Least-squares collocation with covariance-matching constraints

Transcription

1 J Geodesy DOI 1.17/s ORIGINAL ARTICLE Least-sqares collocation with covariance-matching constraints Christopher Kotsakis Received: 18 Agst 26 / Accepted: 28 December 26 Springer-Verlag 27 Abstract Most geostatistical methods for spatial random field (SRF) prediction sing discrete data, inclding least-sqares collocation (LSC) and the varios forms of kriging, rely on the se of prior models describing the spatial correlation of the nknown field at hand over its domain. Based pon an optimal criterion of maximm local accracy, LSC provides an nbiased field estimate that has the smallest mean sqared prediction error, at every comptation point, among any other linear prediction method that ses the same data. However, LSC field estimates do not reprodce the spatial variability which is implied by the adopted covariance (CV) fnctions of the corresponding nknown signals. This smoothing effect can be considered as a critical drawback in the sense that the spatio-statistical strctre of the nknown SRF (e.g., the distrbing potential in the case of gravity field modeling) is not preserved dring its optimal estimation process. If the objective for estimating a SRF from its observed fnctionals reqires spatial variability to be represented in a pragmatic way then the reslts obtained throgh LSC may pose limitations for frther inference and modeling in Earth-related physical processes, despite their local optimality in terms of minimm mean sqared prediction error. The aim of this paper is to present an approach that enhances LSC-based field estimates by eliminating their inherent smoothing effect, while preserving most of their local prediction accracy. Or methodology consists of correcting a posteriori the optimal reslt obtained from LSC in sch a way that the new field estimate matches C. Kotsakis (B) Department of Geodesy and Srveying, The Aristotle University of Thessaloniki, University Box 44, Thessaloniki 54124, Greece kotsaki@topo.ath.gr the spatial correlation strctre implied by the signal CV fnction. Frthermore, an optimal criterion is imposed on the CV-matching field estimator that minimizes the loss in local prediction accracy (in the mean sqared sense) which occrs when we transform the LSC soltion to fit the spatial correlation of the nderlying SRF. Keywords Least-sqares collocation Spatial random field Prediction Smoothing Covariance matching 1 Introdction The prediction of the fnctional vales of a continos spatial random field (SRF), sing a set of observed vales of the same and/or other SRFs, is a fndamental inverse problem in geosciences. The mathematical model describing sch a problem is commonly formlated in terms of the system of observation eqations y i = L i () + v i, i = 1, 2,..., n (1) where (P) denotes the primary random field of interest (P D, with D being a bonded or nbonded, not necessarily Eclidean, spatial domain) that needs to be determined, at one or more points, sing n discrete measrements {y i } which are taken on the same and/or other locations. The symbols L i ( ) correspond to bonded linear or linearized fnctionals of the nknown field, depending on the physical model that relates the observable qantities with the nderlying SRF itself. The additive terms {v i } contain the effect of measrement random noise, inclding possible errors de to model ncertainty in the specification of the field fnctionals L i ( ) that shold not exceed the data noise level.

2 C. Kotsakis Typical examples in geodesy that fall within the realm of the aforementioned SRF prediction scheme inclde the determination of the distrbing gravity potential on or otside the Earth sing varios types of observed gravity field fnctionals, the prediction of stationary or non-stationary ocean circlation patterns from satellite altimetry data, the prediction and modeling of atmospheric fields (tropospheric, ionospheric) from the tomographic inversion of GPS data, the de-noising and optimal separation of geodetic and geophysical signals from a given data record, the generation of error correction meta-srfaces for network-based mobile positioning and GPS-based leveling in a local vertical datm, and the spatio-temporal prediction of crstal deformation fields from geodetic data. Several methods exist in geosciences for tackling problems of spatial signal interpolation and prediction from discrete noisy measrements, and all of them depend to some extent on the way in which some prior information abot the primary nknown SRF is assimilated into the data inversion algorithm. The predominant approach that is followed in geodesy for the optimal soltion of sch problems is least-sqares collocation (LSC) which was introdced by Krarp (1969) in a deterministic context as a rigoros approximation method in separable Hilbert spaces with reprodcing kernels, and formlated in parallel by Moritz (197, 1973) in a probabilistic setting as a statistical prediction techniqe for spatially correlated random variables and stochastic processes; see also Sanso (198, 1986), Dermanis (1976), Tscherning (1986). Most of the conceivable geodetic measrements, as well as nknown parameters and signals, can be simltaneosly handled by the LSC method (Moritz 198; Dermanis 198) which has formed the basis of the integrated geodesy concept as introdced in Eeg and Krarp (1973). Similar methods have also been developed in the general area of geostatistics and spatial data analysis, where varios kriging-type estimators are widely sed for applications of SRF prediction in hydrology, geology, mining engineering, environmental monitoring and applied geophysics (Christakos 1992; Cressie 1993; Matheron 1971); for more details on the similarities and differences between kriging and LSC, see Dermanis (1984), and Regzzoni et al. (25). Althogh originally developed for optimal estimation problems in spatially varying geodetic signals, the LSC method is closely associated with the pioneering work of Kolmogorov (1941) and Wiener (1949) on the interpolation, extrapolation and smoothing of stationary time series. The formalism of Wiener Kolmogorov prediction theory in terms of Forier transforms, spectral filters and inpt otpt (I/O) linear systems has actally been amalgamated in the traditional LSC framework (Sanso and Sideris 1997; Nash and Jordan 1978) and it offers an alternative freqency-domain approach for dealing in a comptationally efficient manner with optimal prediction problems in geodetic applications (see e.g., Eren 1982; Schwarz et al. 199; Sideris 1995). One of the critical aspects in LSC is the smoothing effect on the predicted signal vales û(p), which typically exhibit less spatial variability than the actal tre field (P). As a conseqence, small field vales are overestimated and large vales are nderestimated, ths introdcing a likely conditional bias in the final reslts and possibly creating artifact strctres in SRF maps generated throgh the LSC estimation process. Note that smoothing is a characteristic which is not solely associated with the LSC method and it is shared by most interpolation techniqes aiming at the niqe approximation of an nknown continos fnction from a finite nmber of observed fnctionals. Its merit is that it garantees that the recovered field does not prodce artificial details not inherent or proven by the actal data, which is certainly a reasonable and desirable characteristic for an optimal signal interpolator. However, the se of smoothed SRF images or maps generated by techniqes sch as LSC or kriging provides a shortfall for applications sensitive to the presence of extreme signal vales, patterns of field continity and spatial correlation strctre. While fonded on local optimality criteria that minimize the mean sqared error (MSE) independently for each prediction point, the LSC approach overlooks to some extent a featre of reality that is often important to captre, namely spatial variability. The latter can be considered a global attribte of an optimal field estimate, since it only has meaning in the context of the relationship of all predicted vales to one another in space. As a reslt of the smoothing effect, ordinary LSC estimates do not reprodce either the histogram of the nderlying tre SRF, or the spatial correlation strctre as implied by the adopted model of its covariance (CV) fnction. If the objective for estimating a SRF from its observed fnctionals reqires spatial variability to be represented in a pragmatic way (at least according to a theoretical or empirical model of the signal CV fnction) then the reslts obtained throgh LSC, or other similar techniqes like kriging, may pose limitations for frther inference and modeling in Earth-related physical processes, despite their local optimality in terms of minimm mean sqared prediction error. It is becase of this overly smooth representation of reality that Jornel (199, p. 31) cations against the actal mapping of SRF

