Airborne Gravity Across New Zealand - For An Improved Vertical Datum

It is important to be able to accurately determine the height of a point on the Earth in terms of the Earth's gravitational potential field. These heights predict how water will flow and so they are vital for engineering and surveying purposes. They are determined using a vertical datum which consists of a specif ed height system and a defined reference surface. At present, in New Zealand, the o fficial vertical datum is NZVD2009 which uses a normal-orthometric height system and gravimetric quasigeoid, NZGeoid2009, as the reference surface. The aim of this thesis is to develop a more accurate gravimetric quasigeoid than NZGeoid2009, by incorporating new gravity data and utilising a re fined data processing strategy, to establish a better vertical datum for New Zealand. A new airborne gravimetry data set has been collected which covers the North, South and Stewart Islands of New Zealand with a flight line spacing of 10km. The data were susceptible to short error prone sections of track due to poor (turbulent) flight conditions and mean off sets which separate the recorded gravity data along flight lines by a constant value from neighbouring lines and existing gravity models. The error prone sections of track have been visually identified by assessing the cross track agreement with other flight lines and with the global gravity model EGM2008, and the mean offsets were estimated by a least squares method which takes into consideration the spatially correlated gravity signal. The repeatability of the data was assessed from data collected from five flights along two separate calibration lines. The mean gravity anomaly pro files calculated along the calibration lines each had a standard deviation of around 2.5 mGal. The internal consistency of the data was assessed by evaluating the diff erence between flight line data at intersection points. This accuracy measure was shown to be influenced by the along track filter, anisotropic topography and the relative flight line elevations. After correcting for all these effects the set of all intersecting differences had a standard deviation of approximately 5.9 mGal. From an existing terrestrial gravity database, around 40000 observations have been reprocessed to reduce them to Bouguer gravity anomalies, this was done to ensure consistency in the formulas that have been used. A new national 8 m digital elevation model (DEM) was used to calculate terrain corrections and these were carefully compared with terrain corrections estimated from field observations of the topography to reduce any discrepancies in calculating near zone terrain e ffects. The largest source of error in the terrestrial gravity anomaly data is due to inaccurate height estimates of the marks. The height discrepancies have been estimated by comparing the recorded heights in the database to those determined from the 8 m DEM and have been translated into mGal by calculating the propagated effect on the free air and Bouguer slab corrections. The airborne and terrestrial gravity data, along with a satellite altimetry marine gravity anomaly and existing shipborne gravity data, were assimilated by least squares collocation with a logarithmic covariance function to appropriately deal with the downward continuation of the airborne data, and gridded at 1 arc-minute resolution in the geographical region 25° (S) to 60 ° (S) and 160° (E) to 190° (E). 1 arc-minute block averaged heights were then used to calculate a reverse Bouguer slab correction, which when applied to the gravity data gave a gridded Faye anomaly. Different noise level variances were assigned to the separate data sets to optimally combine them. Forty six of the most contemporary global gravity models (from 2008 onwards) have each been compared to 1422 leveling and GNSS derived quasigeoid height anomalies. Overall the Eigen-6C4 model fitted the leveling and GNSS derived quasigeoid height anomalies best with a root mean squared error of 5.29cm. The Eigen-6C4 gravity model was subtracted from the gridded Faye anomaly (remove) and Stokes integral was evaluated on the residual gravity anomaly grid. A, theoretically optimum, modified Stokes kernel has been used and the modification degree L and spherical cap for the integration Ψ₀ were varied over the ranges L = 20; 40; 60; ..., 320 and Ψ₀ = 1° ; 1:5° ; 2° ; 2:5° ; 3° . The Eigen-6C4 geoid undulations were then added back to the residual geoid undulation grids and the primary indirect topographic effect was restored to obtain 80 quasigeoids for each L and Ψ₀ parameter variation. The optimal parameter choice was determined to be L = 280 and Ψ₀ = 1:5 which had the best agreement with the leveling and GNSS derived quasigeoid height anomalies with a standard deviation of 3.8cm and root mean squared residual of 4.8cm of the differences. This is a 1.25cm improvement on NZGeoid2009. The quasigeoid was also assessed closely in three main urban areas, Auckland, Wellington and Christchurch, where the majority of large scale engineering projects and surveying takes place in New Zealand. Here there were 123, 169 and 125 data points and the standard deviations of the differences were 3.976, 3.385 and 2.071cm and root mean squared differences of 3.58,4.388 and 4.572 cm respectively. This gives an average accuracy of 3.1 cm standard deviation in urban areas which is 1.5 cm better than the average for NZGeoid2009

