Robert Kingdon

Poisson Downward Continuation Solution by the Jacobi Method

Poisson Downward Continuation Solution by the Jacobi Method Downward continuation is a continuing problem in geodesy and geophysics. Inversion of the discrete form of the Poisson integration process provides a numerical solution to the problem, but because the B matrix that defines the discrete Poisson integration is not always well conditioned the solution may be noisy in situations where the discretization step is small and in areas containing large heights. We provide two remedies, both in the context of the Jacobi iterative solution to the Poisson downward continuation problem. First, we suggest testing according to the upward continued result from each solution, rather then testing between successive solutions on the geoid, so that choice of a tolerance for the convergence of the iterative method is more meaningful and intuitive. Second, we show how a tolerance that reflects the conditioning of the B matrix can regularize the solution, and suggest an approximate way of choosing such a tolerance. Using these methods, we are able to calculate a solution that appears regular in an area of Papua New Guinea having heights over 3200 m, over a grid with 1 arc-minute spacing, based on a very poorly conditioned B matrix.

Testing Stokes-Helmert geoid model computation on a synthetic gravity field: experiences and shortcomings

We report on testing the UNB (University of New Brunswick) software suite for accurate regional geoid model determination by use of Stokes-Helmert’s method against an Australian Synthetic Field (ASF) as “ground truth”. This testing has taken several years and has led to discoveries of several significant errors (larger than 5mm in the resulting geoid models) both in the UNB software as well as the ASF. It was our hope that, after correcting the errors in UNB software, we would be able to come up with some definite numbers as far as the achievable accuracy for a geoid model computed by the UNB software. Unfortunately, it turned out that the ASF contained errors, some of as yet unknown origin, that will have to be removed before that ultimate goal can be reached. Regardless, the testing has taught us some valuable lessons, which we describe in this paper. As matters stand now, it seems that given errorless gravity data on 1′ by 1′ grid, a digital elevation model of a reasonable accuracy and no topographical density variations, the Stokes-Helmert approach as realised in the UNB software suite is capable of delivering an accuracy of the geoid model of no constant bias, standard deviation of about 25 mm and a maximum range of about 200 mm. We note that the UNB software suite does not use any corrective measures, such as biases and tilts or surface fitting, so the resulting errors reflect only the errors in modelling the geoid.

Does Poisson’s downward continuation give physically meaningful results?

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.

Effects of Hypothetical Complex Mass-Density Distributions on Geoidal Height

Geoid computation according to the Stokes-Helmert scheme requires accurate modelling of the variations of mass-density within topography. Current topographical models used in this scheme consider only horizontal variations, although in reality density varies three-dimensionally. Insufficient knowledge of regional three-dimensional density distributions prevents evaluation from real data. In light of this deficiency, we attempt to estimate the order of magnitude of the error in geoidal heights caused by neglecting the depth variations by calculating, for artificial but realistic mass-density distributions, the difference between results from 2D and 3D models.

Optimal Combination of Satellite and Terrestrial Gravity Data for Regional Geoid Determination Using Stokes-Helmert’s Method, the Auvergne Test Case

The precise regional geoid modelling requires combination of terrestrial gravity data with satellite-only Earth Gravitational Models (EGMs). In determining the geoid using the Stokes-Helmert approach, the relative contribution of terrestrial and satellite data to the computed geoid can be specified by the Stokes integration cap size defined by the spherical distance ψ0 and the maximum degree l0 of the EGM-based reference spheroid. Larger values of l0 decrease the role of terrestrial gravity data and increase the contribution of satellite data and vice versa for larger values of ψ0. The determination of the optimal combination of the parameters l0 and ψ0 is numerically investigated in this paper. A numerical procedure is proposed to find the best geoid solution by comparing derived gravimetric geoidal heights with those at GNSS/levelling points. The proposed method is tested over the Auvergne geoid computation area. The results show that despite the availability of recent satellite-only EGMs with the maximum degree/order 300, the combination of l0 = 160 and ψ0 = 45 arc-min yields the best fitting geoid in terms of the standard deviation and the range of the differences between the estimated gravimetric and GNSS/levelling geoidal heights. Depending on the accuracy of available ground gravity data and reference geoidal heights at GNSS/levelling points, the optimal combination of these two parameters may be different in other regions.

