The first Australian gravimetric quasigeoid model with locationspecific uncertainty estimates
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Abstract
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 altimeterderived 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 \({\sim }\)280,000 land gravity observations, a newer altimeterderived marine gravity anomaly grid, and terrain corrections at \(1^{\prime \prime }\times 1^{\prime \prime }\) 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 \({\sim }100\) mm in regions of steep horizontal gravity gradients or the mountains, and are commensurate with external estimates.
Keywords
Gravimetric quasigeoid Error propagation Australia1 Introduction
We describe the computation of the gravimetric quasigeoid component of AUSGeoid2020, herein called Australian gravimetric quasigeoid 2017 (AGQG2017), as well as the computation of error estimates as a function of location (also called geographic specificity by Pavlis and Saleh 2005). These locationspecific errors have been propagated from a combination of uncertainties in EGM2008 (Pavlis et al. 2012, 2013), land and altimeterderived gravity anomalies, and terrain corrections (McCubbine et al. 2017). In a companion paper (in prep), we describe the geometric component of AUSGeoid2020, where AGQG2017 has been distorted by least squares prediction to fit the Australian Height Datum (AHD; Roelse et al. 1971) on land.
AUSGeoid09 (Brown et al. 2011) included a geometric component where the AGQG2009 gravimetric quasigeoid model (Featherstone et al. 2011) was distorted to fit the AHD using crossvalidated least squares prediction (Featherstone and Sproule 2006). This yielded a surface for the more direct transformation of GNSSderived ellipsoidal heights to the AHD (cf. Milbert 1995; Featherstone 1998, 2008; Smith and Roman 2001), allowing Australian land surveyors to realise AHD heights directly using GNSS, rather than having to apply postsurvey adjustments as with previous Australian quasi/geoid models.
Geoscience Australia (GA), the national geodetic agency, is in the process of moving from the Geocentric Datum of Australia 1994.0 (GDA94) to GDA2020, which is based on the International Terrestrial Reference Frame 2014 (ITRF2014; Altamimi et al. 2016) extrapolated to epoch 2020.0 using Australian ITRF2014 station velocities. This datum change will cause horizontal geodetic coordinates to move \({\sim }1.8\) m in a northeasterly direction and ellipsoidal heights to increase by \({\sim }90\) mm. For full threedimensional (3D) implementation of GDA2020, there remains the need to compute an accompanying model of the separation between the AHD and GRS80, as was done for AUSGeoid09, to be called AUSGeoid2020.
Though the geometric component of AUSGeoid09 could be remodelled for AUSGeoid2020 using AGQG2009 and the colocated GNSSAHD heights now available over Australia, we would also like to profit from the following new data: (i) an additional \({\sim }\)280,000 land gravity observations; (ii) retracked satellite altimeterderived marine gravity anomalies that include Jason1 and CryoSat2 data (Sandwell and Smith 2009; Garcia et al. 2014; Sandwell et al. 2013, 2014); (iii) the \(1^{\prime \prime }\times 1^{\prime \prime }\) (\(\sim \)30 m) resolution DEMH digital elevation model (Gallant et al. 2011) derived from the Shuttle Radar Topography Mission (STRM; Farr et al. 2007); and (iv) improved numerical integration routines (Hirt et al. 2011) in the 1DFFT implementation of Stokes’s integral (Haagmans et al. 1993).
GNSS users of AUSGeoid09 have had to rely on a wholeofregion uncertainty estimate of 30–50 mm taken from the average of nationwide residuals versus GNSSAHD (Brown et al. 2011). This single uncertainty estimate is unrealistic in many regions. This is particularly the case in the denserpopulated Australian coastal zones, where AUSGeoid09 has had to rely on altimeterderived gravity anomalies offshore, which are poor in the coastal zone (e.g., Claessens 2012), and where the shiptrack gravity data are sparse and unreliable (Featherstone 2009). Therefore, we have estimated quasigeoid (height anomaly) errors as a function of location (cf. Pavlis and Saleh 2005; Huang and Véronneau 2013) by propagating uncertainties from EGM2008, land and altimeterderived gravity anomalies, and terrain corrections in a remove–compute–restore (RCR) approach (Sect. 3).
2 Gravimetric quasigeoid computations
2.1 Preliminaries
All previous Australian geoid or quasigeoid models (cited in Featherstone et al. 2001, 2011) have focussed only on the Australian mainland and Tasmania. However, Australia administers several offshore territories and a 200nauticalmile maritime boundary around each. Since the 1DFFT requires a rectangular grid, we extended the computation area to \(8^{\circ }\)S, \(61^{\circ }\)S, \(93^{\circ }\)E, \(174^{\circ }\)E, resulting in a grid of approximately 15.5 million \(1^{\prime }\times 1^{\prime }\) nodes, roughly twice that used for AGQG2009.
