Global Ultra-High Resolution Topographic Gravity ModellingThis project seeks to continue the investigators pioneering work on theoretical and practical foundations of the rapidly evolving discipline of ultra-high resolution topographic gravity modelling. Being of crucial importance for geodesy, geophysics and engineering, among others, the project aims to provide accurate, reliable and efficient estimation of topographic gravity. Based on landmark discoveries and experiences with the development of the currently unmatched Global Gravity Model plus (GGMplus), work in 2019 continues to focus on the identification, quantification and effective mitigation of discretisation errors. Findings from previous studies, showed that information on discretisation errors play a key role in establishing a practical balance between accuracy and computational speed. Ultimately, we aim to use findings of this project to provide global topographic gravity at unprecedented fine spatial scales (at 90 m or below) enabling a future update of GGMplus
Principal investigatorMichael Kuhn M.Kuhn@curtin.edu.au
Area of scienceEngineering, Geosciences
Applications usedFortran compiler
The availability of ultra-high resolution topographic gravity, e.g. gravitational effect generated by detailed information of the topographic masses, is crucial for industry and government allowing instant reduction of gravity surveys. In geodesy, exploration geophysics, engineering and other related areas precise topographic gravity provide reference values to address important research questions and test computational techniques. The computational methodology to derive topographic gravity follows classical Newtonian gravity forward modelling (Newtonian integration) where (global) volume integrals are replaced by discretised numerical integration using an envelope of elementary mass elements. The rigorous solution requires the evaluation of gravitational effects generated by a large number (~million) of globally distributed mass elements in order to obtain the accurate gravitational effect at a single location. Depending on the accuracy level, this calculation can take ~0.1 to 10 seconds for a single location using a standard PC (e.g. CPU with ~3GHz). While it is not a problem to perform this calculation for one or a few locations it becomes numerically very challenging when dealing with millions or even billions of locations within regional or global modelling at ultra-high spatial resolution.
Due to the high computational burden, ultra-high resolution topographic gravity modelling in the past was restricted to local areas only. However, as computation points can be calculated independently of each other, when applying discretised Newtonian integration, massive parallel computation can easily be applied by partitioning the overall task in many smaller tasks, e.g. the independent calculation of a sub-set of computation points. In order to realize parallel computation we divide the original ultra-high resolution computation point data into a series of smaller tiles defining a subset of computation points that can be calculated completely independent of any other tile. To calculate all tiles we use the parallelism approach by packing multiple serial jobs. The preparation of all tiles and their computation is realised through a shell script while the Newtonian integration is performed using a specifically developed Fortran 77 (serial) code. Importantly, following this approach, very precise gravitational effects can be obtained that are suitable to serve as reference values to test theories and other (often approximate) techniques as has been demonstrated by the investigators in the past.
Work in 2019 continued a series of important studies initiated over several years ago and made possible through continued support by the Pawsey Centre. In 2019 this continued work culminated into a series studies (in part already initiated in 2018; see 2018 report) focusing on theoretical and practical aspects related to the determination of global ultra-high resolution topographic gravity and the calculation and provision of a new global model of spherical gravimetric terrain corrections (SRTM2gravity) at an unprecedented spatial resolution of 3-arc-seconds (~90 m). In particular we continued to look at aspects related to convergence/divergence issues when dealing with topographic gravity using spectral domain techniques (Bucha, Hirt and Kuhn 2019a), studied theoretical and practical aspects of residual terrain modelling (RTM) a faster technique to derive topographic gravity (Hirt et al. 2019a, Bucha, Hirt and Kuhn 2019b, 2019c, Bucha et al. 2019) and completed the derivation of global ultra-high resolution topographic gravity culminating in the SRTM2gravity model (Hirt et al. 2019b).
Already reported in the 2018 report but now fully published (on-line version was already available in August 2018) Pawsey Centre resources helped to develop a new technique able to mitigate divergence problems related to spectral domain techniques commonly used to model topographic gravity (Bucha, Hirt and Kuhn 2019a). The new approach relies on the Runge–Krarup theorem and a mathematical technique called iterative downward continuation. While the former is a theorem stating that at least in theory topographic gravity should be able to be derived divergence-free the latter provides the mathematical framework to achieve this. Based on the very rugged lunar topography (more challenging than Earth’s topography), it has been numerically demonstrated that the newly developed technique can effectively mitigate divergence effects when compared to divergence-free reference values obtained through global Newtonian integration at a global 5-arc-minute resolution. Importantly, the numerical test to confirm theoretical developments was only possible through the provision of supercomputing resources provided by the Pawsey Centre.
