matrix in this study. new strategy to use the TR in a sequential way is simply Since the matrix A is derived after discretization of the developed and named sequential TR (STR), which is ap- integral formula (1a), e. g. based on the simple quadra- plied for recovering the gravity anomalies at sea level ture method, then it will be ill-conditioned. This means from the SGG data. that the condition number of A will be large so that by inverting the system of equations (2), the errors of ob- servables are amplified and destroy the solution. This unwanted property can be controlled by regularization 2 Second-order radial derivative of which means to neglect the high frequencies of the solu- Stokes’ formula tion. Numerous methods have been presented for solving such an ill-posed problem. These methods are so-called The second-order radial derivative of geopotential is the regularization. The next section will present an overview simplest and the most important element of the gravita- for some of them. tional tensor. It is simple because its mathematical for- mulation is easier than the other gradients, and it is im- portant as it has the most power with respect to the other gradients. Let the following estimator for this gradient at 3 A conceptual overview of regularization satellite level be (Reed 1973): methods R T (P ) = S (r, ψ)∆g (Q)dσ , (1a) The regularization methods are divided into two catego- rr 4π ∫∫ rr σ0 ries of direct and iterative. Iterative methods are impor- tant as they avoid the direct inversion of the system of 2 2 where Trr (P ) = ∂ T ∂r , T stands for disturbing potential, equations. This is the reason that the iterative methods R is the mean radius of the Earth, Dg (Q) is the gravity are recommended for large systems. In spite of the it- anomaly at the integration point Q, y is the geocentric erative methods, the direct methods solve the system by angle between the computation and integration points P direct inversion of its coefficients matrix, which is a com- and Q, s0 cap size of integration as we are integrating over plicated problem when it is large and ill-conditioned. In a spherical cap (which is an assumption with respect to the the following sections, some of the direct and iterative realistic full scale inversion problem) and refer to it later methods are reviewed. 2 2 in the numerical example, and Srr (r, ψ) = ∂ S (r, ψ) ∂r (where S (r, ψ) is the extended Stokes’ function) is the kernel of integral (Reed 1973, Eq. (5.35)): 3.1 Direct methods 2 t 3 3(1 − t ) 4 1 + t 2 10 This subsection presents1 − ttwocos ψwell-known+ D methods of = − − − − − + − + Srr (r, ψ) 2 (1 t cos ψ) 5 3 3 18DTSVD2 3andt co sTRψ for15 solving6 ln ill-conditioned system of equa- R D D D D 2 tions. Both of these methods have a shortcoming of 2 t 3 3(1 − t ) 4 1 + t 2 10 1 − t cos ψ + D working with large matrices.