2.2 Gravity data
We experimented with the effect on the quasigeoid of computing the gravity anomalies with respect to an a priori gravimetric quasigeoid model (cf. Amos and Featherstone 2009). The GNSSlevelling data (Sect. 2.4) were used to calculate a geometric component (cf. Brown et al. 2011) that models the difference between the a priori gravimetric quasigeoid and the AHD. This is dominated by a north–south tilt (cf. Featherstone and Filmer 2012) but with some regional distortions. The geometric component was converted to a freeair gravity term and modifiedStokesintegrated using the parameters in Sect. 2.6 to determine the effect on the quasigeoid. The maximum magnitude is less than 3 mm. This is due to (1) the highpass filtering properties of a modified Stokes kernel over a small (0.5degree) integration cap (cf. Vaníček and Featherstone 1998), and (2) the isotropic (azimuth independent) modified Stokes kernel is blind to a tilt in the gravity anomalies over the integration cap. That is, a tilt that manifests as positive values over one half of the cap and equal negative values over the other half cancel out and thus make no contribution to the quasigeoid.
For each record in the GA database, we calculated and applied normal gravity (Somigliana–Pizzetti formula, GRS80 ellipsoid (Moritz 1980)) and secondorder free air and Bouguer plate corrections to the point gravity values to obtain simple planar Bouguer gravity anomalies. The tensioned spline (Smith and Wessel 1990) routine in the Generic Mapping Tools (GMT; Wessel et al. 2013) with a tension factor of 0.25 was used for the interpolation of point Bouguer gravity anomalies. We used the “reconstruction” technique (Featherstone and Kirby 2000) to generate Molodensky freeair gravity anomalies on the topography (Fig. 2a), using the \(1^{\prime \prime }\times 1^{\prime \prime }\) DEMH data from GA. We added a \(1^{\prime \prime }\times 1^{\prime \prime }\) grid of planar terrain corrections (Fig. 2b; McCubbine et al. 2017) to these to give Faye anomalies and then blockaveraged these onto a \(1^{\prime }\times 1^{\prime }\) grid in GMT for subsequent quasigeoid computations.
2.3 Subtleties of EGM synthesis
2.3.1 Synthesis on the topography
For AGQG2009 (Featherstone et al. 2011), we used the harmonic_synth.f FORTRAN software (http://earthinfo.nga.mil/GandG/wgs84/gravitymod/egm2008/index.html) to synthesise ellipsoidal blockaveraged gravity anomalies and quasigeoid heights for use in the RCR technique. After receiving reports of poorer performance in the Great Dividing Range in southeastern Australia (e.g., Rexer et al. 2013; Sussanna et al. 2016), we discovered that EGM2008 had been synthesised on the ellipsoid instead of on the topography as demanded by Molodensky theory. However, this poorer performance is more likely to be explained by erroneous Australian land gravity data that have now been corrected in the GA database (Sect. 2.6; Fig. 8).
Synthesising highdegree gravity field functionals on the topography dramatically slows the performance of harmonic_synth.f for a \(1^{\prime }\times 1^{\prime }\) grid over the extents defined in Sect. 2.1, because the numerically efficient recursion routines (Holmes and Featherstone 2002) cannot be implemented on an irregular surface. Therefore, we used the isGrafLab.m Matlab\(^{\mathrm{TM}}\) software (Bucha and Janák 2013, 2014) instead, which is capable of computing gravity field functionals from spherical harmonic coefficients on an irregular surface quite efficiently using the gradient approach (Hirt 2012) even for very high spherical harmonic degrees. This requires ellipsoidal heights of the topography, computed as follows.
We blockaveraged the \(1^{\prime \prime }\times 1^{\prime \prime }\) DEMH data onto a \(1^{\prime }\times 1^{\prime }\) grid (the same resolution as AGQG2017) to yield the computation point locations for the spherical harmonic synthesis on the topography. These blockaveraged heights need to be converted to ellipsoidal heights for isGrafLab, thus requiring an iterative scheme. We first synthesised a \(1^{\prime }\times 1^{\prime }\) quasigeoid grid on the ellipsoid using isGrafLab and then added the blockaveraged DEMH heights to this to obtain a first estimate of the ellipsoidal heights of the topography. We then recomputed the quasigeoid at this first estimate of the ellipsoidal height of the topography and added the DEMH heights to gain a second estimate of the ellipsoidal height of the topography. This iteration converged quickly (\({<}1\) mm difference between the first and second iterations).
Figure 3 shows the differences between calculating gravity anomalies and quasigeoid heights from EGM2008 at the ellipsoidal height of the topography versus on the surface of the GRS80 ellipsoid. The differences reach maximum magnitudes of 42.162 mGal and 0.363 m, respectively. We ran an experiment over the reasonably mountainous island of Tasmania (maximum elevation of \({\sim }1600\) m) to identify the magnitude of error that has resulted in AGQG2009 from the theoretically incorrect synthesising EGM2008 on the ellipsoid (Appendix A). It transpires that the RCR technique is rather forgiving of this consistently made mistake. Nevertheless, it remains theoretically superior to synthesise the EGM on the topography for quasigeoid determination, and this was done for AGQG2017.
