Roadmap to a mutually consistent set of offshore vertical reference frames

This thesis presents a combined approach for the realization of the (quasi-)geoid as a height reference surface and the vertical reference surface at sea (chart datum). This approach, specifically designed for shallow seas and coastal waters, provides the relation between the two vertical reference surfaces without gaps down to the coast. It uses a shallow water hydrodynamic model which provides water levels relative to a given (quasi-)geoid. The latter requires that the hydrodynamic model is vertically referenced to the same (quasi-)geoid. Vice versa, the hydrodynamic model is also used to realize a (quasi-)geoid by providing corrections to the dynamic sea surface topography, which are used to reduce radar altimeter-derived sea surface heights to the (quasi-)geoid. The coupled problem of vertically referencing the hydrodynamic model and computing the (quasi-)geoid is solved iteratively. After convergence of the iteration process, the vertically referenced hydrodynamic model is used to realize the chart datum. In this way, consistency between the chart datum and (quasi-)geoid is ensured. The feasibility and performance of this approach is demonstrated for the Dutch North Sea and mainland. It is shown that the differences between modeled and observed instantaneous and mean dynamic sea surface topography is 8-10 cm and 5.8 cm, respectively, for the Dutch North Sea. On land, it is shown that the methodology provides a (quasi-)geoid which has a lower standard deviation than the European Gravimetric Geoid 2008 (EGG08) and the official Netherlands (quasi-)geoid NLGEO2004-grav when compared to GPS-levelling data. The standard deviation at 81 GPS-levelling points is below 1 cm; no correction surface is needed. Finally, it is shown that the chart datum (lowest astronomical tide, LAT) agrees with the observed chart datum at 92 onshore tide gauges to within 21.5 cm standard deviation. This study also examines the impact of instantaneous dynamic sea surface topography (DT) corrections to be applied to altimeter-derived sea surface slopes on the quasi-geoid in the shallow and coastal waters of the North Sea. It is found that the steric and surge parts of the DT mainly contribute to improvements in the signal-to-noise ration at longer wavelengths down to 100-200 km and that the improvements increase towards the southern North Sea. It is also found that the shallow water hydrodynamic model provides better tidal corrections compared to a global ocean tide model, which are most pronounced in the southern North Sea and affect almost the entire spectrum. In terms of quasi-geoid heights, the differences are very small differences (mostly below ±2 cm). This is explained by the fact that altimeter-derived (quasi-)geoid slopes hardly contribute to the estimated quasi-geoid if shipboard gravity data are included. The last question treated in this thesis is whether a spherical Slepian basis representation enables to obtain spectral consistency between a high- and low-resolution data set (following recent studies, this question is treated in the context of mean dynamic topography (MDT) estimation by computing the difference between a high-resolution mean sea level (MSL) model obtained from satellite altimetry and a low-resolution gravimetric geoid). The answer is no; a Slepian representation of the low-resolution MSL signal suffers from broadband leakage. Furthermore, it is shown that a meaningful definition of a low-resolution MSL over incomplete spherical domains involves orthogonal basis functions with additional properties that Slepian functions do not possess. One of these sets of orthogonal basis functions are computed using the Gram-Schmidt orthogonalization for spherical harmonics. For the oceans, an orthogonal basis could be constructed only for resolutions equivalent to a spherical harmonic degree 36. The computation of a basis with a higher resolution failed due to inherent instabilities. More research is needed to solve the instability problem.Geoscience and Remote SensingCivil Engineering and Geoscience

Towards a combined estimation of Greenland’s ice sheet mass balance using GRACE and ICESat data