3 Least-sqares collocation with covariance-matching constraints prediction reslts obtained from a linear nbiased estimator with minimm MSE: In all rigor, estimates based on a local accracy criterion sch as kriging shold only be tablated; to map them is to entice the ser to read on the map patterns of spatial continity that may be artifacts of the data configration and the kriging smoothing effect. Considering the previos argments, the objective of this paper is to present a prdent approach that enhances LSC-based field estimates by eliminating their inherent smoothing effect, while preserving most of their local prediction accracy. Or approach consists of correcting a posteriori the optimal reslt obtained from the LSC techniqe for the inversion of Eq. (1), in sch a way that the new field estimate matches the spatial correlation strctre implied by the signal CV fnction that was sed to constrct the initial LSC soltion. In contrast to stochastic simlation techniqes which provide mltiple eqiprobable signal realizations according to some CV model of spatial variability (Christakos 1992; Detsch and Jornel 1998), the methodology presented herein gives a niqe field estimate that is statistically consistent with a prior model of its spatial CV fnction. The niqeness condition is imposed thogh an optimal criterion that minimizes the loss in local prediction accracy (in the MSE sense) which occrs when we transform the LSC soltion to match the spatial correlation of the nderlying nknown SRF. The paper is organized as follows: in Sect. 2 a review of the traditional LSC techniqe for the optimal estimation of an nknown SRF is presented, and an analysis of its inherent smoothing effect is given. The proposed de-smoothing approach for LSC-based field estimates is introdced in Sect. 3, and some related nmerical tests are presented in Sect. 4 sing sets of simlated gravity data. Finally, Sect. 5 concldes the paper by presenting some ideas for ftre work, along with a few remaining open qestions that need frther stdy. 2 Ordinary least-sqares collocation 2.1 General concept Based on the general observation model of Eq. (1), let s briefly review the LSC techniqe as a means to obtain an optimal SRF estimate û( ), at an arbitrary nmber of prediction points {P i }, from a set of discrete noisy measrements. The symbol denotes the random vector that contains the vales of the primary SRF ( ) at the selected prediction points, while the vector û contains their corresponding estimates obtained from the LSC method. Denoting by s i = L i () the signal part in the available data, the system (1) can be written in vector form as y = s + v (2) where y, s and v are n-dimensional random vectors containing the known measrements, and the nknown signal and noise vales, respectively, at all observation points {P i }. The LSC scheme reqires some prior information in the form of axiliary hypotheses placed on the first and second order moments of the signal and measrement noise. In particlar, the additive signal and noise components in (2) are considered ncorrelated with each other (a crcial simplification that is reglarly applied in practice), and of known statistical properties in terms of their given expectations and co-variances. Assming that the spatial variability of the primary SRF (e.g., the distrbing potential in most physical geodesy applications) is described by a known CV fnction C (P, Q) = E{((P) m (P))((Q) m (Q))} (3) with m (P) = E{(P)} being the spatial trend of, then the elements of the CV matrix of the signal vector s are determined throgh a straightforward application of co-variance propagation (Moritz 198) C s (i, j) = L i L j C (P i, P j ) (4) where L i and L j correspond to the fnctionals associated with the ith and jth observation, respectively. In the same way, the cross-cv matrix between the primary field vales (at the selected prediction points) and the observed signal vales (at the observation points) is obtained as C s (i, j) = L j C (P i, P j) (5) The CV matrix of the data noise is also considered known, based on the availability of an appropriate stochastic model describing the statistical behavior of the zero-mean measrement errors. C v (i, j) = E{v i v j }=σ vi v j (6) Within the LSC framework it is not reqired, in principle, to impose any particlar constraints on the fnctional form of the signal CV fnction C (P, Q), apart from the fact that it corresponds to a positivedefinite bivariate kernel. In practice, a stationarity modeling assmption is sally adopted in order to infer the signal and/or the noise covariance strctre from available data records and to simplify the overall SRF prediction process. An additional postlate on the spatial trend m (P) of the primary SRF is often employed as an axiliary hypothesis for the LSC inversion of Eqs. (1) or(2). In fact, varios LSC prediction algorithms may arise in