Descriptive statistics (m) of the levelled heights from four different LSAs and their differences (7224 points on the Australian mainland)
AAHD  MC_NOC  MC_NC  CARS  AHD minus MC_NOC  AHD minus MC_NC  AHD minus CARS  MC_NOC minus MC_NC  

Max  2229.480  2229.634  2229.806  2229.609  0.400  0.402  0.368  0.029 
Min  0.400  0.511  0.509  0.381  \(\)0.911  \(\)0.896  \(\)0.464  \(\)0.173 
Mean  214.464  214.598  214.592  214.493  \(\)0.134  \(\)0.128  \(\)0.029  0.006 
STD  ±251.783  ±251.768  ±251.774  ±251.764  ±0.118  ±0.114  ±0.161  ±0.010 
2.3.2 Blockaveraged ellipsoidal gravity anomalies
The above procedure uses the existing outputs of the isGrafLab software to compute ellipsoidal gravity anomalies via Eq. (6), thus avoiding having to recode isGrafLab. However, they are point values and not blockaverage values required for the RCR technique (cf. Hirt and Claessens 2011). In contrast to gravity anomalies in spherical approximation, blockaveraged values of ellipsoidal gravity anomalies cannot be computed through Paul (1978) recurrence relations for integrals of associated Legendre functions (Hirt and Claessens 2011). Therefore, we computed a \(1^{\prime \prime }\times 1^{\prime \prime }\) grid of point ellipsoidal gravity anomalies and then blockaveraged these onto a \(1^{\prime }\times 1^{\prime }\) grid (Fig. 2a) to be subtracted from the land and altimeterderived gravity anomalies.
2.4 GNSSlevelling data
The GNSSlevelling data play three roles in the computation of AUSGeoid2020: they are used to (1) select the EGM to be used in the RCR technique; (2) determine the combination of parameter choices in the AGQG2017 gravimetric quasigeoid computations, namely degree of FEO kernel modification and integration cap radius; and (3) generate the geometric component to provide the more direct transformation of GNSS heights to AHD heights (cf. Brown et al. 2011). A nearnationwide dataset of GNSSlevelling points was provided by Australian State and Territory geodetic agencies (Sect. 2.6; Fig. 6d).
We used a highresolution shoreline map (Wessel and Smith 1996) to identify GNSSlevelling sites on islands, which technically are separate vertical datums (e.g., Filmer and Featherstone 2012; Amos and Featherstone 2009): Tasmania (71 points), Lord Howe Island (1 point) and nearcoastal islands (185 points). These sites were excluded from the tests in roles 1 and 2 above, but will be included in role 3 because the AHD is defined as local mean sea level on these islands, despite the likely presence of vertical offsets in the sea level connections to the mainland (Lord Howe and Tasmania). We detected around 20 consistently identified outliers from comparisons with multiple quasigeoid models (Tables 3, 4), leaving 7224 GNSSlevelling sites for use in roles 1 and 2.

LSA 1 (MC_NOC) is a minimally constrained (MC) LSA holding only the Johnston origin station in central Australia fixed arbitrarily to its AHD height and using the truncated Rapp (1961) normalorthometric correction (NOC) as used in the AHD. The Australian height systems are reviewed by Featherstone and Kuhn (2006).

LSA 2 (MC_NC) is a minimally constrained LSA holding only the Johnston origin station fixed arbitrarily to its AHD height and using the normal correction (NC). The NC system is theoretically consistent with the quasigeoid. Gravity values at ANLN benchmarks required for the NC were derived from EGM2008, as in Filmer et al. (2010).

Adjustment 3 (CARS) is a constrained LSA, where the 30 mainland tide gauges that were held fixed to MSL in the AHD LSA are now held fixed to the MSL corrected for mean dynamic topography (MDT; aka sea surface topography) from the CSIRO’s atlas of regional seas (CARS2009) climatology (Dunn and Ridgway 2002; Ridgway et al. 2002). The MSL was observed for three years in the 1960s and the MDT for the past 50 years with bias to more recent years (Featherstone and Filmer 2012). This extra constraint to MDTcorrected MSL, which is theoretically aligned with the quasi/geoid, alleviates large (spatial) scale distortions in the ANLN that propagate into the MC LSAs. Normal corrections were applied as per LSA 2.
