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Solutions to Linear Inverse Problems on the Sphere by Tikhonov Regularization, Wiener Filtering and Spectral Smoothing and Combination − a Comparison Research Article

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Solutions to Linear Inverse Problems on the Sphere by Tikhonov Regularization, Wiener Filtering and Spectral Smoothing and Combination − a Comparison Research Article Journal of Geodetic Science • 2(1) • 2012 • 31-37 DOI: 10.2478/v10156-011-0021-z • Solutions to Linear Inverse Problems on the Sphere by Tikhonov Regularization, Wiener filtering and Spectral Smoothing and Combination − A Comparison Research Article L. E. Sjöberg1∗ 1 Division of Geodesy and Geoinformatics, Royal institute of Technology, 100 44 Stockholm, Sweden Abstract: Solutions to linear inverse problems on the sphere, common in geodesy and geophysics, are compared for Tikhonov's method of regularization, Wiener filtering and spectral smoothing and combination as well as harmonic analysis. It is concluded that Wiener and spectral smoothing, although based on different assumptions and target functions, yield the same estimator. Also, provided that the extra information on the signal and error degree variances is available, the standard Tikhonov method is inferior to the other methods, which, in contrast to Tikhonov's approach, match the spectral errors and signals in an optimum way. We show that the corresponding Tikhonov matrix for optimum regularization can only be determined approximately. Moreover, as Tikhonov's method solves an integral equation, it is less computationally efficient than the other methods, which use forward integration. Also harmonic analysis uses direct integration and is not hampered, as previous methods, with spectral leakage. Spectral combination, in addition to filtering, has the advantage of combining different data sets by least squares spectral weighting. Keywords: Ill-posed problems • regularization • Tikhonov • Wiener filter • spectral smoothing • harmonic analysis © Versita sp. z o.o. Received 25 October 2011; accepted 23 November 2011 1. Introduction Another example is that of determining the gravity anomaly at sea level from satellite gradiometry data. On the sphere such a problem can frequently be expressed as a linear Fredholm integral equation of the 1st kind (e.g., Chambers 1976): Geophysicists and physical geodesists are frequently confronted with linear inverse problems, which can be solved in various ways. An inverse problem generally deals with the problem of (1a) M{K (P;Q)w˜ } = g˜(P) ; converting observations to information w about a physical or other type of system. Frequently the problem is ill-posed, implying that where the available observations are not sufficient to determine a unique solution for w, or, even if a unique solution exists, it can only be ZZ determined approximately. A typical linear inverse problem is that 1 (1b) M{} = {}dσ ; of converting gravity related observations on or above the Earth's 4π σ surface to estimates of the density distribution inside the Earth. σ being the unit sphere and kn are known spectral coefficients ∗E-mail: [email protected] of the kernel function , which relates the observations K (P;Q) 32 Journal of Geodetic Science at the running point and the sought parameters at the This condition can be satisfied either by truncating the unknown g˜ Q w computational point P. It goes without further discussion that spectrum of w to a finite degree, say, nmax , by smoothing the solving an integral equation (inverse problem) is a much more coefficients kn or both. In the first case, despite of the truncation, difficult problem than that of just computing an integral formula the solution will be affected also by high-degree signal and noise of (forward problem). the observations (spectral leakage; Trampert and Snieder 1996). In Note. Throughout the paper the observation , where the second case, by discretizing Eq. (1a), one implicitly smoothes g˜ = g + ε g is the observation signal and ε is assumed to be a random error the solution space to a finite set, corresponding to the selected with expectation zero. block-size on the sphere. Approximately, by choosing the block- This study will be limited to solving Eq. (1a) in the case that the sizeν◦xν◦, theresolutionofthesolutionwillbelimitedtoharmonic kernel function is separable in the form of a series of Legendre's degree ◦(Kaula's rule of thumb). For applications nmax = 180/ν polynomials , i.e. of the Picard condition in geodesy, see e.g. Sj?berg (1979a) and Pn(cosψ) Martinec (1998, pp. 106 and 116). ∞ Below we will study the solutions of Eq. (1a) by Tikhonov regular- X (2) K (p; Q) = (2n + 1)knPn(cosψ) ; ization, Wiener filter and spectral smoothing and combination. n=0 2. Tikhonov