4 C. Kotsakis practice, depending on how we treat the signal de-trending problem. For the prpose of this paper and withot any essential loss of generality, it will be assmed that we deal only with zero-mean SRFs and signals (E{ (P)} =, E{s} =). More general cases can be handled either throgh a remove-restore approach by first sbtracting a known deterministic trend from the available data and then restoring it back to the LSC field estimate, or throgh the simltaneos estimation of the field trend within the LSC algorithm sing a sitable parameterization for the qantities E{(P)} and/or E{s}; see Dermanis (199) for more details. Based on the previos hypotheses, the LSC estimator of the primary SRF at all selected prediction points {P i } is given by the well known matrix formla û = C s (C s + C v ) 1 y (7) which corresponds to the linear nbiased soltion with minimm mean sqared prediction error (Moritz 198; Sanso 1986). 2.2 LSC smoothing effect The inherent smoothing effect in LSC prediction can be easily identified from the CV strctre of its optimal reslt. Applying co-variance propagation to the SRF estimate û in Eq. (7), we obtain the reslt Cû = C s (C s + C v ) 1 C T s (8) which generally differs from the CV matrix C of the original SRF at the particlar set of prediction points, i.e., C (i, j) = C (P i, P j ) = C û(i, j) (9) Moreover, if we consider the vector of the prediction errors e = û, it holds that Cû = C C e (1) where the error CV matrix is given by the eqation (Moritz 198) C e = C C s (C s + C v ) 1 C T s (11) The fndamental relationship in Eq. (1) conveys the meaning of the smoothing effect in LSC, which essentially acts as an optimal low-pass filter to the inpt data. The spatial variability of the LSC prediction errors, in terms of their variances and co-variances, is exactly eqal to the deficit in spatial variability of the LSC estimator û with respect to the original SRF. Note that the decrease in spatial variability between the estimated and the original tre signal, according to Eq. (1), is valid also for the case of noiseless data (C v = ). The only exception occrs when the grop of prediction points is identical to the grop of data points, and the observed signals are of the same type as the primary nknown SRF ( = s, C s = C s = C ). In this special case we have that C e = and Cû = C, which manifests the well known data reprodction property of LSC in the presence of noiseless measrements (Moritz 198). An example of the smoothing effect that can take place when sing the LSC techniqe for SRF prediction is given in Fig. 1. This particlar example refers to a standard interpolation problem, namely the constrction of a gravity anomaly grid from irreglarly distribted noiseless point data. The image shown in Fig. 1a is the realization of a free-air gravity anomaly field which has been simlated within an 5 5 km 2 area with a niform sampling resoltion of 2 km, according to the Hirvonen model of a planar isotropic CV fnction (Meier 1981) C (P, Q) = 1 + C o ( rpq a ) 2 where C o = 75 mgal 2, r PQ is the planar distance between points P and Q, and the parameter a is selected sch that the correlation length of the gravity anomaly field is eqal to 8.5 km. The locations of the irreglar sampling points for this specific experiment are plotted in Fig. 1b, whereas the reslting LSC-based grid with the corresponding prediction errors are shown in Fig. 1c, d, respectively. The LSC smoothing effect can be frther identified in the histograms of the tre (simlated) and predicted gravity anomaly grids (Fig. 2), as well as in their corresponding sample statistics that are listed in Table 1. Remark 1 The smoothing effect according to Eq. (1) inflicts a kind of model denial in the optimal LSC soltion. That is becase the vector estimate û obtained from Eq. (7) reftes its fndamental bilding component, namely the CV strctre of the nderlying tre SRF. In this respect, the merit of LSC as a stochastic interpolation techniqe for the optimal recovery of a continos field from a finite nmber of observed fnctionals, sing its known CV fnction C (P, Q), can be challenged based on the argment that the spatio-statistical strctre of the SRF ( ) is not preserved throgh the optimal estimation process. Remark 2 The LSC prediction algorithm is not affected by the spatial distribtion of the prediction points. If we denote by P i and P j two different points where the optimal prediction of the primary SRF ( ) is soght, then the LSC method yields the following reslts, separately

5 Least-sqares collocation with covariance-matching constraints Fig. 1 Plots of the tre free-air gravity anomaly signal (pper left), the locations of the noiseless irreglar point data (pper right), the LSC signal soltion obtained from the irreglar point data (lower left), and the actal LSC prediction errors (lower right) Fig. 2 Histograms of the tre free-air gravity anomaly signal (pper graph) andthe corresponding LSC soltion sing a sample of noiseless irreglar point data (lower graph) Histogram of the tre signal realization Histogram of the LSC signal soltion Table 1 Statistics of the tre gridded vales and the predicted gridded vales sing the LSC method with noiseless irreglar point data (all vales in mgals) Tre grid LSC-predicted grid for each point û(p i ) = ct i (C s + C v ) 1 y (12) and û(p j ) = ct j (C s + C v ) 1 y (13)