Descriptive statistics (m) of the formally propagated errors in the GNSSlevelling heights (7,224 points on the Australian mainland)
AHD height error \(\sigma (H)\)  GDA2020 ellipsoidal height error \(\sigma (h)\)  Geometric quasigeoid error \(\sigma ( {\zeta _\mathrm{geom} } )=\sqrt{\sigma ^{2}( h )+\sigma ^{2}( H )}\)  

Max  ±0.124  ±0.164  ±0.171 
Min  ±0.001  ±0.001  ±0.003 
Mean  ±0.038  ±0.009  ±0.040 
Descriptive statistics (m) of the differences among three degree2190 EGMs for four variants of the Australian levelling network (7224 points on the Australian mainland)
AHD  MC_NOC  

EGM2008  EIGEN6C4  GECO  EGM2008  EIGEN6C4  GECO  
Max  0.619  0.626  0.590  0.409  0.469  0.418 
Min  \(\)0.791  \(\)0.757  \(\)0.753  \(\)0.871  \(\)0.838  \(\)0.833 
Mean  \(\)0.061  \(\)0.045  \(\)0.063  \(\)0.195  \(\)0.179  \(\)0.197 
STD  ±0.187  ±0.203  ±0.195  ±0.156  ±0.169  ±0.164 
MC_NC  CARS  

EGM2008  EIGEN6C4  GECO  EGM2008  EIGEN6C4  GECO  
Max  0.406  0.471  0.419  0.326  0.375  0.338 
Min  \(\)0.869  \(\)0.835  \(\)0.830  \(\)0.627  \(\)0.597  \(\)0.622 
Mean  \(\)0.189  \(\)0.173  \(\)0.191  \(\)0.090  \(\)0.074  \(\)0.092 
STD  ±0.158  ±0.171  ±0.165  ±0.093  ±0.093  ±0.094 
2.5 Choice of EGM for the RCR technique
Two degree2190 EGMs have been released since EGM2008: EIGEN6C4 (Förste et al. 2015) and GECO (Gilardoni et al. 2015), both of which include GRACE (Tapley et al. 2004) and GOCE (Drinkwater et al. 2003) data. [EGM2008 only includes GRACE data]. The standard deviations in Table 3 show that there is little difference among these EGMs when compared to the 7224 GNSSlevelling data (all EGM values were synthesised on the topography (cf. Sect. 2.1)), with EGM2008 providing marginally lower STDs. However, bearing in mind the inability of the GNSSlevelling data to discriminate these millimetrescale differences among quasigeoid models (Sect. 2.4), the choice of EGM2008 is somewhat arbitrary.
2.6 Modified Stokes parameter sweeps
The 7224 GPSlevelling sites were also used to choose the optimal combination of Featherstone et al. (1998) FEO kernel modification degree (M) and spherical integration cap radius \((\psi _0 )\) in the 1DFFT modified Stokes integration. To add some robustness to the parameter choice, these parameter sweeps were compared with all four variants of the levelling data (AHD, MC_NOC, MC_MC, CARS), without and with solving for a bias and tilted plane (Fig. 5). The tilted plane was included because the minimally constrained adjustments still exhibit a (smaller than AHD) tilt due to systematic levelling errors, and the bias accounts for a combination of the poorly determined zerodegree term and mean offset of the AHD from the ‘true’ geoid potential, \(\hbox {W}_{0}\).
From Fig. 5, and as was found for AGQG2009 (Featherstone et al. 2011) and AUSGeoid98 (Featherstone et al. 2001), the unmodified spherical Stokes kernel is inappropriate because it does not filter out longwavelength errors from the land and altimeter data (cf. Vaníček and Featherstone 1998). The optimal integration radius is clearly \(\psi _0 =0.5\) degrees from all comparisons in Fig. 5. The degree of modification (“FEOmod” in the Fig. 5 legends) is indistinguishable among degrees ranging from \(M=40\) to \(M=140\), so we chose \(M=40\) as there is no empirical evidence to choose a higher degree of modification. Also, recall that the ability of the GNSSlevelling data to discriminate among quasigeoid solutions is only \({\pm }40\) mm (Sect. 2.4), so there is quite some leeway in these parameter choices.
Descriptive statistics (m) of the differences among gravimetric quasigeoid models for four variants of the Australian levelling network (7224 points on the Australian mainland)
AHD  MC_NOC  

EGM2008  AGQG2009  AGQG2017  EGM2008  AGQG2009  AGQG2017  
Max  0.619  0.660  0.591  0.409  0.507  0.408 
Min  \(\)0.791  \(\)0.774  \(\)0.788  \(\)0.871  \(\)0.854  \(\)0.868 
Mean  \(\)0.061  \(\)0.044  \(\)0.070  \(\)0.195  \(\)0.178  \(\)0.204 
STD  ±0.187  ±0.186  ±0.186  ±0.156  ±0.154  ±0.156 
MC_NC  CARS  

EGM2008  AGQG2009  AGQG2017  EGM2008  AGQG2009  AGQG2017  
Max  0.406  0.483  0.399  0.326  0.372  0.310 
Min  \(\)0.869  \(\)0.851  \(\)0.865  \(\)0.627  \(\)0.612  \(\)0.629 
Mean  \(\)0.189  \(\)0.172  \(\)0.198  \(\)0.090  \(\)0.072  \(\)0.099 
STD  ±0.158  ±0.155  ±0.157  ±0.093  ±0.094  ±0.094 
We also attempted to determine the optimal parameter combination using a set of 1080 historical vertical deflections observed in the 1950s and 1960s. The precision of these observations is unknown, but they are probably only \({\pm }1^{\prime \prime }\). Therefore, they were unable to neither confirm nor refute these parameter choices made by comparisons with the GNSSlevelling.