The Greenland ice sheet is sensitive to climate change. Global heating is expected to result in ice mass losses that will contribute to global sea level rise. For this reason monitoring Greenland’s ice mass balance is of utmost importance. Data of both the Ice, Cloud, and land Elevation Satellite (ICESat) laser altimetry mission and the Gravity Recovery and Climate Experiment (GRACE) gravity mission are used to create two independent estimates of Greenland’s ice sheet mass balance over the full measurement period of about 2003 until 2007. For ICESat data, a processing strategy is developed that uses the elevation differences of geometrically overlapping footprints of both crossing and repeated tracks. The dataset is cleaned using quality flags defined by the Geoscience Laser Altimeter System (GLAS) science team. (The GLAS is the sole scientific instrument on ICESat). The cleaned dataset reveals some strong, spatially correlated signals that are shown to be related to physical phenomena like melting glaciers. On the other hand, strong correlation is also visible between the observed elevation differences and the combined effect of roughness and surface slope. Different processing strategies applied to different sets of laser campaigns are used to convert the observed temporal elevation differences to mass changes for 6 different drainage systems, further divided into a region above and below 2000 meter elevation. Here all available laser campaigns are used and outliers are removed using N-sigma thresholding. Both a uniform and non-uniform weighting scheme,used to estimate the elevation changes with respect to a reference epoch, is evaluated. The non-uniform weighting scheme is developed to account for the influence of roughness and surface slope, but it turns out that also signals of interest are sometimes suppressed. In order to obtain our final estimates based on ICESat data, the uniform weighting scheme is applied. For the whole of Greenland the estimated mass change rate is equal to -142.6 Gton/year. This value can be mainly attributed to strong mass losses in the region below 2000 meter elevation. On the other hand we show that for different processing strategies this value ranges between approximately 0 and -200 Gton/year. In general, the obtained results confirm trends discovered by other authors who use altimetry. Differences can be explained by different time spans of the used datasets, but mainly by differences in sampling of the data in the region below 2000 meter. Furthermore, GRACE based monthly variations of the Earth’s gravity field as processed by CNES, CSR, DEOS and GFZ are used to estimate the mass change rate for North and South Greenland. Here, both a Gaussian filter, for half-widths of 300, 500 and 800 km, and a Wiener filter is used. It turns out that the Gaussian filter with a half-width of 500 km has the best performance. The final estimates obtained after application of this filter for the different GRACE solutions range between -60.9 and -93.1 Gton/year. The differences in estimates among different GRACE solutions can be mainly explained by differences in processing strategies used by the processing centers to obtain the monthly gravity fields. Only for the DEOS solutions, these differences can also be attributed to the different time span of this dataset. In any case the estimates are low compared with recently published GRACE estimates, which can be explained by an unaccounted leakage effect in our estimates. The unaccounted leakage effect also mainly explains the differences between estimates based on ICESat and GRACE data. Due to their global coverage and high temporal resolution both the ICESat and GRACE mission have improved the estimations of Greenland’s ice sheet mass balance. Further improvements are possible when both datasets are combined. Hence an attempt is made for a joint inversion of both datasets. Depending on the used GRACE solution the estimated combined mass change rates range between -114.3 and -124.7 Gton/year.GeomaticsDepartment of Earth Observation and Space Systems (DEOS)Aerospace Engineerin

Impact of systematic errors in gravity and heights on a quasi-geoid model for the Netherlands and Belgium

In this study, we quantified systematic errors in surface gravity anomalies, which were caused by systematic errors in gravity and heights of the gravity stations, and computed their impact on the quasi-geoid model of the Netherlands and Belgium. We found that 70% of the gravity datasets have statistically significant biases ranging from −2 mGal to 1.5 mGal. The primary impact of the biases are long-wavelength systematic distortions in the quasi-geoid model with a peak-to-peak amplitude of 8 cm. We also found systematic errors in the height networks of the Netherlands and Belgium, which cause corresponding errors in the heights of the gravity stations. They range from −3.0 cm to 1.7 cm and −12.0 cm to 5.0 cm, respectively. They also introduce errors in the transformation parameters to EVRF2007 of several centimetres. However, the impact of the height errors on the quasi-geoid model is negligible with a peak-to-peak amplitude of less than 0.1 cm.Physical and Space Geodes

Variance–covariance analysis of two high-resolution regional least-squares quasi-geoid models

This paper investigates the full variance–covariance (VC) matrix of two high-resolution regional quasi-geoid models, utilizing a spherical radial basis function parameterization. Model parameters were estimated using weighted least-squares techniques and variance component estimation (VCE) for data weighting. The first model, known as the “RCR model,” is computed through the remove–compute–restore method, incorporating various local gravity and radar altimeter datasets. The second model, the “combined model,” includes the GOCO05s satellite-only global geopotential model as an additional dataset with a full-noise VC matrix. Validation of the noise VC matrix scaling for each quasi-geoid model is achieved by comparing observed and formal noise standard deviations of differences between geometric and gravimetric height anomalies at GPS height markers in the Netherlands. Analysis of the noise VC matrix of height anomalies at grid nodes reveals significantly smaller formal noise standard deviations for the RCR model compared to the combined model. This difference is attributed to VCE assigning larger weights to the GOCO05s dataset, which exhibits greater noise standard deviations for the specific spatial scales used. Additionally, the formal noise standard deviations of height anomaly differences, relevant for GNSS-heighting, favor the RCR model. However, the disparity between the two models is smaller than implied by the height anomaly noise standard deviations. This is due to the combined model’s noise autocorrelation function displaying a longer correlation length (67 km) in contrast to the RCR model’s (17 km). Consequently, the combined model exhibits a greater reduction in noise variance for height anomaly differences relative to white noise compared to the RCR model.Physical and Space Geodes

A methodology for least-squares local quasi-geoid modelling using a noisy satellite-only gravity field model

