Band 8, Heft 1

A large systematic difference (ranging from −20 cm to +130 cm) was found between NAVD 88 (North AmericanVertical Datum of 1988) and the pure gravimetric geoid models. This difference not only makes it very difficult to augment the local geoid model by directly using the vast NAVD 88 network with state-of-the-art technologies recently developed in geodesy, but also limits the ability of researchers to effectively demonstrate the geoid model improvements on the NAVD 88 network. Here, both conventional regression analyses based on various predefined basis functions such as polynomials, B-splines, and Legendre functions and the Latent Variable Analysis (LVA) such as the Factor Analysis (FA) are used to analyze the systematic difference. Besides giving a mathematical model, the regression results do not reveal a great deal about the physical reasons that caused the large differences in NAVD 88, which may be of interest to various researchers. Furthermore, there is still a significant amount of no-Gaussian signals left in the residuals of the conventional regression models. On the other side, the FA method not only provides a better not of the data, but also offers possible explanations of the error sources. Without requiring extra hypothesis tests on the model coefficients, the results from FA are more efficient in terms of capturing the systematic difference. Furthermore, without using a covariance model, a novel interpolating method based on the relationship between the loading matrix and the factor scores is developed for predictive purposes. The prediction error analysis shows that about 3-7 cm precision is expected in NAVD 88 after removing the systematic difference.

In 1962 A. Bjerhammar introduced the method of analytical continuation in physical geodesy, implying that surface gravity anomalies are downward continued into the topographic masses down to an internal sphere (the Bjerhammar sphere). The method also includes analytical upward continuation of the potential to the surface of the Earth to obtain the quasigeoid. One can show that also the common remove-compute-restore technique for geoid determination includes an analytical continuation as long as the complete density distribution of the topography is not known. The analytical continuation implies that the downward continued gravity anomaly and/or potential are/is in error by the so-called topographic bias, which was postulated by a simple formula of L E Sjöberg in 2007. Here we will numerically test the postulated formula by comparing it with the bias obtained by analytical downward continuation of the external potential of a homogeneous ellipsoid to an inner sphere. The result shows that the postulated formula holds: At the equator of the ellipsoid, where the external potential is downward continued 21 km, the computed and postulated topographic biases agree to less than a millimetre (when the potential is scaled to the unit of metre).

The Bayesian approach allows an intuitive way to derive the methods of statistics. Probability is defined as a measure of the plausibility of statements or propositions. Three rules are sufficient to obtain the laws of probability. If the statements refer to the numerical values of variables, the so-called random variables, univariate and multivariate distributions follow. They lead to the point estimation by which unknown quantities, i.e. unknown parameters, are computed from measurements. The unknown parameters are random variables, they are fixed quantities in traditional statistics which is not founded on Bayes’ theorem. Bayesian statistics therefore recommends itself for Monte Carlo methods, which generate random variates from given distributions. Monte Carlo methods, of course, can also be applied in traditional statistics. The unknown parameters, are introduced as functions of the measurements, and the Monte Carlo methods give the covariance matrix and the expectation of these functions. A confidence region is derived where the unknown parameters are situated with a given probability. Following a method of traditional statistics, hypotheses are tested by determining whether a value for an unknown parameter lies inside or outside the confidence region. The error propagation of a random vector by the Monte Carlo methods is presented as an application. If the random vector results from a nonlinearly transformed vector, its covariance matrix and its expectation follow from the Monte Carlo estimate. This saves a considerable amount of derivatives to be computed, and errors of the linearization are avoided. The Monte Carlo method is therefore efficient. If the functions of the measurements are given by a sum of two or more random vectors with different multivariate distributions, the resulting distribution is generally not known. TheMonte Carlo methods are then needed to obtain the covariance matrix and the expectation of the sum.

It is well known that the success in precise determinations of the gravimetric geoid height (N) and the orthometric height (H) rely on the knowledge of the topographic mass distribution. We show that the residual topographic bias due to an imprecise information on the topographic density is practically the same for N and H, but with opposite signs. This result is demonstrated both for the Helmert orthometric height and for a more precise orthometric height derived by analytical continuation of the external geopotential to the geoid. This result leads to the conclusion that precise gravimetric geoid heights cannot be validated by GNSS-levelling geoid heights in mountainous regions for the errors caused by the incorrect modelling of the topographic mass distribution, because this uncertainty is hidden in the difference between the two geoid estimators.