Figure 6a shows the residual terraincorrected freeair gravity anomalies after the removal of the blockaveraged EGM2008 ellipsoidal gravity anomalies synthesised on the topography (Sect. 2.3). Figure 6b shows residual quasigeoid undulations from the \(M=40\) FEOmodified Stokes integration over \(\psi _0 =0.5\) degrees. Figure 6c shows the AGQG2017 gravimetric quasigeoid model after restoration of the EGM2008 quasigeoid heights synthesised on the topography. Figure 6d shows the quasigeoid height differences at the 7224 GNSSAHD stations. The north–south tilt and regional distortions in the AHD necessitate the geometric component (cf. Brown et al. 2011) for the transformation of GNSS ellipsoidal heights to the AHD, until the time that the AHD is redefined.
Table 4 shows that, when assessing the gravimetric quasigeoid models against the Australian GNSSlevelling data, there is only a very marginal improvement upon EGM2008. The first qualifier is the uncertainties in the GNSSlevelling data, so a millimetrescale change among gravimetric models is insignificant when compared to the \({\pm }40\) mm mean STD of the GNSSAHD data (Table 2). The second qualifier is the spatial distribution of the GNSSlevelling data, with relatively few stations at high elevations (only 141 of 7224 greater than 1 km and 806 greater than 500 m) and sparse GNSSlevelling coverage in central and northern Australia, where most new land gravity data have been collected (cf. Figs. 6d, 1b). We considered thinning the GNSSlevelling data, but this became subjective because the standard errors from the LSA do not describe the \({\sim }0.5\) m systematic errors in the AHD. Nevertheless, AGQG2017 has used newer and higherresolution (notably the DEM) data sources than its predecessors.
The differences in Fig. 7b are due to a combination of the synthesis of EGM2008 (Sect. 2.3), and the use of the \(1^{\prime \prime }\times 1^{\prime \prime }\) DEMH model for both the reconstruction technique (Featherstone et al. 2001) and computation of terrain corrections (McCubbine et al. 2017). In addition, Claessens et al. (2009) identified the possibility of erroneous Australian gravity data in this region, which have since been corrected in the GA gravity database (Fig. 8). The poorer performance of AUSGeoid09 in this region (cf. Rexer et al. 2013; Sussanna et al. 2016) is therefore attributed to a superposition of all the above effects, but the erroneous land gravity data appears to be the major contributor.
3 Quasigeoid error propagation
3.1 Preliminaries
No previous Australian geoid or quasigeoid model has been accompanied by an error grid, instead relying on an overall error estimate taken from the mean of residuals to GNSSlevelling data (Sect. 1). Notwithstanding least squares collocation (LSC), few previous studies have propagated uncertainties through Stokes’s integral (e.g., Sideris and She 1995; Pavlis and Saleh 2005; Huang et al. 2007). LSC was not used for the Australian gravimetric quasigeoid computations simply because of the sheer size and extent of the Australian datasets (Sect. 2.1).
Sideris and She (1995) provide error propagation formulas for the 1DFFT (Haagmans et al. 1993) when using the RCR technique over a region. Huang et al. (2007) provide a spectral form when using the RCR technique and a modified integration kernel over a spherical integration cap. Common to both these approaches is the need to use a satelliteonly EGM so that errors in the terrestrial gravity data are independent and thus combined without the need to consider correlations. However, when using a combined EGM such as EGM2008, this assumption breaks down. Therefore, we have devised an alternative approach to error propagation (Sect. 3.3).
3.2 Terrestrial gravity error grid
3.3 Quasigeoid error propagation
This novel approach to quasigeoid error propagation allows us not only to avoid the need to use a satelliteonly EGM to avoid correlations, but also to avoid the need to use the error degree variances of the geopotential coefficients that only give a single global estimate of the EGM errors. We essentially implement a RCR on the errors by replacing the gravity anomaly errors used in EGM2008 by our own regional estimates on a denser grid, thus reducing the omission and the commission errors in EGM2008.
3.4 External validations of the error grid
Our error propagation suggests quasigeoid errors of 50–60 mm across most of mainland Australia, increasing to \(\sim \)100 mm in regions of steep horizontal gravity gradients or in the mountains (Fig. 12b). To gauge whether our error estimates are realistic, we compare our values to those reported for Canada by Huang and Véronneau (2013). The maximum AHD elevation is \(\sim \)2230 m and the formally propagated quasigeoid error is \(\sim \)100 mm in the vicinity of this mountainous region (southeastern Australia). The maximum elevation in Canada is 5959 m and Huang and Véronneau (2013, p. 785) report geoid error values of 300 mm in the Canadian Rocky mountains. Making the gross assumption that the errors scale linearly with height, these error estimates appear to be reasonably consistent (i.e., \(5959 \times 100/2230=267\) mm).
