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Physical heights is one of the most important topics in physical geodesy. Their original concept, introduced in the 19-th century, defined physical heights as lengths of plumblines of the Earth's gravity field between the geoid and points of interest. There are orthometric heights of surface points, that have been traditionally estimated by spirit levelling and measured gravity; however, the knowledge of the density distribution of topographic masses (masses between the geoid and Earth's surface) is required that significantly affects their determinability. This was also the main reason why a new type of physical heights was proposed in the mid of the 20-th century. Normal heights approximate orthometric heights in a sense that the Earth's gravity field is replaced by the normal gravity field, an analytic model based on the theory of an equipotential ellipsoid. This height system has been introduced since that time in different countries in Europe and beyond. Contrary to the classical height system based on orthometric heights, its counterpart based on normal heights may have slightly different definitions. Moreover, normal heights are often defined as heights of points above the quasigeoid. This contribution reviews alternative definitions of normal heights and respective height systems. It is argued that both orthometric and normal heights refer to the geoid. In the case physical heights are estimated by satellite positioning, normal heights must be computed through the height anomaly estimated at each point of interest, whether it is below, at or above the Earth's surface. On the contrary, orthometric heights of all points along the same plumbline, be it below, at or above the Earth's surface, are estimated by introducing one value of the geoid height. Normal heights of surface points can be estimated by spirit levelling easier than orthometric heights as no topographic mass density hypothesis is required; however, one has to keep in mind the gravity field approximation used both for their definition and realization. [ABSTRACT FROM AUTHOR]

We improve the precision and computation speed of the fully-normalized associated Legendre functions (fnALFs) for ultra-high degrees and orders of spherical harmonic transforms. We take advantage of their numerical behaviour of and propose two new methods for solving an underflow/overflow problem in their calculation. We specifically discuss the application of the two methods in the fixed-order increasing-degree recursion computation technique. The first method uses successive ratios of fnALFs and the second method, called the Midway method, starts iteration from tiny initial values, which are still in the range of the IEEE double-precision environment, rather than from sectorial fnALFs. The underflow/overflow problem in the successive ratio method is handled by using a logarithm-based method and the extended range arithmetic. We validate both methods using numerical tests and compare their results with the X-number method in terms of precision, stability, and speed. The results show that the relative precision of the proposed methods is better than 10−9 for the maximum degree of 100000, compared to results derived by the high precision Wolfram's Mathematica software. Average CPU times required for evaluation of fnALFs over different latitudes demonstrate that the two proposed methods are faster by about 10–30% and 20–90% with respect to the X-number method for the maximum degree in the range of 50–65000. [ABSTRACT FROM AUTHOR]

The uncertainties of the geoidal heights estimated from ground gravity data caused by their spatial distribution and noise are investigated in this study. To test these effects, the geoidal heights are estimated from synthetic ground gravity data using the Stokes- Helmert approach. Five different magnitudes of the random noise in ground gravity data and three types of their spatial distribution are considered in the study, namely grid, semigrid and random. The noise propagation is estimated for the two major computational steps of the Stokes-Helmert approach, i.e., the downward continuation of ground gravity and Stokes's integration. Numerical results show that in order to achieve the 1-cm geoid, the ground gravity data should be distributed on the grid or semi-grid with the average angular distance less than 2′. If they are randomly distributed (scattered gravity points), the 1-cm geoid cannot be estimated if the average angular distance between scattered gravity points is larger than 1′. Besides, the noise of the gravity data for the tree types of their spatial distribution should be below 1 mGal to estimate the 1-cm geoid. The advantage of interpolating scattered gravity points onto the regular grid, rather than using them directly, is also investigated in this study. Numerical test shows that it is always worth interpolating the scattered points to the regular grid except if the scattered gravity points are sparser than 5′. [ABSTRACT FROM AUTHOR]