The paper is about a methodology to combine a noisy satellite-only global gravity field model (GGM) with other noisy datasets to estimate a local quasi-geoid model using weighted least-squares techniques. In this way, we attempt to improve the quality of the estimated quasi-geoid model and to complement it with a full noise covariance matrix for quality control and further data processing. The methodology goes beyond the classical remove–compute–restore approach, which does not account for the noise in the satellite-only GGM. We suggest and analyse three different approaches of data combination. Two of them are based on a local single-scale spherical radial basis function (SRBF) model of the disturbing potential, and one is based on a two-scale SRBF model. Using numerical experiments, we show that a single-scale SRBF model does not fully exploit the information in the satellite-only GGM. We explain this by a lack of flexibility of a single-scale SRBF model to deal with datasets of significantly different bandwidths. The two-scale SRBF model performs well in this respect, provided that the model coefficients representing the two scales are estimated separately. The corresponding methodology is developed in this paper. Using the statistics of the least-squares residuals and the statistics of the errors in the estimated two-scale quasi-geoid model, we demonstrate that the developed methodology provides a two-scale quasi-geoid model, which exploits the information in all datasets.Physical and Space Geodes

How to deal with the high condition number of the noise covariance matrix of gravity field functionals synthesised from a satellite-only global gravity field model?

The posed question arises for instance in regional gravity field modelling using weighted least-squares techniques if the gravity field functionals are synthesised from the spherical harmonic coefficients of a satellite-only global gravity model (GGM), and are used as one of the noisy datasets. The associated noise covariance matrix, appeared to be extremely ill-conditioned with a singular value spectrum that decayed gradually to zero without any noticeable gap. We analysed three methods to deal with the ill-conditioned noise covariance matrix: Tihonov regularisation of the noise covariance matrix in combination with the standard formula for the weighted least-squares estimator, a formula of the weighted least-squares estimator, which does not involve the inverse noise covariance matrix, and an estimator based on Rao’s unified theory of least-squares. Our analysis was based on a numerical experiment involving a set of height anomalies synthesised from the GGM GOCO05s, which is provided with a full noise covariance matrix. We showed that the three estimators perform similar, provided that the two regularisation parameters each method knows were chosen properly. As standard regularisation parameter choice rules do not apply here, we suggested a new parameter choice rule, and demonstrated its performance. Using this rule, we found that the differences between the three least-squares estimates were within noise. For the standard formulation of the weighted least-squares estimator with regularised noise covariance matrix, this required an exceptionally strong regularisation, much larger than one expected from the condition number of the noise covariance matrix. The preferred method is the inversion-free formulation of the weighted least-squares estimator, because of its simplicity with respect to the choice of the two regularisation parameters.Physical and Space GeodesyNovel Aerospace Material

Exact closed-form expressions for the complete RTM correction

We present exact, closed-form expressions for the complete RTM correction and the harmonic correction to disturbing potential, gravity disturbance, gravity anomaly, and height anomaly. They need to be applied in quasi-geoid modelling whenever data points are buried inside the masses after residual terrain model (RTM) reduction and analytically downward-continued functionals of the disturbing potential at the original locations of the data points are required. Compared to recent work of the authors published in this journal, no Taylor series enter the expressions and numerical instabilities of the harmonic downward continuation from the RTM surface to the Earth’s surface are avoided as are inaccuracies in the free-air upward continuation from the Earth’s surface to the RTM surface caused by a lack of precise information about higher-order derivatives of the disturbing potential. The new expressions can easily be implemented in any existing RTM software package and do not require additional computational resources. For two test areas located in western Norway and the Auvergne in France, we compute the complete RTM correction and the harmonic correction to the afore-mentioned functionals of the disturbing potential. Overall, all harmonic corrections are non-negative with maximum values of 1.54 m 2/ s 2, 263.0 μ Gal, 263.9 μ Gal, and 15.7 m (Norway) and 1.55 m 2/ s 2, 263.3 μ Gal, 263.3 μ Gal, and 15.8 cm (Auvergne) for disturbing potential, gravity disturbance, gravity anomaly, and height anomaly, respectively. The medians are 0.02 m 2/ s 2, 33.6 μ Gal, 33.7 μ Gal, and 0.3 cm (Norway) and 0.01 m 2/ s 2, 19.2 μ Gal, 19.2 μ Gal, and 0.1 cm (Auvergne). We also show that the nth Taylor polynomials used in the recent work of the authors published in this journal may have large remainders depending on the topography in the vicinity of the evaluation point no matter how n is chosen. Finally, we show that the commonly used expression for the harmonic correction to gravity anomaly introduced in 1981 is almost exact, though it was derived along a completely different line of reasoning. The errors do not exceed 49 μ Gal in both test areas. Moreover, the errors have a negligible impact on the computed height anomalies in one-centimetre quasi-geoid modelling, as the mean error does not exceed a few μ Gal in both test areas.Physical and Space Geodes

Systematic errors observed in CryoSat-2 elevation swaths on mountain glaciers and their implications