Another check comes from a comparison of the broadscale quasigeoid error estimates deduced from variance component estimation (Filmer et al. 2014). Their error estimate was ±79 mm for ellipsoidal height minus AGQG2009 using 277 GNSS points. The average internal error estimate of the GNSS ellipsoidal heights from the Bernese processing for the dataset used for Filmer et al. (2014) was ±26 mm (scaled by a factor of 10 to provide a more realistic error estimate, e.g., Rothacher 2002), so the quasigeoid component of this error is \({\sim }{\pm }\)75 mm based on variance propagation.
The final check comes from a comparison of our quasigeoid error estimates with the formal errors in the GNSSlevelling data (Sect. 2.4) on a pointbypoint basis. We used bicubic interpolation of the error grid to the locations of the GNSSlevelling stations and compared them to the formal error at each of the 7224 points. The mean absolute difference is \({\sim }{\pm }\)25 mm, with the gravimetrically propagated errors being larger than the formal errors in the GNSSlevelling data. The apparently better GNSSlevelling errors are partly due to the disproportionately large number of GNSS points in centraleastern and southwestern Australia (Fig. 6d), where the formal errors are smaller.
3.5 Critique of the EGM2008 error grid around Australia
Figure 10a shows a band of relatively low (<1 mGal) marine gravity anomaly error estimates around the Australian coast for EGM2008, which is unique to Australia. When these are propagated through Eq. (25) for a 0.5degree cap and \(M=40\) FEO modification, the corresponding quasigeoid error estimates are less than ±10 mm (Fig. 11a). This is contrary to the knowledge that altimeterderived gravity anomalies are of lower accuracy in the coastal zone (e.g., Deng et al. 2002; Volkov et al. 2007); also see Fig. 11b.
Therefore, we have propagated the grav_error.img.23.1 grid of altimeterderived gravity anomaly errors so as to increase the AGQG2017 error estimates (\({\sim }{\pm }\)50 mm) above those of EGM2008 in the coastal zone, expecting this to be more realistic. In other areas, the EGM2008 errors are reduced because the higherresolution grid reduces the omission error (and part of the commission error) when using the formally propagated terrestrial gravity data errors (cf. Figs. 10b, 12b).
4 Concluding remarks
We have described the computation of the gravimetric quasigeoid AGQG2017, as well as the computation of locationdependent error estimates that have been propagated from the combined uncertainties in EGM2008, land and altimeterderived gravity anomalies, and terrain corrections. AGQG2017 has included an additional \(\sim \)280,000 land gravity observations over AGQG2009 (1,764,351 in total), retracked satellite altimeterderived marine gravity anomalies that include Jason1 and CryoSat2 data, the \(1^{\prime \prime }\times 1^{\prime \prime }\) (\(\sim \)30 m) resolution DEMH digital elevation model for reconstruction of mean freeair gravity anomalies on the topography, computation of terrain corrections that include formally propagated error estimates, and improved numerical integration routines.
Computational refinements were made to the synthesis of EGM gravity field functionals on the topography as is demanded by Molodensky theory (Sect. 2.3). Three separate outputs of the isGrafLab software were combined to generate blockaveraged ellipsoidal gravity anomalies, also on the topography. Four least squares readjustments of the Australian levelling data (Sect. 2.4) were used to more robustly select the EGM2008 model for the RCR technique (Sect. 2.5) and parameter sweeps for the degree of kernel modification and integration cap radius (Sect. 2.6). These indicated that a CARS2009 MDTconstrained ANLN is the best currently available levelling adjustment for testing quasigeoid models in Australia. However, the mean error of the GNSSlevelling data is \(\sim \)40 mm, making it a less useful tool to discriminate among gravimetric quasigeoid models. Historical deflections of the vertical were not able to assist during the parameter sweeps because of their low precision.
We present a new and alternative technique for the propagation of locationdependent errors without the need to rely on global estimates of the error degree variance of a satelliteonly EGM. Instead, the error grids of quasigeoid heights and gravity anomalies were taken from EGM2008 and used in a RCRtype procedure, where errors in the land gravity anomalies, altimeter gravity anomalies and terrain corrections were propagated through the same modified Stokes integral used to compute AGQG2017 (Sect. 3). The gravimetric quasigeoid errors (one sigma) are 50–60 mm across most of the Australian landmass, increasing to \(\sim \)100 mm in regions of steep horizontal gravity gradients or the mountains (Fig. 12b). Our error estimates were validated externally using three separate approaches, indicating them to be realistic.