6 C. Kotsakis The cross-cv vectors c i and c j are determined throgh co-variance propagation from the basic CV fnction C (P, Q), i.e., c i = [ L 1 C (P i, P 1) L 2 C (P i, P 2) L n C (P i, P n) ] T (14) [ T c j = L 1 C (P j, P 1) L 2 C (P j, P 2) L n C (P j n)], P (15) The vales that are obtained from the prediction formlae (12) and (13) are not directly inflenced by the relative spatial position of points P i and P j. More importantly, these vales remain naffected by the nmber and/or the geometry of other prediction points where additional field estimates may be compted. Note that the vector formla in Eq. (7) originates essentially by stacking each individal LSC estimate û( ) for all desired prediction points. The LSC reslts from Eqs. (12) and (13) are both affected by the common data-dependent factor (C s + C v ) 1 y. Frthermore, the cross-cv vectors in Eqs. (14) and (15) are both generated from the same fndamental CV kernel, ths creating an implicit interdependence on the vales û(p i ) and û(p j ). However, what is lacking from the LSC framework is an explicit constraint that wold jointly control the nmerical vales of the field estimates û( ) at all prediction points, in a manner that their spatial variation wold be statistically consistent with the CV fnction C (P, Q) of the primary SRF. As it will be seen in the next section, the formlation of sch a constrained SRF prediction approach can lead to an adaptive field estimator which, in contrast to the traditional LSC soltion û, emlates the spatial variation patterns that are inherent into the CV fnction model C (P, Q) for the particlar distribtion of prediction points {P i }. Depending on the specific problem at hand, the preservation of sch patterns in the spatial variation of the primary SRF may be critical for its trthfl approximation from discrete noisy data and the modeling or the comptation of other qantities of interest that depend on it. 3 An optimal de-smoothing scheme for least-sqares collocation The main isse raised in the previos section is the inability of the classic LSC algorithm to reprodce the spatial variability of the primary SRF that needs to be predicted on the basis of a finite set of discrete observations. The term spatial variability is sed here to describe the joint statistical behavior of the SRF vales, as this is imposed and dictated by an a-priori CV fnction model. It shold be noted that for increasing data sampling resoltion, the reslt of LSC prediction converges to the tre field (Tscherning 1978) and ths its inherent smoothing effect ceases to be a problem. However, whether the smoothing effect in LSC is a problem or not in a particlar application depends on the relationship between the actal data resoltion and the degree of roghness of the primary nknown field, as well as on the data noise level. A data sampling resoltion level that may be considered sfficient for one problem, may not be sfficient for another problem with a less smooth field and/or more noisy data. Moreover, the jstification for the effectiveness of any estimation or prediction method, inclding LSC and other least-sqares techniqes, shold not be based solely on its behavior nder some ideal conditions that ensre nice properties of mathematical convergence, bt it mst also consider what happens in an arbitrary general sitation where the resoltion of the available data set may not match the spatial variability of the primary nknown field that is dictated by its CV fnction. Or objective in this paper is to develop a postprocessing correction algorithm that can be applied to any optimal field estimate obtained throgh LSC for the prpose of redcing its inherent smoothing effect, while sstaining most of its local prediction accracy. In general terms, we seek a de-smoothing transformation to act pon the LSC estimator û =R(û) (16) so that the CV strctre of the primary SRF (P) is recovered. This means that the transformation R( ) shold garantee that Cû = C (17) where C is the CV matrix that is formed from the CV fnction C (P, Q) of the nknown SRF; see Eq. (9). In addition, the prediction errors e = û associated with the field estimator û shold remain small in some sense, so that the new soltion can provide not only a CV-adaptive representation for the SRF variation patterns, bt also locally accrate predicted vales on the basis of the given data. For this prpose, the formlation of the de-smoothing operator R( ) shold additionally incorporate an optimality principle by minimizing, for example, the trace of the new error CV matrix C e. 3.1 Optimal linear de-smoothing Let s introdce a straightforward linear approach to modify the LSC estimator û by setting û = Rû (18)

7 Least-sqares collocation with covariance-matching constraints where R is a sqare filtering matrix that needs to be determined according to some optimal properties for the new estimator û. The field estimate obtained from Eq. (18) shold reprodce the CV strctre of the primary SRF, in the sense that Cû = C for the given spatial distribtion of all prediction points {P i }. As a reslt, the filtering matrix R shold satisfy the CV-matching constraint RCûR T = C (19) where C and Cû correspond to the known CV matrices of the tre and the LSC-predicted SRFs, respectively. The assessment of the prediction accracy of the new soltion û can be made throgh its error CV matrix C e = E{(û )(û ) T } (2) which, taking into accont Eq. (18), yields. C e = RCûR T + C RCû C û R T (21) Using the following relations that are always valid for the LSC estimator (assming that there is zero correlation between the observed signals s and the measrement noise v) C = Cû + C e (22) Cû = Cû (23) the new error CV matrix C e can be finally expressed as C e = C e + (I R)Cû(I R) T (24) where C e is the error CV matrix of the sal LSC soltion. Evidently, the pointwise prediction accracy of the modified soltion û will always be worse than the prediction accracy of the original LSC soltion û, regardless of the form of the matrix R. This is expected since LSC provides the best (in the MSE sense) linear predictor from the available measrements, which cannot be frther improved by additional linear, or even nonlinear in the case of normally distribted SRFs, operations. Or aim here is to determine an optimal filtering matrix R that will satisfy the CV-matching constraint in (19), while minimizing the loss of the MSE prediction accracy in the sense that trace(c e C e ) = trace(δc e ) = minimm (25) According to Eq. (24), the residal matrix δc e = (I R)Cû(I R) T (26) represents the part of the error CV matrix of the new estimator û which depends on the choice of the filtering matrix R. ˆ, C ˆ LSC soltion R =? ˆ, Cˆ CV-matching soltion Fig. 3 De-smoothing of the LSC estimator û as a filtering operation in a SISO linear system Remark 3 To obtain an idea abot the degradation in the MSE prediction accracy cased by the modified estimator û, nder the presence of the CV-matching constraint (19), let s consider the case where the filtering matrix takes the general form R = I+δR,withδR being a non negative-definite matrix (i.e., all eigenvales of R are larger, or eqal, than one). Then, it follows from Eq. (24) that trace(c e ) = trace(c e + Cû + RCûR T RCû CûR T ) = trace(c e ) + trace(cû) + trace(rcûr T ) trace(rcû) trace(cûr T ) (27a) Taking into accont the matrix decomposition R = I + δr, the CV-matching constraint (19), and the well known properties trace(ab) = trace(ba) = trace((ab) T ),we have trace(c e ) = trace(c e ) + trace(cû) + trace(c ) 2trace(RCû) = trace(c e ) + trace(cû) + trace(c ) 2trace(Cû) 2trace(δRCû) = trace(c e ) + trace(c ) trace(cû) 2trace(δRCû) = trace(c e ) + trace(cû) + trace(c e ) trace(cû) 2trace(δRCû) = 2trace(C e ) 2trace(δRCû). (27b) Since both δr and Cû are non negative-definite matrices, then it holds that trace(δrcû) (Theobald 1974, p. 14). Ths, we infer that trace(c e ) 2trace(C e ) (28) Note that the above ineqality is tre for every filtering matrix that satisfies the CV-matching constraint (19) and has the form R = I + δr, withδr being a non negative-definite matrix. 3.2 An alternative SISO-type formlation The optimal de-smoothing process that was presented in the previos section can be formlated in a different, yet eqivalent, way as follows. Using a single-inpt singleotpt (SISO) linear system approach, we take the LSC

