Volume 1, Issue 2 -

Numerical evaluation of mean values of topographical effects The main problem treated in this paper is the determination of accurate mean values of the topographical effects from point values known on a regular geographical grid. Three kinds of topographical effects are studied: terrain correction, condensed terrain correction and direct topographical effect. The relation between the terrain roughness and optimal density of the points to be used in the computations is investigated in five morphologically different areas of Canada. The error of the geoid caused by the inaccuracy of the mean values computed from a variable number of points in a cell is estimated. These errors are then compared against the one centimetre target to give us the sufficient minimum number of points needed for the averaging. The mean terrain effects are computed from the point values as a simple average over a particular cell. Point values are assumed to be errorless so that the accuracy of the mean values is a function of the density of the point values only. The mentioned one centimetre criterion is applied in the sense of the Chebyshev norm. It has been observed that the relation between the number of points needed for the averaging and the terrain roughness as quantified by the terrain RMS is almost linear. After estimating the two parameters of this linear relation, seven minimally required grid densities are suggested for different intervals of terrain roughness. The results are applied to produce maps of a minimal density of points needed for sufficiently accurate determination of mean topographical effects for Canada.

A comparison of model estimates of ocean-tide loading displacements in New Zealand We analyze the regional accuracy of the global ocean tide models TPXO7.2, GOT00.2, NAO. 99b, FES2004, and EOT10a using the tide-gauge records from 7 stations along the coastline of New Zealand. The comparison reveals that TPXO7.2 provides the bestfit with the tide-gauge data in this part of the world. The TPXO. 7.2 ocean tide model is then used for computing and analysis of the vertical and horizontal ocean-tide loading displacements in New Zealand. The ocean-tide loading is calculated for the semi-diurnal and diurnal tidal harmonic constituents M 2 , S 2 , N 2 , K 2 , K 1 , O 1 , P 1 , and Q 1 , the long-period tidal components M f and M m , and for the shallow-water (nonlinear) components M 4 , MS 4 , and MN 4 . The charts of the ocean tide (offshore) and the ocean-tide loading (onshore) are compiled on a 5×5 arc-min geographical grid at the study area of New Zealand. The results reveal that the major contribution to the ocean-tide loading displacements up to a few centimeters is attributed to the semi-diurnal and diurnal harmonic constituents. The contribution of the long-period tidal components is to about 1 mm, and the effect of the shallow-water components is less than 0.2 mm. The maxima of vertical and horizontal displacements are typically of the same order of magnitude with larger amplitudes of vertical motions. The maximum amplitudes of vertical motions up to 5.6 cm are found in the upper north region of the North Island. The maximum horizontal motions up to 2.1 cm are in the northwest part of the South Island The ocean-tide loading displacements in New Zealand are finally compared with the crustal deformations due to geodynamic, atmospheric and other phenomena.

External calibration of GOCE accelerations to improve derived gravitational gradients The aim of ESA's satellite mission GOCE is to determine the Earth's gravity field with high accuracy and resolution. To achieve this aim, GOCE carries a gravitational gradiometer that needs calibration. Existing global gravity field models in combination with GOCE star sensor data may be used to synthesize reference differential accelerations with which the common and differential accelerations, as derived from the gradiometer measurements, can be calibrated. We present a new method in which the data are transformed from the time to the frequency domain, which allows accounting for the coloured noise on the measurements. The weight matrix is iteratively adjusted and we apply our method to real GOCE data. With our method, the gravitational gradient trace significantly reduces as compared with the currently available in-flight calibrated measurements.