Finally, we question the band of low EGM2008 gravity anomaly error estimates around the Australian coast, instead increasing the AGQG2017 quasigeoid error estimate based on propagation of errors from the grav_error.img.23.1 grid. On land away from the coasts, the EGM2008 error is reduced through the use of Australian land gravity and DEM data on a highresolution \(1^{\prime \prime }\times 1^{\prime \prime }\) grid.
Notes
Acknowledgements
The current work has been supported financially by the Cooperative Research Centre for Spatial Information, whose activities are funded by the Business Cooperative Research Centres Programme, and by Geoscience Australia. Previous works (e.g., software and algorithm development) that have been used here have been supported financially by the Australian Research Council (ARC) over the past two decades through Grant Numbers A39938040, A00001127, DP0211827 and DP0663020. We thank Scripps Institution of Oceanography (University of California), the US National Oceanographic and Atmospheric Administration and the US National GeospatialIntelligence Agency for permission to use the marine gravity anomalies from Sandwell et al. (2014). All maps and charts in this paper were produced using GMT (Wessel et al. 2013) in the Lambert conformal conic projection with two standard parallels or the Mercator projection (Fig. 8; Appendix A). Nicholas Brown publishes this paper with the permission of the CEO, Geoscience Australia. Finally, we thank the three anonymous reviewers for their constructive and complimentary critiques on the firstsubmitted version of this manuscript
References
 Altamimi Z, Rebischung P, Métivier L, Xavier C (2016) ITRF2014: a new release of the international terrestrial reference frame modeling nonlinear station motions. J Geophys Res Solid Earth 121(8):6109–6131. doi: 10.1002/2016JB013098 CrossRefGoogle Scholar
 Amos MJ, Featherstone WE (2009) Unification of New Zealand’s local vertical datums: iterative gravimetric quasigeoid computations. J Geod 83(1):57–68. doi: 10.1007/s001900080232y CrossRefGoogle Scholar
 Andersen OB, Knudsen P, Berry PAM (2010) The DNSC08GRA global marine gravity field from double retracked satellite altimetry. J Geod 84(3):191–199. doi: 10.1007/s0019000903559 CrossRefGoogle Scholar
 Brown NJ, Featherstone WE, Hu G, Johnston GM (2011) AUSGeoid09: a more direct and more accurate model for converting ellipsoidal heights to AHD heights. J Spat Sci 56(1):27–37. doi: 10.1080/14498596.2011.580498 CrossRefGoogle Scholar
 Bucha B, Janák J (2013) A MATLABbased graphical user interface program for computing functionals of the geopotential up to ultrahigh degrees and orders. Comput Geosci 56:186–196. doi: 10.1016/j.cageo.2013.03.012 CrossRefGoogle Scholar
 Bucha B, Janák J (2014) A MATLABbased graphical user interface program for computing functionals of the geopotential up to ultrahigh degrees and orders: Efficient computation at irregular surfaces. Comput Geosci 66:219–227. doi: 10.1016/j.cageo.2014.02.005 CrossRefGoogle Scholar
 Claessens SJ (2006) Solutions to ellipsoidal boundary value problems for gravity field modelling. PhD thesis, Curtin University, Perth, AustraliaGoogle Scholar
 Claessens SJ (2012) Evaluation of gravity and altimetry data in Australian coastal regions. IAG Symp 136:435–442. doi: 10.1007/9783642203381_52 Google Scholar
 Claessens SJ, Featherstone WE, Anjasmara IM, Filmer MS (2009) Is Australian data really validating EGM2008, or is EGM2008 just in/validating Australian data? Newton’s Bull 4:207–251Google Scholar
 Deng XL, Featherstone WE, Hwang C (2002) Estimation of contamination of ERS2 and POSEIDON satellite radar altimetry close to the coasts of Australia. Mar Geod 25(4):249–271. doi: 10.1080/01490410290051572 CrossRefGoogle Scholar
 Drinkwater MR, Floberghagen R, Haagmans R, Muzi D, Popescu A (2003) GOCE: ESA’s first earth explorer core mission, in earth gravity field from space—from sensors to earth sciences, Springer, pp 419–432. doi: 10.1007/9789401713337_36
 Dunn J, Ridgway KR (2002) Mapping ocean properties in regions of complex topography. Deep Sea Res I 49(3):591–604. doi: 10.1016/S09670637(01)000693 CrossRefGoogle Scholar
 Farr TG, Rosen PA, Caro E, Crippen R, Duren R, Hensley S, Kobrick M, Paller M, Rodriguez E, Roth L, Seal D, Shaffer S, Shimada J, Werner M, Oskin M, Burbank D, Alsdorf D (2007) The shuttle radar topography mission. Rev Geophys 45(2):RG2004. doi: 10.1029/2005RG000183 CrossRefGoogle Scholar