8 C. Kotsakis Fig. 4 Plots of the tre gravity anomaly signal (pper left), the noisy observed signal (pper right), the CV-matching signal soltion (lower left), and the LSC signal soltion (lower right) Signal realization Noisy (observed) signal CV-adaptive filtering soltion 6 LSC soltion estimate û as the known inpt to a SISO system which is implemented in terms of an arbitrary sqare filtering matrix R (see Fig. 3). The otpt of the SISO system corresponds to a new estimate û of the primary SRF, whose properties are controlled by the system matrix. We want to impose a specific CV strctre on the otpt field û, which mst reprodce the spatial variation patterns of the nderlying primary SRF at all prediction points {P i }. Hence, the filtering matrix of the SISO system mst satisfy the same CV-matching constraint that was already given in Eq. (19). In contrast to the optimal principle in (25) that minimizes the loss of the MSE prediction accracy for the new estimate û with respect to the original LSC soltion, we shall now introdce a different criterion for the niqe determination of the filtering matrix R. In particlar, we reqire that the differences between the original LSC soltion and the new CV-adaptive field estimate are as small as possible. This reqirement is formlated in terms of the following minimization principle which enforces an optimal fit, in the mean sqared sense, between the inpt and the otpt signals in the SISO system of Fig. 3 û û 2 = E{(û û) T (û û)} =minimm. (29) The above principle is algebraically eqivalent to the one given in Eq. (25). Indeed, if we se the following property which holds for every zero-mean random vector x and symmetric matrix Q (Koch 1999, p. 134) E{x T Qx} =trace(qc x ). we have that E{(û û) T (û û)} =E{(Rû û) T (Rû û)} = E{(û T R T û T )(Rû û)} = E{û T (R T I)(R I)û} = E{û T (I R) T (I R)û} = trace((i R) T (I R)Cû} = trace((i R)Cû(I R) T } = trace(c e C e ) = trace(δc e ). (3) Conseqently, the desired filtering matrix R for implementing or CV-adaptive prediction scheme according to (18) and (19) yields two basic properties for the field soltion û, namely: (i) (ii) minimm loss in the MSE prediction accracy with respect to the original LSC soltion, and maximm statistical agreement, in the mean sqared sense, between the CV-adaptive soltion and the LSC soltion.

9 Least-sqares collocation with covariance-matching constraints Fig. 5 Histograms of the tre gravity anomaly signal (pper graph), the LSC signal soltion (middle graph), and the CV-matching signal soltion (lower graph) 6 4 Histogram of the tre signal realization Histogram of the LSC signal soltion Histogram of the CV-adaptive signal soltion The optimal filtering matrix The soltion to the optimization problem that was formlated in the last two sections, namely the determination of the filtering matrix R that satisfies the CVmatching constraint (19) and also minimizes the loss in the prediction accracy for the linear estimator û,is analytically given in the appendix. Note that the following reslt was originally derived in Eldar (21, 23) in a completely different context than the one discssed in this paper, focsing on applications sch as matched-filter detection, qantm signal processing and sbspace signal whitening; see also Eldar and Oppenheim (23). The optimal filtering matrix will be given by the eqation R = C 1/2 (C 1/2 CûC 1/2 ) 1/2 C 1/2 (31) or eqivalently, R = C 1/2 û (C 1/2 û C C 1/2 û )1/2 C 1/2 û (32) Table 2 Statistics of the tre gravity anomaly signal, the LSC signal soltion and the CV-matching signal soltion (all vales in mgals) Tre grid LSC filtered signal CV-matching filtered signal Using the matrix identity ST 1/2 S 1 = (STS 1 ) 1/2,we can also obtain the following eqivalent matrix expressions from the previos eqations: R = (C Cû) 1/2 C (33) R = (C Cû) 1/2 C 1 û (34) R = C 1 û (C ûc ) 1/2 (35) R = C (CûC ) 1/2 (36) The nmerical implementation of any of the last for formlae reqires the se of an appropriate algorithm