regularization where ψ is the geocentric angle between the points P and Q. Inserting Eq. (2) into Eq. (1a) and interchanging summation and One method for regularization of an ill-posed problem originates integration one obtains: with Phillips (1962) and A N Tikhonov in 1963 (see Tikhonov and Arsenin 1977). By this method, Eq (1a) is first discretized into a ∞ ∞ matrix observation equation, where we assume that the system X X (3) knw˜n(P) = g˜(P) = g˜n(P) ; is over-determined, i.e. the number of observations is larger than n=0 n=0 the number of unknowns. The result is: where Kw g − g (8) ( ) (" # ) ˜ = ˜ = w P w Q ˜n( ) 2n + 1 ˜ ( ) (4) = M Pn(cosψ) g P π g Q K, w, g and are the design matrix, vectors of unknowns, ˜n( ) 4 ˜( ) where ˜ ˜ observations and residuals, respectively. Assuming that the resid- are the so-called Laplace harmonics of and . Although the w˜ g˜ uals are random with expectation zero, and that there are no Laplace harmonics are functions of position, below we will usually correlations among the individual residuals, the related Tikhonov not specify this unless necessary for the understanding. problem is to minimize the target function From Eq. (3) we may identify a relation between the unknown w˜ and the known g as ˜ T wT T w (9) E{ } + Γ Γ or g˜n if and only if (5) knw˜n = g˜n w˜n = kn =6 0 ; for some choice of the Tikhonov matrix . For I, where kn Γ Γ = α α is a small positive constant and I is the unit matrix, the solution to and these relations hold also for the error free harmonics and w˜n the minimization is given by the modified normal matrix equation . g˜n In this study we will always assume that kn 6 for all degrees. T T = 0 K K 2I K (10a) Then, at least tentatively, one may come up with a solution for the ( + α )w˜ = g˜; unknown as with the solution ∞ ∞ X X g˜n(P) (6) w˜ (P) = w˜n(P) = : kn n n T T =0 =0 K K 2I −1K g (10b) wˆ = ( + α ) ˜ However, this series does not necessarily converge, but in order to do so, must be smoother than −1. More precisely, a square where the matrix term I stabilizes the original least squares g˜n kn α2 integrable solution for exists if only if the Picard condition solution obtained for . As the stabilization has the less w˜ α = 0 (Courant and Hilbert 1953, p. 160; Hansen 1998, p. 9) is satisfied, wanted effect of making the solution biased, the size of α should i.e. be a compromise between the bias and the expected observation ∞ X g 2 error propagation, and it must be sufficiently large to match the ˜n < ∞ : (7) kn computer capacity to solve Eq. (10a). n=0 Journal of Geodetic Science 33 Applying singular value decomposition, matrix K can be decom- where h is an arbitrary linear kernel function, the expected mean posed into square error (MSE) of the estimator becomes K UDVT (11) = m2 E{ w P − w P 2} where U and V are matrices containing all the eigen-vectors Ui ˜ = ( ˜ ( ) ( )) = and V of K, and D is a diagonal matrix constructed by the singular 2 i σw (P) − 2MQ{h(P;Q)cgw (Q; P)}+ T values (λi = the square roots of the eigen-values of K K). As the (14) MQ[h(P;Q)Mx {h(P;X)cg˜g˜ (Q; X)}] ; eigen-vectors are orthonormal, it follows that Eq. (10a) has the solution where and are the cross- and auto-covariance functions cgw cg˜g˜ between the signals marked by the subscripts, and 2 is the T σi (P) V D2 2I −1DU g wˆ = ( +α ) ˜ = variance of w. The minimum of the MSE is obtained for h satisfying q UT g q UT gV the Wiener-Hopf equation (e.g., Sj?berg 1979b): X λi i ˜ X i ˜ i Vi fi (12) 2 = λi + α2 λi i=1 i=1 (15) cwg(P;Q) = Mx {h˜(P;X)cg˜g˜ (Q; X)} ; where the filter factor 2 2 2 smoothes the solution for fi = λi /(λi +α ) w. By taking the statistical expectation of Eq. (12) and inserting the yielding the MSE expected value for from Eq. (8), it follows that each component g˜ ˆ of the computed vector is biased by 2 2 2 . 2 2 (16) wi −α wi/(λi + α ) m˜ = σw (P) − MQ[h˜(P;Q)Mx {h˜(P;X)cg˜g˜ (Q; X)}] : Note. Here we discuss only the simple Tikhonov regularization by Eq. (10a). Other important methods can be found, e.g., Assuming that the covariance functions are homogeneous and in Hansen (1998, Sect. 5.1). In statistical literature Tikhonov's isotropic, they can be written in the spectral forms method is known as ridge regression, e.g., Marquardt (1970). Xu and Rummel (1994) presented such a technique, by introducing cg˜g˜ (X;Q) = cgg(X;Q) + cεε(X;Q) = more than one regularization parameter, based on the criterion ∞ X of minimizing the trace of the mean square error of the solution, 2 (17a) (cn + σn )Pn(cosψ); to determine gravity potential harmonic coefficients from satellite n=0 gravimetric data. See also Sect.6. Let us finally mention that one simple way of smoothing the and Tikhonov type of solution is to limit the number of unknowns in Eq.
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