Michael G. Sideris

Gravity field convolutions without windowing and edge effects

A new set of formulas has been developed for the computation of geoid undulations and terrain corrections by FFT when the input gravity anomalies and heights are mean gridded values. The effects of the analytical and the discrete spectra of kernel functions and that of zero-padding on the computation of geoid undulations and terrain corrections are studied in detail.Numerical examples show that the discrete spectrum is superior to the analytically-defined one. By using the discrete spectrum and 100% zero-padding, the RMS differences are 0.000 m for the FFT geoid undulations and 0.200 to 0.000 mGal for the FFT terrain corrections compared with results obtained by numerical integration.

A new, high-resolution geoid for Canada and part of the U.S. by the 1D-FFT method*

A new, high-resolution and high-precision geoid has been computed for the whole of Canada and part of the U.S., ranging from 35°N to about 90°N in latitude and 210°E to 320°E in longitude. The OSU91A geopotential model complete to degree and order 360 was combined with a 5′ × 5′ mean gravity anomaly grid and 1km × 1km topographical information to generate the geoid file. The remove-restore technique was adopted for the computation of terrain effects by Helmerts condensation reduction. The contribution of the local gravity data to the geoid was computed strictly by the 1D-FFT technique, which allows for the evaluation of the discrete spherical Stokes integral without any approximation, parallel by parallel. The indirect effects of up to second order were considered. The internal precision of the geoid, i.e. the contribution of the gravity data and the model coefficients noise, was also evaluated through error propagation by FFT. In a relative sense, these errors seem to agree quite well with the external errors and show clearly the weak areas of the geoid which are mostly due to insufficient gravity data coverage. Comparison of the gravimetric geoid with the GPS/levelling-derived geoidal heights of eight local GPS networks with a total of about 900 stations shows that the absolute agreement with respect to the GPS/levelling datum is generally better than 10 cm RMS and the relative agreement ranges, in most cases, from 4 to 1 ppm over short distances of about 20 to 100km, 1 to 0.5 ppm over distances of about 100 to 200 km, and 0.5 to 0.1 ppm for baselines of 200 to over 1000 km. Other existing geoids, such as UNB90, GEOID90 and GSD91, were also included in the comparison, showing that the new geoid achieves the best agreement with the GPS/levelling data.

Analysis of Gravity Recovery and Climate Experiment time-variable mass redistribution signals over North America by means of principal component analysis

Four years of data provided by the NASA/German Aerospace Center Gravity Recovery and Climate Experiment (GRACE) satellite mission are analyzed over North America using principal component analysis (PCA). Three hydrology models [Global Land Data Assimilation System (GLDAS), Climate Prediction Center (CPC), and Land Dynamics (LaD)] are used to analyze the water mass changes over the same area and time period. The GRACE-observed and the hydrology models mass changes are compared spatially and temporally, and good agreement is observed. Two signal modes are found to represent more than 65% of the GRACE-observed mass variability. The first mode represents mainly mass changes related to the snow accumulation and melting and has maximum amplitude in the western Cordillera and Quebec-Labrador regions. The second mode comprises long-term positive mass changes in central and eastern Canada and negative mass changes in Alaska. In addition, two more spatiotemporal patterns that explain 14% of the GRACE-observed mass variability are extracted and studied, but no definite relation to hydrology is established. While the GLDAS model agrees very well with the GRACE observations, it is found that the CPC model also provides useful information for validating the GRACE-observed mass changes in North America. On the basis of the results of this study, we can state that principal component analysis is a useful technique for extracting and validating regional hydrology signals from GRACE gravity field data. The main advantage of PCA is the capability to extract interannual and nonperiodic mass changes in addition to long-term and periodic variations.

On the use of heterogeneous noisy data in spectral gravity field modeling methods

AbstractSpectral methods have been a standard tool in physical geodesy applications over the past decade. Typically, they have been used for the efficient evaluation of convolution integrals, utilizing homogeneous, noise-free gridded data. This paper answers the following three questions:(a)Can data errors be propagated into the results?(b)Can heterogeneous data be used?(c)Is error propagation possible with heterogeneous data? The answer to the above questions is yes and is illustrated for the case of two input data sets and one output.Firstly, a solution is obtained in the frequency domain using the theory of a two-input, single-output system. The assumption here is that both the input signals and their errors are stochastic variables with known PSDs. The solution depends on the ratios of the error PSD and the signal PSD, i.e., the noise-to-signal ratios of the two inputs. It is shown that, when the two inputs are partially correlated, this solution is equivalent to stepwise collocation.Secondly, a solution is derived in the frequency domain by a least-squares adjustment of the spectra of the input data. The assumption is that only the input errors are stochastic variables with known power spectral density functions (PSDs). It is shown that the solution depends on the ratio of the noise PSDs.In both cases, there exists the non-trivial problem of estimating the input noise PSDs, given that we only have available the error variances of the data. An effective but non-rigorous way of overcoming this problem in practice is to approximate the noise PSDs by simple stationary models.

FFT-evaluation and applications of gravity-field convolution integrals with mean and point data

Due to the fact that the spectrum of a convolution is the product of the spectra of the two convolved functions, the convolution integrals of physical geodesy can be evaluated very efficiently by the use of the fast Fourier transform (FFT) provided that gravity and/or terrain data are available on a regular grid. All Fourier transform-based methods usually treat the gridded data as point values despite the fact that these discrete values may have been obtained by averaging and they represent mean values over the whole area of a grid element. In the frequency domain, this fact can be taken into account very easily, because the spectra of mean and point data are related via a two-dimensional (2D) sinc function. The paper shows explicitly this relationship using the convolution integrals for the evaluation of geoid undulations, deflections of the vertical, and gravity and gradiometry terrain effects. Numerical tests are presented, indicating that the differences in the two approaches may become significant when highly accurate results are wanted. The application of the2D sinc function in the evaluation, update, and inversion of other convolution integrals is briefly discussed as well.