10 C. Kotsakis Fig. 6 Plots of the actal prediction errors in the LSC signal soltion (pper graph), and the CV-matching signal soltion (lower graph) 4 2 Estimation error from LSC soltion Estimation error from CV-adaptive soltion Table 3 Statistics of the actal predictions errors in the LSC signal soltion and the CV-matching signal soltion (all vales in mgals) LSC prediction errors CV-matching prediction errors for the comptation of the sqare root of a generally non-symmetric matrix; see, e.g., Higham (1987, 1997, 26), Iannazzo (26), and Smith (23). Following the general expression given in (24), the error CV matrix of the CV-matching field estimator û that corresponds to the previos eqivalent optimal choices for the filtering matrix R will be given by the formla C e = C e + Cû + C (CûC ) 1/2 (C Cû) 1/2 (37) which can be sed for the assessment of the prediction error originating from the de-smoothing process û = Rû that is applied to the standard LSC soltion. 4 Nmerical tests A nmber of simlation experiments have been worked ot to demonstrate the performance of the CV-adaptive field estimator û in contrast to the classic LSC estimator û. The examples presented in this section refer to a standard SRF prediction problem that often appears in many geodetic applications, namely the optimal denoising of observed gravity field fnctionals. The first simlation regards a signal (x), with x [, 1], thoght of as a free-air gravity anomaly profile obtained from airborne gravity measrements. In particlar, we have simlated a 4 Hz ensemble of a zero-mean SRF (x) based on the CV fnction model C (P, Q) = 1 + C o ( rpq a ) 2 (38) where C o = 225 mgal 2, r PQ is the distance between points P and Q, and the parameter a is selected sch that the correlation length of the gravity anomaly profile is 1 km. The observed data record is obtained by adding white noise to the simlated signal vales, with the noise variance set to 1, 25 mgal 2. Two filtering soltions for (x) have been determined based on the aforementioned simlation setting. The first field estimate comes from the sal LSC algorithm that is applied directly to the available data (a standard Wiener filter in this particlar case), while the second field estimate corresponds to the CV-matching optimal soltion that was described in the previos sections. The two signal soltions, along with the original SRF and the observed noisy data, are illstrated in Fig. 4. Asit can be seen from these plots, the CV-matching soltion emlates mch closer than the LSC soltion the

11 Least-sqares collocation with covariance-matching constraints Fig. 7 Plots of the tre gravity anomaly signal (pper left), the noisy observed signal (pper right), the CV-matching signal soltion (lower left), and the LSC signal soltion (lower right) spatial variability and the range of vales of the original SRF (x), despite LSC s optimal pointwise prediction accracy. The improved replication of the spatial variability of the nknown SRF by the CV-matching soltion û,in contrast to the smoothed representation obtained by the LSC reslt û, can also be seen in the histogram plots in Fig. 5, as well as in the signal statistics in Table 2. The actal prediction errors in the LSC soltion and the CV-matching soltion (i.e., e = û and e = û, respectively), are plotted in Fig. 6, while their corresponding statistics are given in Table 3. From the spatial behavior and the magnitde of their prediction errors, we conclde that the performance of both soltions is similar in terms of average prediction accracy, with the LSC reslt being marginally better than the CV-matching field estimate, as shold be expected. The second example is essentially a repetition of the previos noise filtering experiment, with the difference being that now a two-dimensional gravity anomaly SRF is considered. In particlar, sing the same signal and noise simlation parameters and the same CV fnction model of Eq. (38), a two-dimensional free-air gravity anomaly grid of tre and observed vales is simlated over a 5 5 km 2 with a niform resoltion of 1 km (see Fig. 7). The LSC and the CV-matching filtering soltions obtained from this test are also shown in Fig.7, while their corresponding histograms are presented in Fig. 8, and their actal prediction errors in Fig. 9. Similarly to the previos example, we see again the improvement achieved by the CV-matching soltion in the representation of the spatial variability patterns and the range of signal vales, at the cost of a small increase in the signal prediction errors; see also Tables 4 and 5. As a final note, let s mention that the improvement obtained from the CV-matching prediction algorithm over the LSC soltion, with respect to the representation of spatial signal variability and the reprodction of the global statistics of the nderlying SRF, becomes more evident in cases with particlarly low signal-to-noise ratio (SNR), while for high SNR sitations both field estimates give practically identical reslts. A characteristic example is shown in Figs. 1 and 11 for two noise filtering cases with high and low SNR vales, respectively. Specifically, in Fig. 1 we see the reslts obtained from the CV-matching and the LSC prediction methods when the ratio between the signal and the noise standard deviation in the simlation of a free-air gravity anomaly field is set to SNR = 1.5, while in Fig. 11 the corresponding reslts are shown when the simlation experiment takes place with SNR =.1. From the vales of the statistics given in the corresponding Tables 6, 7, 8 and 9, we confirm the speriority of the CV-matching soltion when the SNR of the problem is low, and the eqivalency with the LSC soltion when the SNR is high.

12 C. Kotsakis Fig. 8 Histograms of the tre gravity anomaly signal (pper graph), the LSC signal soltion (middle graph), and the CV-matching signal soltion (lower graph). The plots refer to the 2D noise filtering experiment Fig. 9 Plots of the actal prediction errors in the LSC signal soltion (pper graph), and the CV-matching signal soltion (lower graph). The plots refer to the 2D noise filtering experiment

13 Least-sqares collocation with covariance-matching constraints Fig. 1 Plots of the tre gravity anomaly signal (pper left), the noisy observed signal (pper right), the CV-matching signal soltion (lower left), and the LSC signal soltion (lower right), for the case where SNR = 1.5 Table 4 Statistics of the tre gravity anomaly signal, the LSC signal soltion and the CV-matching signal soltion for the twodimensional noise filtering experiment (all vales in mgals) Tre grid LSC filtered signal CV-matching filtered signal Table 5 Statistics of the actal predictions errors in the LSC signal soltion and the CV-matching signal soltion for the two-dimensional noise filtering experiment (all vales in mgals) LSC prediction errors CV-matching prediction errors Conclsions Most geostatistical methods for SRF prediction problems sing discrete data, inclding LSC and the varios forms of kriging, rely on the se of prior models that describe the spatial correlation of the nknown field at hand over its domain. Bilt pon an optimal criterion of maximm local accracy, LSC provides an nbiased field estimate that has the smallest mean sqared prediction error, at every comptation point, among any other linear prediction method sing the same data. However, LSC field estimates do not reprodce the spatial variability of the corresponding nknown SRFs which is implied by their adopted CV fnctions. From a theoretical viewpoint, this can be considered as a critical drawback in the sense that the spatial correlation strctre of the SRF (e.g., the distrbing potential in the case of gravity field modeling) is not preserved dring its optimal estimation process. On the practical side, the LSC algorithm acts as a low-pass filter which smoothens the spatial variability patterns originating from the adopted, either theoretical or empirical, CV fnction model of the nknown field. Althogh smoothing is not necessarily a bad sideeffect in signal interpolation and prediction problems, there is an apparent paradox in the idea that the optimality criteria sed in the LSC method give field estimates that do not achieve the global statistics of the nderlying SRF, ths sggesting that local prediction accracy (in the sense of nbiased signal estimates with minimm pointwise MSE) and global field mapping (in the sense of preserving the spatial correlation strctre of the nknown signal according to a CV fnction model) cannot be both optimally accomplished. Generally speaking, this dichotomy may be a LSC ser s analoge to the well known Heisenberg s ncertainty principle. Whether or not this paradox represents something more fndamental, the crrent needs for SRF prediction in geodesy and geosciences make it clear that the objective(s) in every application mst be clearly defined in advance (e.g., maximm local accracy, preservation of global statistics, preservation of non-stationary patterns in spatial variability, etc.) since no single method