Detecting and mitigating ocean tidal loading displacements in the Bay of Fundy using GPS Tidal induced displacement is one of the systematic errors that contribute to the scatter in the geodetic measurements derived from the GPS system. This paper focuses on ascertaining and reducing tidal induced errors due to ocean tide loading (OTL) at two GPS sites, namely CGSJ (Coast Guard Saint John, New Brunswick) and DRHS (Digby High School, Nova Scotia), established under the Princess of Acadia project in Saint John, New Brunswick and in Digby, Nova Scotia, respectively. Baseline solutions were obtained by processing 3 and 24 hourly GPS data for the period of one month, using differential positioning software DIPOP and its client end GUI, FACE v2.0. The observed differential variations of GPS sites were compared to the OTL modeled differential variations along the baselines for statistical analysis. The predictive differential ocean tide loading induced variations of the baselines were modeled with global ocean tide model FES 95.2 (0.5 by 0.5 resolution) supplemented with a higher resolution regional tide model by Pagiatakis (0.25 by 0.25 resolution). After modeling for tidal effects the solutions for both the height component and baseline length show daily repeatability better than 1.5 mm for baselines ranging from 87.5 km to 170 km. The tidal model used in the investigation explains the observed motion considerably well, with correlation coefficients of greater than 0.70 between modeled and observed curves. Elimination of tidal effects has resulted in day-to-day rms reduction of height better than 80 percent.

Comparisons of geoid models over Alaska computed with different Stokes' kernel modifications Various Stokes kernel modification methods have been developed over the years. The goal of this paper is to test the most commonly used Stokes kernel modifications numerically by using Alaska as a test area and EGM08 as a reference model. The tests show that some methods are more sensitive than others to the integration cap sizes. For instance, using the methods of Vaníček and Kleusberg or Featherstone et al. with kernel modification at degree 60, the geoid decreases by 30 cm (on average) when the cap size increases from 1° to 25°. The corresponding changes in the methods of Wong and Gore and Heck and Grüninger are only at the 1 cm level. At high modification degrees, above 360, the methods of Vaníček and Kleusberg and Featherstone et al become unstable because of numerical problems in the modification coefficients; similar conclusions have been reported by Featherstone (2003). In contrast, the methods of Wong and Gore, Heck and Grüninger and the least-squares spectral combination are stable at any modification degree, though they do not provide as good fit as the best case of the Molodenskii-type methods at the GPS/Leveling benchmarks. However, certain tests for choosing the cap size and modification degree have to be performed in advance to avoid abrupt mean geoid changes if the latter methods are applied.

A theoretical study on the bottlenecks of GPS phase ambiguity resolution in a CORS RTK Network Crucial to the performance of GPS Network RTK positioning is that a user receives and applies correction information from a CORS Network. These corrections are necessary for the user to account for the atmospheric (ionospheric and tropospheric) delays and possibly orbit errors between his approximate location and the locations of the CORS Network stations. In order to provide the most precise corrections to users, the CORS Network processing should be based on integer resolution of the carrier phase ambiguities between the network's CORS stations. One of the main challenges is to reduce the convergence time, thus being able to quickly resolve the integer carrier phase ambiguities between the network's reference stations. Ideally, the network ambiguity resolution should be conducted within one single observation epoch, thus truly in real time. Unfortunately, single-epoch CORS Network RTK ambiguity resolution is currently not feasible and in the present contribution we study the bottlenecks preventing this. For current dual-frequency GPS the primary cause of these CORS Network integer ambiguity initialization times is the lack of a sufficiently large number of visible satellites. Although an increase in satellite number shortens the ambiguity convergence times, instantaneous CORS Network RTK ambiguity resolution is not feasible even with 14 satellites. It is further shown that increasing the number of stations within the CORS Network itself does not help ambiguity resolution much, since every new station introduces new ambiguities. The problem with CORS Network RTK ambiguity resolution is the presence of the atmospheric (mainly ionospheric) delays themselves and the fact that there are no external corrections that are sufficiently precise. We also show that external satellite clock corrections hardly contribute to CORS Network RTK ambiguity resolution, despite their quality, since the network satellite clock parameters and the ambiguities are almost completely uncorrelated. One positive is that the foreseen modernized GPS will have a very beneficial effect on CORS ambiguity resolution, because of an additional frequency with improved code precision.