The Nordic Geodetic Commission (NKG) has launched a joint NKG GNSS Analysis Centre that aims to routinely produce high qualityGNSS solutions for the common needs of the NKG and the Nordic and Baltic countries. A consistent and densified velocity field is needed for the constraining of the gla-cial isostatic adjustment (GIA) modelling that is a key component of maintaining the national reference frame realisations in the area. We described the methods of the NKG GNSS Analysis Centre including the defined processing setup for the local analysis centres (LAC) and for the combination centres.We analysed the results of the first 2.5 years (2014.5-2016). The results showed that different subnets were consistent with the combined solution within 1-2 mm level. We observed the so called network effect affecting our reference frame alignment. However, the accuracy of the reference frame alignment was on a few millimetre level in the area of the main interest (Nordic and Baltic Countries). TheNKGGNSS AC was declared fully operational in April 2017.

Monitoring deformation of man-made structures is very important to prevent them from a risk of collapse and save lives. Such a process is also used for monitoring change in historical objects, which are deforming continuously with time. An example of this is the Vasa warship, which was under water for about 300 years. The ship was raised from the bottom of the sea and is kept in the Vasa museum in Stockholm. A geodetic network with points on the museum building and the ship’s body has been established and measured for 12 years for monitoring the ship’s deformation. The coordinate time series of each point on the ship and their uncertainties have been estimated epoch-wisely. In this paper, our goal is to statistically analyse the ship’s hull movements. By fitting a quadratic polynomial to the coordinate time series of each point of the hull, its acceleration and velocity are estimated. In addition, their significance is tested by comparing them with their respective estimated errors after the fitting. Our numerical investigations show that the backside of the ship, having highest elevation and slope, has moved vertically faster than the other places by a velocity and an acceleration of about 2 mm/year and 0.1 mm/year 2 , respectively and this part of the ship is the weakest with a higher risk of collapse. The central parts of the ship are more stable as the ship hull is almost vertical and closer to the floor. Generally, the hull is moving towards its port and downwards

This study provides additional information about the impact of atmospheric pressure on sea level variations. The observed regularity in sea level atmospheric pressure depends mainly on the latitude and verified to be dominantly random closer to the equator. It was demonstrated that almost all the annual and semiannual sea level variations at 27 globally distributed tide gauge stations can be attributed to the regional/local atmospheric forcing as an inverted barometric effect. Statistically significant non-linearities were detected in the regional atmospheric pressure series, which in turn impacted other sea level variations as compounders in tandem with the lunar nodal forcing, generating lunar sub-harmonics with multidecadal periods. It was shown that random component of regional atmospheric pressure tends to cluster at monthly intervals. The clusters are likely to be caused by the intraannual seasonal atmospheric temperature changes,which may also act as random beats in generating sub-harmonics observed in sea level changes as another mechanism. This study also affirmed that there are no statistically significant secular trends in the progression of regional atmospheric pressures, hence there was no contribution to the sea level trends during the 20 th century by the atmospheric pressure.Meanwhile, the estimated nonuniform scale factors of the inverted barometer effects suggest that the sea level atmospheric pressure will bias the sea level trends inferred from satellite altimetry measurements if their impact is accounted for as corrections without proper scaling.

The aim of this paper is to present a quantitative analysis of the adequacy of the main currently existing combined Global Geopotential Models (GGMs) for modeling normal-geoid heights throughout Brazil. As major advances have been reached since mid-2016 in the combined GGMs elaboration and development, the main objective of this analysis is to verify if, in fact, the most recent models present superior or equivalent performance to the most performant previous models. The analysis was based on comparisons between normal-geoid height values obtained fromGNSS/leveling solutions and values calculated from GGMs XGM2016, GOCO05C, EIGEN-6C4 and EGM2008, according to different geopotential functionals - geoid height and height anomaly - and in different degrees of development, always through the relative method. This procedure was applied to 997 stations which carry information of both ellipsoidal and normal-orthometric heights, located all over Brazil. As a main result, it was observed the superior performance of the recent combined GGMs, GOCO05C and XGM2016, when compared to the older models, EIGEN-6C4 and EGM2008, when all of them are developed up to degree 720, the maximum degree of the recent models; and a approximate equality of results when all of the models are used in their individual maximum degrees.