14 C. Kotsakis Fig. 11 Plots of the tre gravity anomaly signal (pper left), the noisy observed signal (pper right), the CV-matching signal soltion (lower left), and the LSC signal soltion (lower right), for the case where SNR =.1 Table 6 Statistics of the tre gravity anomaly signal, the LSC signal soltion and the CV-matching signal soltion (all vales in mgals), for the case where SNR = 1.5 Tre grid LSC filtered signal CV-matching filtered signal Table 9 Statistics of the actal predictions errors in the LSC signal soltion and the CV-matching signal soltion (all vales in mgals), fir the case where SNR =.1 LSC prediction errors CV-matching prediction errors Table 7 Statistics of the tre gravity anomaly signal, the LSC signal soltion and the CV-matching signal soltion (all vales in mgals), for the case where SNR =.1 Tre grid LSC filtered signal CV-matching filtered signal Table 8 Statistics of the actal predictions errors in the LSC signal soltion and the CV-matching signal soltion (all vales in mgals) for the case where SNR = 1.5 LSC prediction errors CV-matching prediction errors exists that allows the simltaneos optimal reaching of all possible objectives in a given prediction problem. Choosing the best soltion based solely on a minimm MSE criterion may be considered a prejdiced approach that does not necessarily lead to the most realistic pictre of physical reality. The SRF prediction method presented in this paper is not as locally accrate as LSC, bt is more globally representative for an nknown signal with a known CV fnction. Or methodology employs a post-processing filter that is applied to the sal LSC soltion and yields a niqe field estimate that reprodces the global statistics of the primary SRF based on a CV-matching constraint. Besides having the same spatial CV strctre with the primary SRF, the new field estimate retains most of the local prediction accracy associated with the original LSC soltion by minimizing the increase of the error variance at each prediction point. In that sense, the proposed techniqe offers an interesting alternative to LSC for SRF filtering problems in cases where the final reslt shold follow a given CV fnction model. It can provide a sefl post-processing tool, in particlar, for physical geodesy estimation problems with low SNR and/or high degree of smoothness in their reglarized least-sqares soltion (e.g., Wiener filtering of airborne gravity data, downward contination, etc.), for calibration or validation of signal and/or error CV models sing discrete data, for compting stochastic simlations (according to some a-priori CV model) of varios gravity field fnctionals that need to be additionally conditioned pon

15 Least-sqares collocation with covariance-matching constraints actal data vales, and finally for adjsting heterogeneos gravity field data sets according to a common CV model of spatial variability. Acknowledgements The athor wold like to acknowledge the valable sggestions and comments by C.C. Tscherning and D. Roman, which led to an improved revised version of this paper. Appendix: Determination of the optimal filtering matrix We want to determine the optimal filtering matrix R that is sed for obtaining a new SRF estimate û from the original LSC soltion û according to û = Rû, which satisfies the CV-matching constraint Cû = RCûR T = C (A1) and also minimizes the loss in MSE prediction accracy as stated by the optimal principle (see Sect. 3.1) ϕ = trace[(i R)Cû(I R) T ]=minimm (A2) where C and Cû are known n n positive-definite matrices, with Cû = C in general. Note that the MSE principle in the last eqation is eqivalent to the following expression ϕ = E{(û û) T (û û)} =minimm (A3) as explained in Sect Let s apply an axiliary invertible transformation to the LSC field estimate û and the corresponding CVmatching field estimate û = Rû (note that there is a different transformation matrix applied to each of the vectors û and û ): ŵ = C 1/2 û (A4) ŵ = C 1/2 û (A5) In this way, the transformed SRF estimates are related throgh the filtering eqation ŵ = R w ŵ (A6) where the matrix R w is R w = C 1/2 RC 1/2 (A7) BasedonEqs.(A4) and (A5), we have that Cŵ = C 1/2 CûC 1/2 (A8) Cŵ = C 1/2 Cû C 1/2 (A9) Taking into accont the CV-matching constraint (A1), the last eqation yields the following whitening constraint for the CV matrix of the transformed vector ŵ Cŵ = C 1/2 C C 1/2 = I (A1) Also, sing the transformation formlae in (A4) and (A5), the MSE principle from (A3) can be now expressed as ϕ = E{(C 1/2 ŵ C 1/2 ŵ) T (C 1/2 ŵ C 1/2 ŵ)} = E{ŵ T }C ŵ + ŵ T C 1 ŵ ŵ T ŵ ŵ T ŵ } = E{(ŵ T ŵ + ŵ T ŵ w ˆ T ŵ ŵ T ŵ ) +(ŵ T C ŵ + ŵ T C 1 ŵ ŵ T ŵ ŵ T ŵ)} = E{(ŵ ŵ) T (ŵ ŵ)}+e{ŵ T C ŵ +ŵ T C 1 ŵ ŵ T ŵ ŵ T ŵ)} = minimm (A11) Note that both vectors ŵ and ŵ have zero means, since the LSC soltion û (and ths also û = Rû) is also a zeromean random vector based on the initial assmptions stated in Sect Hence, the previos MSE principle can finally take the form ϕ = E{(ŵ ŵ) T (ŵ ŵ)}+trace(c Cŵ ) + trace(c 1 C ŵ) tracecŵ tracecŵ = minimm (A12) or, by sbstitting from (A8) and (A1), ϕ = ϕ + trace(c + C 1/2 CûC 1/2 I C 1/2 CûC 1/2 ) = minimm (A13) where ϕ = E{(ŵ ŵ) T (ŵ ŵ)} (A14) Since C and Cû are both known and fixed matrices not depending on R, we conclde that finding the optimal filtering matrix R that minimizes the qantity ϕ in (A3), sbject to the CV-matching constraint (A1), is eqivalent to finding the filtering matrix R w = C 1/2 RC 1/2 that minimizes the qantity ϕ in (A14), sbject to the CV-matching whitening constraint (A1). Let s consider the orthogonal decomposition of the CV matrix Cŵ given in (A8) Cŵ = C 1/2 CûC 1/2 = VDV T (A15) where V is a nitary matrix whose colmns correspond to the orthonormal eigenvectors of Cŵ, and D is a diagonal matrix that contains the corresponding eigenvales. If we apply the following invertible transformation to the random vectors ŵ and ŵ ẑ = V T ŵ ẑ = V T ŵ (A16) (A17)