The Effect of Noise on Geoid Height in Stokes-Helmert Method

Noises are an inevitable part of gravity observations and they can affect the accuracy of the height datum if they are not treated properly in geoid determination. To provide data for geodetic boundary value problems, surface gravity observations must be transferred harmonically down to the geoid, which is called Downward Continuation (DC). Fredholm integral of the first kind is one of the physically meaningful ways of DC, where the Poisson kernel is used to evaluate the data on the geoid. DC behaves inherently as a high pass filter so it magnifies existing noise in Helmert gravity anomalies on geoid (free air anomalies after applying the Helmert’s second condensation method); although the results on the geoid will be later smoothed by evaluating the Stokes’s integral so the noise is less pronounced in the final geoid heights. The effect of noise in Stokes-Helmert geoid determination approach is numerically investigated in this study. The territory of Iran, limited to 44–62° longitude and 24–40° latitude, is considered as the area of interest in this study. The global gravity model EGM2008, up to degree/order 2160, is used to synthesize the free air gravity anomalies on a regular grid on topography and are then transferred to Helmert space using available Digital Elevation Models (DEMs). Different levels of noise are added to the data and the effects of noise are investigated using the SHGeo software package, developed at the University of New Brunswick (UNB). Results show that if the downward continuation of 5*5 arc-min surface points is required, the standard deviation of differences between “noisy” and “clean” data on the geoid will increase by 15% with respect to the corresponding standard deviation on topography. These differences will increase for denser grid resolutions. For example, the noise of ∆g on geoid will increase up to 100% if 1*1 arc-min points are used. The results of evaluating the Stokes integral show smoother results in terms of noise in the data. For example, 2 mGal noise in the gravity anomalies on a 5*5 arc-min grid can cause 1.5 cm of error in the geoid heights. This value is smaller when denser grids are used. Despite increasing noise in downward continuation steps, the results show smaller error in geoid heights if gravity anomalies are located on a denser grid.

Rigorous Evaluation of Gravity Field Functionals from Satellite-Only Gravitational Models Within Topography

Currently, extensive work is being done in the field of geodesy on producing better gravitational models using purely space-based techniques. With the large datasets spanning a long timeframe, thanks to the GOCE and GRACE missions, it is now possible to compute high quality global gravitational models and publish them in a convenient form: spherical harmonics. For regional geoid modeling, this is advantageous as these models provide a useful reference which can be improved with terrestrial observations. In order for these global models to be usable below the topographical surface, certain considerations are required; topographical masses cause the function that describes the gravity potential to be non-harmonic in the space between the topographical surface and the geoid. This violates the mathematical assumptions behind solid spherical harmonics.

Computation of precise geoid model of Auvergne using current UNB Stokes-Helmert’s approach

Abstract The aim of this paper is to show a present state-of-the-art precise gravimetric geoid determination using the UNB Stokes-Helmert’s technique in a simple schematic way. A detailed description of a practical application of this technique in the Auvergne test area is also provided. In this paper, we discuss the most problematic parts of the solution: correct application of topographic and atmospheric effects including the lateral topographical density variations, downward continuation of gravity anomalies from the Earth surface to the geoid, and the optimal incorporation of the global gravity field into the final geoid model. The final model is tested on 75 GNSS/levelling points supplied with normal Molodenskij heights, which for this investigation are transformed to rigorous orthometric heights. The standard deviation of the computed geoid model is 3.3 cm without applying any artificial improvement which is the same as that of the most accurate quasigeoid.