Josef Sebera

On computing ellipsoidal harmonics using Jekeli’s renormalization

Gravity data observed on or reduced to the ellipsoid are preferably represented using ellipsoidal harmonics instead of spherical harmonics. Ellipsoidal harmonics, however, are difficult to use in practice because the computation of the associated Legendre functions of the second kind that occur in the ellipsoidal harmonic expansions is not straightforward. Jekeli’s renormalization simplifies the computation of the associated Legendre functions. We extended the direct computation of these functions—as well as that of their ratio—up to the second derivatives and minimized the number of required recurrences by a suitable hypergeometric transformation. Compared with the original Jekeli’s renormalization the associated Legendre differential equation is fulfilled up to much higher degrees and orders for our optimized recurrences. The derived functions were tested by comparing functionals of the gravitational potential computed with both ellipsoidal and spherical harmonic syntheses. As an input, the high resolution global gravity field model EGM2008 was used. The relative agreement we found between the results of ellipsoidal and spherical syntheses is 10−14, 10−12 and 10−8 for the potential and its first and second derivatives, respectively. Using the original renormalization, this agreement is 10−12, 10−8 and 10−5, respectively. In addition, our optimized recurrences require less computation time as the number of required terms for the hypergeometric functions is less.

Matlab script for 3D visualizing geodata on a rotating globe

We present a Matlab package for visualizing global data on a 3D sphere, whose rotation can be animated. Planetary elevation data sets such as geoid height or Earth topography can easily be represented through a slightly exaggerated, colored 3D relief, and then saved either as images or animations. All necessary parameters for the 3D visualization and animation are described and their usage is demonstrated on examples. Among other things, users are shown how to easily create their own color scales. In principle, any geoscientific scalar data given on a global grid of longitudes and latitudes can be visualized with this package. The package requires only the basic module of Matlab, running on an ordinary PC or notebook, and it is available for free download at http://www.asu.cas.cz/~bezdek/vyzkum/rotating_3d_globe/. HighlightsA Matlab package is presented for visualizing global data on 3D sphere.Possibly, rotation of the 3D sphere can be animated.For all examples shown, Matlab code is provided on the package website.Any global data can be visualized, e.g. planetary topography.The package is available for free download.

Satellite gravity gradient grids for geophysics

The Gravity field and steady-state Ocean Circulation Explorer (GOCE) satellite aimed at determining the Earth’s mean gravity field. GOCE delivered gravity gradients containing directional information, which are complicated to use because of their error characteristics and because they are given in a rotating instrument frame indirectly related to the Earth. We compute gravity gradients in grids at 225 km and 255 km altitude above the reference ellipsoid corresponding to the GOCE nominal and lower orbit phases respectively, and find that the grids may contain additional high-frequency content compared with GOCE-based global models. We discuss the gradient sensitivity for crustal depth slices using a 3D lithospheric model of the North-East Atlantic region, which shows that the depth sensitivity differs from gradient to gradient. In addition, the relative signal power for the individual gradient component changes comparing the 225 km and 255 km grids, implying that using all components at different heights reduces parameter uncertainties in geophysical modelling. Furthermore, since gravity gradients contain complementary information to gravity, we foresee the use of the grids in a wide range of applications from lithospheric modelling to studies on dynamic topography, and glacial isostatic adjustment, to bedrock geometry determination under ice sheets.

Spherical integral formulas for upward/downward continuation of gravitational gradients onto gravitational gradients

New integral formulas for upward/downward continuation of gravitational gradients onto gravitational gradients are derived in this article. They provide more options for continuation of gravitational gradient combinations and extend available mathematical apparatus formulated for this purpose up to now. The starting point represents the analytical solution of the spherical gradiometric boundary value problem in the spatial domain. Applying corresponding differential operators on the analytical solution of the spherical gradiometric boundary value problem, a total of 18 integral formulas are provided. Spatial and spectral forms of isotropic kernels are given and their behaviour for parameters of a GOCE-like satellite is investigated. Correctness of the new integral formulas and the isotropic kernels is tested in a closed-loop simulation. The derived integral formulas and the isotropic kernels form a theoretical basis for validation purposes and geophysical applications of satellite gradiometric data as provided currently by the GOCE mission. They also extend the well-known Meissl scheme.

Assessment of Systematic Errors in the Computation of Gravity Gradients from Satellite Altimeter Data

With satellite radar altimetry, the oceanic geoid can be determined with high precision and resolution. Double differentiation of these data along satellite altimeter ground tracks yields along-track gravity gradients that can be used to compute vertical gravity gradients at ground track crossovers. One way to counteract the noise amplification due to the differentiation is to smooth the data using smoothing splines. Although the effect of satellite altimeter data noise has been investigated to some extent, the associated systematic errors have not been assessed so far. Here we show that some of the systematic errors cannot be neglected. In particular, we found that the negligence of the dynamic ocean topography (DOT) may introduce errors that are greater than the measurement noise induced errors. If the gravity gradients are to be used for GOCE validation, then also in this case the DOT may not be neglected as the signal at GOCE altitude of 260 km may be above the GOCE requirements. In addition, we show that the altimetry derived gravity gradients cannot be compared one-to-one with those in a local Cartesian frame. The differences are small compared with the total signal, but they may be larger than the satellite altimetry induced stochastic errors and may be above the GOCE requirements. The cubic splines second derivative truncation error requires the use of 10 Hz altimeter data for the computation of gravity gradients at the Earths surface, while 1 Hz data are sufficient for validation at GOCE altitude.

Orbit Tuning of Planetary Orbiters for Accuracy Gain in Gravity Field Mapping

gradiometer on board) to the 16:1 resonance. To avoid the decrease in A, we have to choose the orbit in such a way that orderB of the lowest-order resonance, which will occur, will be higher than the highest degree Lmax of spherical harmonic expansion of the potential already known for the particular body. For the Earth, Lmax is now 150 [European improved gravitymodel of theEarth by new techniques (EIGEN-5S)] for satellite-only solutions and 2190

A new noise reduction method for airborne gravity gradient data

Airborne gravity gradient (AGG) measurements offer an increased resolution and accuracy compared to terrestrial measurements. But interpretation and processing of AGG data are often challenging as levelling errors and survey noise affect the data, and these effects are not easily recognised in the gradient components. We adopted the classic method of upward continuation in the noise reduction using the noise level estimates by the AGG system. By iteratively projecting the survey data to a lower level and upward continuing the data back to the survey height, parts of the high-frequency signal are suppressed. The filter, which is defined by this approach, is directly dependent on the noise level of the AGG data, the maximum number of iterations and the iterative step. We demonstrate the method by applying it to both synthetic data and real AGG data over Karasjok, Norway, and compare the results to the directional filtering method. The results show that the iterative filter can effectively reduce high-frequency noise in the data. A new noise reduction method that iteratively projects data to a lower height and upward continuing the data back to the survey height is described. This method can significantly improve the signal-to-noise ratio of noisy gravity gradient data, and has been successfully applied to both synthetic and real data.