16 C. Kotsakis then we have immediately that Cẑ = V T Eq. (A15) CŵV = D (A18) Cẑ = V T Eq. (A1) Cŵ V = V T V = I (A19) so that both vectors ẑ and ẑ are ncorrelated. Frthermore, it holds that ẑ = R z ẑ (A2) where the matrix R z is related with the previos filtering matrix R w according to the eqation R z = V T R w V (A21) Also, de to the orthogonality of the vector transformation applied in (A16) and (A17), the qantity ϕ can be eqivalently expressed as ϕ = E{(ẑ ẑ) T (ẑ ẑ)} (A22) Taking into accont that both ẑ and ẑ are zero-mean random vectors (since they are obtained throgh sccessive linear transformations applied to the zero-mean LSC soltion û and the CV-matching field soltion û = Rû, respectively), the above qantity takes the form ϕ = E{ z ˆ T ẑ }+E{ẑ T ẑ} 2E{(ẑ T ẑ } = trace(cẑ ) + trace(cẑ) 2E{(ẑ T ẑ } = trace(i) + trace(d) 2E{(ẑ T ẑ } (A23) The first two terms in the above eqation are fixed and they do not depend on the desired filtering matrix R (or any of its transformed versions R w = C 1/2 RC 1/2 and R z = V T R w V). Note that the diagonal matrix D is niqely specified throgh the eigenvale decomposition in (A15), since C and Cû are both known and given symmetric matrices; see (A15). In this way, the minimization of ϕ with respect to the choice of the filtering matrix R z is eqivalent to the minimization of the term ρ = 2E{(ẑ T ẑ } (A24) Taking into accont the linearity of the expectation operator E{ }, we can express the qantity ρ in the alternative form n ρ = 2 E{ẑ[k]ẑ [k]} (A25) k=1 where ẑ[k] and ẑ [k] denote the k-th elements of the vectors ẑ and ẑ, respectively. In order to find the minimm of ρ we will se the well known cosine ineqality (Papolis 1991, p. 154) E{x 1 x 2 } E{x 2 1 }E{x2 2 } (A26) which is valid for any pair (x 1, x 2 ) of real-valed random variables. It needs to be emphasized that the eqality sign in the previos ineqality holds if and only if the random variables are linearly related, x 2 = ax 1, where a is an arbitrary real scalar. Ptting the elements ẑ[k] and ẑ [k] in place of the arbitrary random variables x 1 and x 2, the cosine ineqality of (A26) yields E{ẑ[k]ẑ [k]} E{(ẑ[k]) 2 }E{(ẑ [k]) 2 } (A27) or eqivalently 2E{ẑ[k]ẑ [k]} 2 E{(ẑ[k]) 2 }E{(ẑ [k]) 2 } (A28) and by smming over all possible vales of the index k, we finally obtain n n 2 E{ẑ[k]ẑ [k]} 2 E{(ẑ[k]) 2 }E{(ẑ [k]) 2 } k=1 or eqivalently ρ 2 n k=1 k=1 E{(ẑ[k]) 2 }E{(ẑ [k]) 2 } (A29) (A3) The eqality sign in (A3)s holds if and only if the cosine ineqality in (A27) becomes an eqality for every vale of the index k, i.e., E{ẑ[k]ẑ [k]} = E{(ẑ[k]) 2 }E{(ẑ [k]) 2 }, k=1, 2,..., n (A31) which obviosly holds if and only if the random variables ẑ[k] and ẑ [k] are linearly related ẑ [k] =λ k ẑ[k], k = 1, 2,..., n (A32) where λ k denotes an arbitrary scalar factor. Hence, we can conclde that the minimization of the term ρ reqires that the random variables ẑ[k] and ẑ [k] shold be related according to above linear eqation. In view of (A32), and considering the fact that Cẑ = D and Cẑ = I [see Eqs. (A18) and (A18)], we easily dedce that the vectors ẑ and ẑ shold be related throgh a diagonal matrix = diag(λ 1,..., λ k,..., λ n ) ẑ = ẑ (A33) sch that = D 1/2. In this way, the matrix R z according to the filtering formla in (A2) becomes R z = D 1/2 (A34) From Eq. (A21), and taking also into accont Eq. (A15), we obtain the filtering matrix R w