Airborne Gravity Across New Zealand - For An Improved Vertical Datum

It is important to be able to accurately determine the height of a point on the Earth in terms of the Earth's gravitational potential field. These heights predict how water will flow and so they are vital for engineering and surveying purposes. They are determined using a vertical datum which consists of a specif ed height system and a defined reference surface. At present, in New Zealand, the o fficial vertical datum is NZVD2009 which uses a normal-orthometric height system and gravimetric quasigeoid, NZGeoid2009, as the reference surface. The aim of this thesis is to develop a more accurate gravimetric quasigeoid than NZGeoid2009, by incorporating new gravity data and utilising a re fined data processing strategy, to establish a better vertical datum for New Zealand. A new airborne gravimetry data set has been collected which covers the North, South and Stewart Islands of New Zealand with a flight line spacing of 10km. The data were susceptible to short error prone sections of track due to poor (turbulent) flight conditions and mean off sets which separate the recorded gravity data along flight lines by a constant value from neighbouring lines and existing gravity models. The error prone sections of track have been visually identified by assessing the cross track agreement with other flight lines and with the global gravity model EGM2008, and the mean offsets were estimated by a least squares method which takes into consideration the spatially correlated gravity signal. The repeatability of the data was assessed from data collected from five flights along two separate calibration lines. The mean gravity anomaly pro files calculated along the calibration lines each had a standard deviation of around 2.5 mGal. The internal consistency of the data was assessed by evaluating the diff erence between flight line data at intersection points. This accuracy measure was shown to be influenced by the along track filter, anisotropic topography and the relative flight line elevations. After correcting for all these effects the set of all intersecting differences had a standard deviation of approximately 5.9 mGal. From an existing terrestrial gravity database, around 40000 observations have been reprocessed to reduce them to Bouguer gravity anomalies, this was done to ensure consistency in the formulas that have been used. A new national 8 m digital elevation model (DEM) was used to calculate terrain corrections and these were carefully compared with terrain corrections estimated from field observations of the topography to reduce any discrepancies in calculating near zone terrain e ffects. The largest source of error in the terrestrial gravity anomaly data is due to inaccurate height estimates of the marks. The height discrepancies have been estimated by comparing the recorded heights in the database to those determined from the 8 m DEM and have been translated into mGal by calculating the propagated effect on the free air and Bouguer slab corrections. The airborne and terrestrial gravity data, along with a satellite altimetry marine gravity anomaly and existing shipborne gravity data, were assimilated by least squares collocation with a logarithmic covariance function to appropriately deal with the downward continuation of the airborne data, and gridded at 1 arc-minute resolution in the geographical region 25° (S) to 60 ° (S) and 160° (E) to 190° (E). 1 arc-minute block averaged heights were then used to calculate a reverse Bouguer slab correction, which when applied to the gravity data gave a gridded Faye anomaly. Different noise level variances were assigned to the separate data sets to optimally combine them. Forty six of the most contemporary global gravity models (from 2008 onwards) have each been compared to 1422 leveling and GNSS derived quasigeoid height anomalies. Overall the Eigen-6C4 model fitted the leveling and GNSS derived quasigeoid height anomalies best with a root mean squared error of 5.29cm. The Eigen-6C4 gravity model was subtracted from the gridded Faye anomaly (remove) and Stokes integral was evaluated on the residual gravity anomaly grid. A, theoretically optimum, modified Stokes kernel has been used and the modification degree L and spherical cap for the integration Ψ₀ were varied over the ranges L = 20; 40; 60; ..., 320 and Ψ₀ = 1° ; 1:5° ; 2° ; 2:5° ; 3° . The Eigen-6C4 geoid undulations were then added back to the residual geoid undulation grids and the primary indirect topographic effect was restored to obtain 80 quasigeoids for each L and Ψ₀ parameter variation. The optimal parameter choice was determined to be L = 280 and Ψ₀ = 1:5 which had the best agreement with the leveling and GNSS derived quasigeoid height anomalies with a standard deviation of 3.8cm and root mean squared residual of 4.8cm of the differences. This is a 1.25cm improvement on NZGeoid2009. The quasigeoid was also assessed closely in three main urban areas, Auckland, Wellington and Christchurch, where the majority of large scale engineering projects and surveying takes place in New Zealand. Here there were 123, 169 and 125 data points and the standard deviations of the differences were 3.976, 3.385 and 2.071cm and root mean squared differences of 3.58,4.388 and 4.572 cm respectively. This gives an average accuracy of 3.1 cm standard deviation in urban areas which is 1.5 cm better than the average for NZGeoid2009