The downward continuation (DWC) of the gravity anomalies from the Earth's surface to the geoid is still probably the most problematic step in the precise geoid determination. It is this step that motivates the quasi-geoid users to opt for Molodenskij's rather than Stokes's theory. The reason for this is that the DWC is perceived as suffering from two major flaws: first, a physically meaningful DWC technique requires the knowledge of the irregular topographical density; second, the Poisson DWC, which is the only physically meaningful technique we know, presents itself mathematically in the form of Fredholm integral equation of the 1st kind. As Fredholm integral equations are often numerically ill-conditioned, this makes some people believe that the DWC problem is physically ill-posed. According to a revered French mathematician Hadamard, the DWC problem is physically well-posed and as such gives always a finite and unique solution. The necessity of knowing the topographical density is, of course, a real problem but one that is being solved with an ever increasing accuracy; so sooner or later it will allow us to determine the geoid with the centimetre accuracy. [ABSTRACT FROM AUTHOR]

We derive expressions for computing the gravitational field (potential and its radial derivative) generated by an arbitrary homogeneous or laterally varying density contrast layer with a variable depth and thickness based on methods for a spherical harmonic analysis and synthesis of gravity field. The newly derived expressions are utilised in the gravimetric forward modelling of major known density structures within the Earth's crust (excluding the ocean density contrast) beneath the geoid surface. The gravitational field quantities due to the sediments and crust components density contrasts, shown in numerical examples, are computed using the 2 × 2 arc-deg discrete data from the global crustal model CRUST2.0. These density contrasts are defined relative to the adopted value of the reference crustal density of 2670 kgm. All computations are realised globally on a 1 × 1 arc-deg geographical grid at the Earth's surface. The maxima of the gravitational signal due to the sediments density contrast are mainly along continental shelf regions with the largest sedimentary deposits. The corresponding maxima due to the consolidated crust components density contrast are over areas of the largest continental crustal thickness with variable geological structure. [ABSTRACT FROM AUTHOR]

In geophysical studies investigating the lithosphere structure, the gravitational field generated by the ocean density contrast (i.e., bathymetry-generated gravitational field) represents a significant amount of the signal to be modelled and subsequently removed from the Earth's gravity field. The ocean density contrast is typically calculated as the difference between the mean density values of the Earth's crust and seawater. The approximation of the actual seawater density distribution by its mean value yields relative errors up to about 2% in computed quantities of the gravitational field. To reduce these errors, a more realistic model of the seawater density distribution is utilized based on the analysis of existing oceanographic data of salinity, temperature, and pressure (depth). We study the accuracy of the bathymetry-generated gravitational field quantities formulated for a depth-dependent model of the seawater density distribution. This density distribution approximates the seawater density variations due to an increasing pressure with depth, whereas smaller lateral density variations caused by salinity, temperature, and other oceanographic factors are not taken into consideration. The error analysis reveals that the approximation of the seawater density by the depth-dependent density model reduces the maximum errors to less than 0.6%. The corresponding depth-averaged errors are below 0.1%. The depth-dependent seawater density model is further facilitated in expressions for computing the bathymetry-generated gravitational field quantities by means of the spherical bathymetric (ocean bottom depth) functions. The numerical realization reveals large differences in the results obtained with and without consideration of the depth-dependent seawater density distribution. The maxima of absolute differences reach 201 m/s and 16.5 mGal in computed values of the potential and attraction, respectively. The application of the depth-dependent seawater density model thus significantly improves the accuracy in the forward modelling of the bathymetric gravitational field quantities. [ABSTRACT FROM AUTHOR]

Spherical harmonic transformation is of practical interest in geodesy for transformation of globally distributed quantities such as gravity between space and frequency domains. The increasing spatial resolution of the latest and forthcoming gravitational models pose true computational challenges for classical algorithms since serious numerical instabilities arise during the computation of the respective base functions of the spherical harmonic expansion. A possible solution is the evaluation of the associated Legendre functions in the Fourier domain where numerical instabilities can be circumvented by an independent frequency-wise scaling of numerical coefficients into a numerically suitable double precision range. It is then rather straightforward to commit global fast data transformation into the Fourier domain and to evaluate subsequently spherical harmonic coefficients. For the inverse, the computation of respective Fourier coefficients from a given spherical harmonic model is performed as an inverse Fast Fourier Transform into globally distributed data points. The two-step formulation turns out to be stable even for very high resolutions as well as efficient when using state-of-the-art shared memory/multi-core architectures. In principle, any functional of the geopotential can be computed in this way. To give an example for the overall performance of the algorithm, we transformed an equiangular 1 arcmin grid of terrain elevation data corresponding to spherical harmonic degree and order 10800. [ABSTRACT FROM AUTHOR]