We have now completed a series of theoretical and practical studies (see also 2018 report) on residual terrain modelling (RTM), a common technique allowing for faster (but approximate) derivation of topographic gravity (Hirt et al. 2019a, Bucha, Hirt and Kuhn 2019b, 2019c, Bucha et al. 2019). The studies culminated in (i) a comprehensive numerical study to identify the most suitable/effective practical realisation (Hirt et al. 2019a) and (ii) development of a new RTM technique (Bucha et al. 2019) removing limitations identified in classical RTM techniques used in (i). For the numerical studies in (i), supercomputing resources of the Pawsey Centre have been used to provide high-resolution reference solutions over selected test areas while the mostly theoretical study in (ii) did not require Pawsey Centre resources but used findings from (i). As a key feature of the new RTM technique, cap-modified spectral gravity forward modelling is applied to rigorously account for near-zone gravity effects (e.g. only from masses close to the computation point). This technique has been introduced and successfully tested by (Bucha, Hirt and Kuhn 2019b) where again high-resolution reference solutions have been provided using Pawsey Centre resources. An extension of this technique has been given by (Bucha, Hirt and Kuhn 2019c) for the full gravity tensor did not directly rely on Pawsey Centre resources.
Based on the RTM technique tested above work on the derivation of ultra-high-resolution topographic gravity has been completed (Hirt et al. 2019b). Results are presented as global models of (i) full-scale gravitational attraction of Earth’s global topography (e.g. gravimetric terrain corrections) and (ii) residual gravity effects (e.g. RTM gravity) containing short-wavelength (< ~10 km) gravitational signals. This is the first-ever successful conversion of global Shuttle Radar Topography Mission (SRTM) elevation data to gravity effects at the unprecedented ultra-high spatial resolution of 3-arc-seconds (~90 m). The methodology used is based on the RTM technique combining spatial and spectral gravity forward modelling. While the numerically challenging calculations have been performed on facilities of the Leibniz Supercomputing Centre of the Bavarian Academy of Sciences (www.lrz.de), Pawsey Centre supercomputing resources have been used to provide rather time consuming reference solutions over six globally distributed test areas. Comparison to the reference solutions as well as ground-truth data revealed an accuracy of the new model well below the 0.2 mGal-level (except in very rugged terrain such as over the Himalaya) being sufficient for most practical applications. The SRTM2gravity model has been made publically available at http://ddfe.curtin.edu.au/models/SRTM2gravity2018 and can readily be used to correct gravimetric surveys in geodesy and geophysics or augment Earth gravitation models to crate ultra-fine gravity maps such as provided by GGMplus (result of a former iVec project).
Overall, the studies outlined above demonstrate how Pawsey Centre supercomputing resources helped to numerically confirm theoretical developments (e.g. divergence-free spectral domain gravity forward modelling and improved RTM techniques) as well as test practical calculations of ultra-high resolution topographic gravity (e.g. SRTM2gravity model
List of Publications
B. Bucha, C. Hirt, M. Kuhn, Divergence-free spherical harmonic gravity field modelling based on the Runge-Krarup theorem: a case study for the Moon. Journal of Geodesy, 93(4), 489-513 (2019a).
B. Bucha, C. Hirt, M. Kuhn, Cap integration in spectral gravity forward modelling: near- and far-zone gravity effects via Molodensky’s truncation coefficients. Journal of Geodesy, 93(1), 65-63 (2019b).
B. Bucha, C. Hirt, M. Kuhn, Cap integration in spectral gravity forward modelling up to the full gravity tensor. Journal of Geodesy, 93(9), 1707-1737 (2019c).
B. Bucha, C. Hirt, M. Yang, M. Kuhn, M. Rexer, Residual terrain modelling (RTM) in terms of the cap-modified spectral technique: RTM from a new perspective. Journal of Geodesy, 93(10), pp. 2089-2108 (2019).
C. Hirt, M. Yang, B. Bucha, M. Kuhn, A numerical study of residual terrain modelling (RTM) techniques and the harmonic correction using ultra-high-degree spectral gravity modelling. Journal of Geodesy, 93(9), 1469-1486 (2019a).
C. Hirt, M. Yang, M. Kuhn, B. Bucha, A. Kurzman, R. Pail, SRTM2gravity: An ultrahigh resolution global model of gravimetric terrain corrections. Geophysical Research Letters, 46(9), 4618-4627 (2019b).