Our awareness of ice caps’ and mountain glaciers’ sensitivity to climate change has driven major advances in the application of remote sensing techniques during the past decade. Regarding ESA’s SARIn altimeter CryoSat-2, processing the full waveform to generate swaths of elevation estimates has become standard practice in regions of complex topographies. This technique provides information on areas where we would be blind otherwise. In this article, we discuss systematic errors and analyze their impact on surface elevation measurements and change rates of two test areas. In particular, we focus on periodically occurring errors in elevation swaths, caused by the superposition of coherent signals from range-ambiguous surfaces. They can lead to measurement errors in excess of 10m, affect most measurements in mountainous regions, are difficult to exclude with established post-processing techniques, and occur repeatedly for satellite revisits introducing a 369-day periodicity—difficult to distinguish from the annual cycle. We show a correlation between derived elevation swaths and the sensor view angle and explore the influence of common data exclusion choices on higher-level products. Our results indicate that these systematic errors hold a substantial share of the error budget and that the choice of thresholds impacts higher-level products. We conclude that error correlations need to be considered to characterize the data accuracy. With the established data editing strategies, systematic errors prevent resolving seasonal mass changes of single mountain glacier basins and impact aggregates over larger areas or longer periods.Physical and Space Geodes

Data requirements for a 5-mm quasi-geoid in the Netherlands

We assess the surface gravity data requirements for a 5-mm quasi-geoid model for the Netherlands mainland and continental shelf in terms of omission and commission errors. The omission error critically depends on the roughness of the topography and bathymetry. For the Netherlands continental shelf, Central and Northern Netherlands, the omission error is well described by the model 0.32d mm, where d is the data spacing in km. For the more hilly Southern Netherlands, the omission error model is 0.92d mm. The commission error depends on the kernel modification, the data spacing, and the data accuracy. When using the spheroidal Stokes kernel, it is well described by 0.277 d σΔg mm, where σΔg is the noise standard deviation of surface gravity data in mGal. An upper bound of the commission error of the state-of-the-art satellite-only gravity model GOCO05S over the Netherlands is e0.03676L–11.419 mm, where L is the maximum degree up to which this model is used. Only if this model is truncated at a sufficiently low degree, e.g., at degree 100, its contribution to the total commission error can be neglected. We determine the total error as the sum of commission and omission errors. Hence, to realize a 5-mm quasi-geoid model for the Netherlands mainland and continental shelf, a data spacing of 3.5 km is needed when assuming a noise standard deviation of 1.5 mGal for surface gravity data. The currently available land-based gravity data fulfill this requirement. This does not apply to the situation at sea, where the density of the shipboard gravity data and the accuracy of the radar altimeter-derived data do not allow the realization of a 5-mm quasi-geoid model.Novel Aerospace MaterialsPhysical and Space Geodes

Estimating the rates of mass change, ice volume change and snow volume change in Greenland from ICESat and GRACE data

The focus of this paper is on the quantification of ongoing mass and volume changes over the Greenland ice sheet. For that purpose, we used elevation changes derived from the Ice, Cloud, and land Elevation Satellite (ICESat) laser altimetry mission and monthly variations of the Earth’s gravity field as observed by the Gravity Recovery and Climate Experiment (GRACE) mission. Based on a stand alone processing scheme of ICESat data, the most probable estimate of the mass change rate from 2003 February to 2007 April equals ?139 ± 68 Gton yr?1. Here, we used a density of 600±300 kgm?3 to convert the estimated elevation change rate in the region above 2000m into a mass change rate. For the region below 2000 m, we used a density of 900±300 kgm?3. Based on GRACE gravity models from half 2002 to half 2007 as processed by CNES, CSR, DEOS and GFZ, the estimated mass change rate for the whole of Greenland ranges between ?128 and ?218 Gton yr?1. Most GRACE solutions show much stronger mass losses as obtained with ICESat, which might be related to a local undersampling of the mass loss by ICESat and uncertainties in the used snow/ice densities. To solve the problem of uncertainties in the snow and ice densities, two independent joint inversion concepts are proposed to profit from both GRACE and ICESat observations simultaneously. The first concept, developed to reduce the uncertainty of the mass change rate, estimates this rate in combination with an effective snow/ice density. However, it turns out that the uncertainties are not reduced, which is probably caused by the unrealistic assumption that the effective density is constant in space and time. The second concept is designed to convert GRACE and ICESat data into two totally new products: variations of ice volume and variations of snow volume separately. Such an approach is expected to lead to new insights in ongoing mass change processes over the Greenland ice sheet. Our results show for different GRACE solutions a snow volume change of ?11 to 155 km3 yr?1 and an ice loss with a rate of ?136 to ?292 km3 yr?1.Geoscience & Remote SensingCivil Engineering and Geoscience