spectral representation of the topographic corrections to gravity field quantities is formulated in terms of spherical height functions. When computing the far-zone contributions to the topographic corrections, various types of the truncation coefficients are applied to a spectral representation of Newton's integral. In this study we utilise Molodensky's truncation coefficients in deriving the expressions for the far-zone contributions to topographic corrections. The expressions for computing the far-zone gravity field contributions corrected for the effect of topography are then obtained by combining the expressions for the far-zone contributions to the gravity field quantities and to the respective topographic corrections, both expressed in terms of Molodensky's truncation coefficients. The numerical examples of the far-zone contributions to the topographic corrections and to the topography-corrected gravity field quantities are given over the study area situated in the Canadian Rocky Mountains with adjacent planes. Coefficients of the global elevation and geopotential models are used as the input data. [ABSTRACT FROM AUTHOR]

Two different approaches for transformation of airborne gravity disturbances, derived from gravity observations at low-elevation flying platforms, into geoidal undulations are formulated, tested and discussed in this contribution. Their mathematical models are based on Green's integral equations. They are in these two approaches defined at two different levels and also applied in a mutually reversed order. While one of these approaches corresponds to the classical method commonly applied in processing of ground gravity data, the other approach represents a new method for processing of gravity data in geoid determination that is unique to airborne gravimetry. Although theoretically equivalent in the continuous sense, both approaches are tested numerically for possible numerical advantages, especially due to the inverse of discretized Fredholm's integral equation of the first kind applied on different data. High-frequency synthetic gravity data burdened by the 2-mGal random noise, that are expected from current airborne gravity systems, are used for numerical testing. The results show that both approaches can deliver for the given data a comparable cm-level accuracy of the geoidal undulations. The new approach has, however, significantly higher computational efficiency. It would be thus recommended for real life geoid computations. Additional errors related to regularization of gravity data and the geoid, and to accuracy of the reference field, that would further deteriorate the quality of estimated geoidal undulations, are not considered in this study. [ABSTRACT FROM AUTHOR]

In this paper, the numerical stability and efficiency of methods of harmonic downward continuation from flying altitudes are treated for sampled gravity field data. The problem is first formulated in its continuous form, i.e. as the inverse solution of the spherical Dirichlet problem, and is then approximated by Gaussian quadrature to yield a finite system of linear equations. The numerical stability of this system is investigated for both error-free gravity data and for the noisy and band-limited gravity measurements usually obtained from airborne gravity surveys. It can be shown that the system becomes ill-conditioned, once the ratio between flying altitude and data sampling rate exceeds a certain limit. It can also be shown that noisy measurements tend to generate a solution that is practically useless, long before the system becomes ill-conditioned. Therefore, instead of treating the general solution of the discrete downward continuation problem, the more modest question is studied, for which range of flying altitudes and sampling rates, the numerical solution of the discrete linear system can be considered as practically useful. ‘Practically useful’ will be defined heuristically as of sufficient accuracy and stability to satisfy the requirements of the user. The question will be investigated for the specific application of geoid computation from gravity data sampled at flying altitudes. In this case, a stable solution with a standard deviation of a few centimeters is required. Typical flight parameters are heights of 2–6 km, a minimum half-wavelength resolution of 2 km, and data noise between 0.5 and 1.5 mGal. Different methods of geoid determination, different solution techniques for the resulting systems of linear equations, and different minimization principles will be compared. As a result operational parameters will be defined which, for a given noise level, will result in a geoid accuracy of a few centimeters for the estimated band-limited gravity field spectrum. [ABSTRACT FROM AUTHOR]