Review of Geoid Prediction Methods in Mountainous Regions

The solutions to the geodetic boundary value problem of predicting geoid undulations from gravity observations are complicated by the non-level observation surface, thus requiring the use of Molodensky’s theory instead of Stokes’ theory. For practical computations, Molodensky’s equations, as well as Stokes’ equation, may be reformulated as convolution integrals that can be efficiently evaluated by Fast Fourier Transform (FFT) techniques. A link between the two approaches, to a first-order approximation, is provided by use of the classical terrain correction, which can also be evaluated by FFT techniques. The terrain correction is also required for terrain reductions, which smooth the gravity data using topographic density assumptions, yielding more reliable gridding of free-air gravity anomalies and smaller and smoother Molodensky corrections. These reductions can be used in a remove-restore fashion as pre- and post-processing steps, analogously to the direct and indirect effects of shifting the topographic masses below the geoid.

Global positioning system testing of geoids computed from geopotential models and local gravity data: A case study

Oithometric height differences obtained from a combination of global positioning system (GPS) ellipsoidal height differences and gravimetric geoid predictions have the potential of replacing costly and time-consuming spirit leveling, especially in remote unsurveyed areas like the Canadian north. In this paper 88 GPS stations along a 900-km-long first-order leveling line around the Great Slave Lake area, Northwest Territories, Canada, are used as control for intercomparison of geoid predictions which used various geopotential models and gravimetric techniques. The global spherical harmonic models tested include satellite-only solutions, such as GEM-9, GEM-L2, GEM-Tl and GEM-T2, as well as combination solutions, such as GEMlOB, RAPP78, RAPP81, GPM2, OSU86F, OSU89A, OSU89B and OSU91 A. These models fit the geoid with standard errors of 30 to 50 cm. Local gravimetric geoid predictions based on available gravity data were carried out using various forms of Stokes integration, fast Fourier transform methods and least squares collocation. For the first two methods, gravity data were gridded on a 5 km by 5 km Cartesian grid and on a 5′ by 10′ geographical grid. No terrain reductions were applied due to the flatness of the topography. Results show that the differences between control and predicted geoid heights have standard deviations at the 20-to 25-cm level, with differences between methods at 2 to 8 cm (1σ). By fitting a four-parameter model to the geoid predictions (corresponding to a GPS, leveling or geoid datum shift and scale change), long-wavelength errors were significantly reduced, with all methods yielding geoid fits of about 7 cm (1σ). The relative accuracies achieved were of the order of 2 ppm for baselines shorter than 200 km, and of the order of 1 ppm for baselines with lengths between 200 km and 700 km. A discussion of possible error sources concludes the paper. Some errors in the original GPS data were detected in the initial phase of the project, and a GPS resurvey verifyed these errors.

Geoid Determination by FFT Techniques

This chapter introduces Fourier-based methods, and in particular the fast Fourier transform (FFT), as a tool for the efficient evaluation of the convolution integrals involved in geoid determination. An attempt was made to make this document as self-contained as possible for the benefit of readers inexperienced in spectral methods. Therefore, the Fourier transform and its properties are presented in the appendix following the chapter (Appendix A), and reference is made to the particular formulas and properties employed in geoid determination.

Solving Molodensky's series by fast Fourier transform techniques

The use of the fast Fourier transform algorithm in the evaluation of the Molodensky series terms is demonstrated in this paper. The solution by analytical continuation to point level has been reformulated to obtain convolution integrals in planar approximation which can be efficiently evaluated in the frequency domain. Preliminary results show that the solution by Faye anomalies is not sufficient for highly accurate deflections of the vertical and height anomalies. The Molodensky solution up to at least the second-order term must be carried out. Part of the unrecovered deflection and height anomaly signal appears to be due to density variations, verifying the essential role of density modelling. A remove-restore technique for the terrain effects can improve the convergence of the series and minimize the interpolation errors.

Assessment of the capabilities of the temporal and spatiotemporal ICA method for geophysical signal separation in GRACE data

We investigate the potential of two independent component analysis (ICA) methods, i.e., the temporal and spatiotemporal ICA, for separating geophysical signals in Gravity Recovery and Climate Experiment (GRACE) data. These methods are based on the assumption of the statistical independence of the signals and thus separate the GRACE-observed mass changes into maximal independent signals. These two ICA methods are compared to the conventional principal component analysis (PCA) method. We test the three methods with respect to their ability to separate a periodic hydrological signal from a trend signal originating in the solid Earth or the cryosphere with simulated and Center for Space Research GRACE mass changes for the time period of January 2003 to December 2010. In addition, we investigate whether the methods are capable of separating hydrological annual and semiannual mass variations. It is shown that both ICA methods are superior to PCA when non-Gaussian mass variations are analyzed. Furthermore, the spatiotemporal ICA resolves successfully the lack of full temporal and spatial independence of the geophysical signals observed by GRACE both in global and regional simulation scenarios. Although the temporal and spatiotemporal ICA are nearly equivalent, both superior to PCA in the global GRACE analysis, the spatiotemporal ICA proves to be more efficient in regional applications by recover more reliably the postglacial rebound trend in North America and the bimodal total water storage variability in Africa.