In geodesy, hypothesis testing is applied to a wide area of applications e.g. outlier detection, deformation analysis or, more generally, model optimisation. Due to the possible far-reaching consequences of a decision, high statistical test power of such a hypothesis test is needed. The Neyman-Pearson lemma states that under strict assumptions the often-applied likelihood ratio test has highest statistical test power and may thus fulfill the requirement. The application, however, is made more difficult as most of the decision problems are non-linear and, thus, the probability density function of the parameters does not belong to the well-known set of statistical test distributions. Moreover, the statistical test power may change, if linear approximations of the likelihood ratio test are applied. The influence of the non-linearity on hypothesis testing is investigated and exemplified by the planar coordinate transformations. Whereas several mathematical equivalent expressions are conceivable to evaluate the rotation parameter of the transformation, the decisions and, thus, the probabilities of type 1 and 2 decision errors of the related hypothesis testing are unequal to each other. Based on Monte Carlo integration, the effective decision errors are estimated and used as a basis of valuation for linear and non-linear equivalents.

The Tibetan Plateau is the largest region of high elevation in the world. The source of water for a number of important rivers, the Himalayan region is vital to the billions of inhabitants of the Asian continent. Over the last fifty years, the climate in the region has warmed more rapidly than anywhere else at the same latitude. Causes and effects, and the geographical details of these alarming warming trends are as yet not fully known. One way of assessing the effects of climate change in this area is to measure the change in glacier volume in the region, but estimates made on the basis of different techniques have not been conclusive to date, and remain difficult to reconcile. We examine the temporal behavior of the mass flux integrated over four distinct groupings of Tibetan glaciers using satellite gravimetry from the Gravity Recovery and Climate Experiment (GRACE). We use a technique known as spatio-spectral localization using spherical Slepian functions to convert global spherical harmonic expansions of the time-dependent geopotential into monthly estimates of mass changes over the Tibetan Plateau. Subsequent reductions are aimed at interpreting this mass change as due to gains or losses in ice mass. We find that (ice) mass has been decreasing on the Tibetan Plateau between 2002 and 2016 but with significant spatial variability throughout the region. Specifically, in the regions of Himalaya, Pamir, Qilian, and Tien Shan, glaciers have been losing ice mass at a rate of −11±3, −1±2, +8±2, and −6±1 Gt/yr, respectively, over the last decade.

The geoid, but not the quasigeoid, is an equipotential surface in the Earth’s gravity field that can serve both as a geodetic datum and a reference surface in geophysics. It is also a natural zero-level surface, as it agrees with the undisturbed mean sea level. Orthometric heights are physical heights above the geoid,while normal heights are geometric heights (of the telluroid) above the reference ellipsoid. Normal heights and the quasigeoid can be determined without any information on the Earth’s topographic density distribution, which is not the case for orthometric heights and geoid. We show from various derivations that the difference between the geoid and the quasigeoid heights, being of the order of 5 m, can be expressed by the simple Bouguer gravity anomaly as the only term that includes the topographic density distribution. This implies that recent formulas, including the refined Bouguer anomaly and a difference between topographic gravity potentials, do not necessarily improve the result. Intuitively one may assume that the quasigeoid, closely related with the Earth’s surface, is rougher than the geoid. For numerical studies the topography is usually divided into blocks of mean elevations, excluding the problem with a non-star shaped Earth. In this case the smoothness of both types of geoid models are affected by the slope of the terrain,which shows that even at high resolutions with ultra-small blocks the geoid model is likely as rough as the quasigeoid model. In case of the real Earth there are areas where the quasigeoid, but not the geoid, is ambiguous, and this problem increases with the numerical resolution of the requested solution. These ambiguities affect also normal and orthometric heights. However, this problem can be solved by using the mean quasigeoid model defined by using average topographic heights at any requested resolution. An exact solution of the ambiguity for the normal height/quasigeoid can be provided by GNSS-levelling.

Predicting sea level rise is essential for current climate discussions. Empirical models put in use to monitor and analyze sea level variations observed at globally distributed tide gauge stations during the last decade can provide reliable predictions with high resolution. Meanwhile, prediction intervals, an alternative to confidence intervals, are to be recognized and deployed in sea level studies. Predictions together with their prediction intervals, as demonstrated in this study, can quantify the uncertainty of a single future observation from a population, instead of the uncertainty of a conceivable average sea level namely a confidence interval, and it is thereby, better suited for coastal risk assessment to guide policy development for mitigation and adaptation responses.