On the Definition and Realization of a Global Vertical Datum A Global Vertical Datum (GVD) is naturally defined by the geoid, and there is a well-established consensus to adopt Gauss-Bessel-Listing's definition of the geoid (i.e. as being the level surface of the Earth's gravity field that best fits the undisturbed sea level). The main problem in defining the geoid is therefore to fix its constant geopotential (W 00 ). Nevertheless, this definition can be interpreted as to fit either the geopotential of sea surface to a constant (W 0 ), or to minimize the height of sea level with respect to the geoid. Although the two interpretations lead to apparently different solutions, we show that they are practically the same. To improve the estimation of W 0 , we propose to weight the included data according to their a priori error estimates. Finally we discuss the use of GNSS/levelling data for vertical datum connections, concluding that such data, although indispensable for regional vertical datum connections, in combination with satellite altimetry over the oceans are practically useless for determining the GVD. Also, such a joint adjustment of the GVD and regional vertical datum biases yields inferior local connections vs. a separate adjustment with fixed GVD.

Positioning and Position Error of Petroleum Wells We present a new model for the estimation of positional uncertainty of petroleum wells. The model uses a heavy tailed normal inverse Gaussian distribution for the errors in Earth's magnetic field reference values. These references are required for calculating the position of a well using magnetic directional surveying. The results show that the normal inverse Gaussian distribution gives a more realistic fit to the Earth's magnetic field reference values than the normal distribution, which is the current state of art. Errors in surveying sensors and reference errors are propagated along the well path to obtain the positioning uncertainty. The positional uncertainty is important for the probability of collisions between wells and for drilling accurate relief wells when an eventual collision has resulted in a blow-out situation. We compare results of the normal model used in the petroleum industry with our proposed model. The comparison is made for anti-collision calculations between drilled wells and a planned well. The results indicate a higher risk of collisions for certain well geometries when using the normal distribution, while a collision is avoided in the normal inverse Gaussian situation. We find clear differences between an approximate normal test and a Monte Carlo test.

Procrustean Statistical Inference of Deformations A two step method has been devised for the statistical inference of deformation changes. In the first step of this method and based on Procrustes analysis of deformation tensors, the significance of the change in a time or space series of deformation tensors is statistically analyzed. In the second step significant change(s) in deformations are localized. In other words, they are assigned to certain parameters of deformation tensor. This is done using the Global Model Test. Because of the key role of Procrustes analysis in the proposed method for the inference of deformation changes, it has been given the name of Procrustean Statistical Inference of Deformations. The method has been implemented to synthetic and real deformations. The 3D-deformation tensors of a regional GPS network in the Kenai Peninsula, for analyzing the spatial variation of deformation tensors or the change of deformation within the study area, and a local GPS network in France, for analyzing the temporal variation of deformation tensors or the change of deformation in time at every point of the network in the study area have been used for illustrating the practical application of the proposed method.

Mean Sea Surface and ocean circulation in North Atlantic and the Arctic Sea High resolution Mean Sea Surface (MSS) model and its error estimation over the study region (56°N<ϕ<82°N, 45°W<λ<33°E) have been determined to evaluate Mean Dynamic Topography (MDT) in the Fram Strait and adjacent seas. Multiple high-latitude observing satellite radar altimetry data were used to determine a new MSS model that, hereafter, is called NTNU MSS. Sea Surface Height (SSH) values of the NTNU MSS model vary between 15 to 70 m over the study region, and the internal consistency or the quality estimate for the model ranges from 1 to 5 cm. External comparisons have been performed to validate the NTNU model applying available MSS models such as KMS04, mean and Standard Deviations (SD) of the differences are 1.7 cm and 8.9 cm, respectively. To assess the compatibility of the NTNU MSS model, residual analysis has been carried out relative to geoid and Oceanographic MDT (OMDT) models. Utilizing OCTAS04 geoid and OCCAM MDT models, the SD and mean of the residuals are 13.1 cm and 10.1 cm, respectively. Finally, the surface geostrophic velocities have been computed and compared with the velocities extracted from the OCCAM and KMS04. Differences between the geostrophic velocities derived from the NTNU Synthetic MDT (SMDT) and the OCCAM OMDT are 4.97 and 2.94 cm/s for the mean and SD values.