Fast-Fourier-based error propagation for the gravimetric terrain correction

We have identified a gap in the literature on error propagation in the gravimetric terrain correction. Therefore, we have derived a mathematical framework to model the propagation of spatially correlated digital elevation model errors into gravimetric terrain corrections. As an example, we have determined how such an error model can be formulated for the planar terrain correction and then be evaluated efficiently using the 2D Fourier transform. We have computed 18.3 billion linear terrain corrections and corresponding error estimates for a 1 arc-second (~30 m) digital elevation model covering the whole of the Australian continent

The first Australian gravimetric quasigeoid model with location-specific uncertainty estimates

We describe the computation of the first Australian quasigeoid model to include error estimates as a function of location that have been propagated from uncertainties in the EGM2008 global model, land and altimeter-derived gravity anomalies and terrain corrections. The model has been extended to include Australia’s offshore territories and maritime boundaries using newer datasets comprising an additional (Formula presented.)280,000 land gravity observations, a newer altimeter-derived marine gravity anomaly grid, and terrain corrections at (Formula presented.) resolution. The error propagation uses a remove–restore approach, where the EGM2008 quasigeoid and gravity anomaly error grids are augmented by errors propagated through a modified Stokes integral from the errors in the altimeter gravity anomalies, land gravity observations and terrain corrections. The gravimetric quasigeoid errors (one sigma) are 50–60 mm across most of the Australian landmass, increasing to (Formula presented.) mm in regions of steep horizontal gravity gradients or the mountains, and are commensurate with external estimates

Gsolve, a Python computer program with a graphical user interface to transform relative gravity survey measurements to absolute gravity values and gravity anomalies

A Python program (Gsolve) with a graphical user interface has been developed to assist with routine data processing of relative gravity measurements. Gsolve calculates the gravity at each measurement site of a relative gravity survey, which is referenced to at least one known gravity value. The tidal effects of the sun and moon, gravimeter drift and tares in the data are all accounted for during the processing of the survey measurements.The calculation is based on a least squares formulation where the difference between the absolute gravity at each surveyed location and parameters relating to the dynamics of the gravimeter are minimized with respect to the relative gravity observations, and some supplied gravity reference site values. The program additionally allows the user to compute free air gravity anomalies, with respect to the GRS80 and GRS67 reference ellipsoids, from the determined gravity values and calculate terrain corrections at each of the surveyed sites using a prism formula and a user supplied digital elevation model.This paper reviews the mathematical framework used to reduce relative gravimeter survey observations to gravity values. It then goes on to detail how the processing steps can be implemented using the software. Keywords: Relative gravity, Python, Least squares, Free air anomalies, Terrain correction