This observational study reports that several globally distributed tide gauge stations exhibit a propensity of statistically significant sea level accelerations during the satellite altimetry era. However, the magnitudes of the estimated tide gauge accelerations during this period are systematically and noticeably smaller than the global mean sea level acceleration reported by recent analyses of satellite altimetry. The differences are likely to be caused by the interannual, decadal and interdecadal sea level variations, which are modeled using a broken trend model with overlapping harmonics in the analyses of tide gauge data but omitted in the analysis of satellite altimetry.

We tested the accuracy and usability of a sparse GNSS reference station network for network RTK (NRTK) using the Finnish permanent GNSS network FinnRef. We modified the configuration of the FinnRef network stations used in NRTK computation. This allowed us to perform the test both inside and outside of the network area using different NRTK methods and two different RTK receivers. In the test area the average distance between the FinnRef stationswas 160 km. As a comparison,we tested also with the commercial Trimnet and HxGN SmartNet positioning services operated by Geotrim Oy and Leica Geosystems Finland, respectively. Tests showed that the horizontal and vertical rms of Trimnet servicewas 16mmand 40 mm, and of HxGN SmartNet service 23mmand 48 mm. The best rms for the sparse NLS (National Land Survey of Finland) Service was 22 mm and 56 mm. These results indicate that a good NRTK solution can be achieved with a sparser network than typically used. This study also indicates, that the methods for NRTK processing can also affect the quality of the solution.

Spherical harmonic expansion is a commonly applied mathematical representation of the earth’s gravity field. This representation is implied by the potential coeffcients determined by using elements/parameters of the field observed on the surface of the earth and/or in space outside the earth in the spherical harmonic expansion of the field. International Centre for Gravity Earth Models (ICGEM) publishes, from time to time, Global Gravity Field Models (GGMs) that have been developed. These GGMs need evaluation with terrestrial data of different locations to ascertain their accuracy for application in those locations. In this study, Bouguer gravity anomalies derived from a total of eleven (11) recent GGMs, using sixty sample points, were evaluated by means of Root-Mean-Square difference and correlation coeficient. The Root-Mean-Square differences of the computed Bouguer anomalies from ICGEMwebsite compared to their positionally corresponding terrestrial Bouguer anomalies range from 9.530mgal to 37.113mgal. Additionally, the correlation coe_cients of the structure of the signal of the terrestrial and GGM-derived Bouguer anomalies range from 0.480 to 0.879. It was observed that GECO derived Bouguer gravity anomalies have the best signal structure relationship with the terrestrial data than the other ten GGMs. We also discovered that EIGEN-6C4 and GECO derived Bouguer anomalies have enormous potential to be used as supplements to the terrestrial Bouguer anomalies for Enugu State, Nigeria.

This article studies the Optimal Biased Kriging (OBK) approach which is an alternative geostatistical method that gives up the unbiasedness condition of Ordinary Kriging (OK) to gain an improved Mean Squared Prediction Error (MSPE). The system of equations for the optimal linear biased Kriging predictor is derived and itsMSPE is compared with that of Ordinary Kriging. A major impediment in implementing this system of equations and performing Kriging interpolation with massive datasets is the inversion of the spatial coherency matrix. This problem is investigated and a novel method, called “homeogram tapering”, which exploits spatial sorting techniques to create sparse matrices for efficient matrix inversion, is described. Finally, as an application, results from experiments performed on a geoid undulation dataset from Korea are presented. A precise geoid is usually the indispensable basis for meaningful hydrological studies over wide areas. These experiments use the theory presented here along with a relatively new spatial coherency measure, called the homeogram, also known as the non-centered covariance function.

In order to accelerate the spherical/spheroidal harmonic synthesis of any function, we developed a new recursive method to compute the sine/cosine series coefficient of the 4π fully- and Schmidt quasi-normalized associated Legendre functions. The key of the method is a set of increasing-degree/order mixed-wavenumber two to four-term recurrence formulas to compute the diagonal terms. They are used in preparing the seed values of the decreasing-order fixed-degree, and fixed-wavenumber two- and three-term recurrence formulas, which are obtained by modifying the classic relations. The new method is accurate and capable to deal with an arbitrary high degree/ order/wavenumber. Also, it runs significantly faster than the previous method of ours utilizing the Wigner d function, say around 20 times more when the maximum degree exceeds 1,000.