 Featherstone WE (1998) Do we need a gravimetric geoid or a model of the base of the Australian Height Datum to transform GPS heights? Aust Surv 43(4):273–280CrossRefGoogle Scholar
 Featherstone WE (2008) GNSSbased heighting in Australia: current, emerging and future issues. J Spat Sci 53(2):115–133. doi: 10.1080/14498596.2008.9635153 CrossRefGoogle Scholar
 Featherstone WE (2009) Only use shiptrack gravity data with caution: a casestudy around Australia. Aust J Earth Sci 56(2):191–195. doi: 10.1080/08120090802547025 CrossRefGoogle Scholar
 Featherstone WE, Filmer MS (2012) The northsouth tilt in the Australian Height Datum is explained by the ocean’s mean dynamic topography. J Geophys Res Oceans 117(C8):C08035. doi: 10.1029/2012JC007974 CrossRefGoogle Scholar
 Featherstone WE, Kirby JF (2000) The reduction of aliasing in gravity anomalies and geoid heights using digital terrain data. Geophys J Int 141(1):204–212. doi: 10.1046/j.1365246X.2000.00082.x CrossRefGoogle Scholar
 Featherstone WE, Kuhn M (2006) Height systems and vertical datums: a review in the Australian context. J Spat Sci 51(1):21–42. doi: 10.1080/14498596.2006.9635062 CrossRefGoogle Scholar
 Featherstone WE, Sproule DM (2006) Fitting AUSGeoid98 to the Australian Height Datum using GPS data and least squares collocation: application of a crossvalidation technique. Surv Rev 38(301):573–582. doi: 10.1179/003962606780732065 CrossRefGoogle Scholar
 Featherstone WE, Evans JD, Olliver JG (1998) A Meisslmodified Vaníček and Kleusberg kernel to reduce the truncation error in gravimetric geoid computations. J Geod 72(3):154–160. doi: 10.1007/s001900050157 CrossRefGoogle Scholar
 Featherstone WE, Kirby JF, Kearsley AHW, Gilliland JR, Johnston GM, Steed J, Forsberg R, Sideris MG (2001) The AUSGeoid98 geoid model of Australia: data treatment, computations and comparisons with GPSlevelling data. J Geod 75(5–6):313–330. doi: 10.1007/s001900100177 CrossRefGoogle Scholar
 Featherstone WE, Kirby JF, Hirt C, Filmer MS, Claessens SJ, Brown NJ, Hu G, Johnston GM (2011) The AUSGeoid09 model of the Australian Height Datum. J Geod 85(3):133–150. doi: 10.1007/s0019001004222 CrossRefGoogle Scholar
 Filmer MS, Featherstone WE (2012) A reevaluation of the offset in the Australian Height Datum between mainland Australia and Tasmania. Mar Geod 35(1):1–13. doi: 10.1080/01490419.2011.634961 CrossRefGoogle Scholar
 Filmer MS, Featherstone WE, Kuhn M (2010) The effect of EGM2008based normal, normalorthometric and Helmert orthometric height systems on the Australian levelling network. J Geod 84(8):501–513. doi: 10.1007/s0019001003880 CrossRefGoogle Scholar
 Filmer MS, Featherstone WE, Claessens SJ (2014) Variance component estimation uncertainty for unbalanced data: application to a continentwide vertical datum. J Geod 88(11):1081–1093. doi: 10.1007/s0019001407446 CrossRefGoogle Scholar
 Förste C, Bruinsma SL, Abrikosov O, Lemoine JM, Marty JC, Flechtner F, Balmino G, Barthelmes F, Biancale R (2015) EIGEN6C4 The latest combined global gravity field model including GOCE data up to degree and order 2190 of GFZ Potsdam and GRGS Toulouse. doi: 10.5880/icgem.2015.1
 Gallant JC, Dowling TI, Read AM, Wilson N, Tickle P, Inskeep C (2011) 1 second SRTM derived digital elevation models user guide, geoscience Australia. www.ga.gov.au/topographicmapping/digitalelevationdata.html
 Garcia E, Sandwell DT, Smith WHF (2014) Retracking CryoSat2, Envisat, and Jason1 radar altimetry waveforms for improved gravity field recovery. Geophys J Int 196(3):1402–1422. doi: 10.1093/gji/ggt469 CrossRefGoogle Scholar
 Gilardoni M, Reguzzoni M, Sampietro D (2015) GECO: a global gravity model by locally combining GOCE data and EGM2008. Stud Geophys Geod 60(2):228–247. doi: 10.1007/s1120001511144 CrossRefGoogle Scholar
 Grafarend EW, Ardalan A, Sideris MG (1999) The spheroidal fixedfree twoboundaryvalue problem for geoid determination (the spheroidal Bruns’ transform). J Geod 73(10):513–533. doi: 10.1007/s001900050263 CrossRefGoogle Scholar
 Haagmans RHN, van Gelderen M (1991) Error variancescovariances of GEMTI: their characteristics and implications in geoid computation. J Geophys Res Solid Earth 96(12):20011–20022. doi: 10.1029/91JB01971 CrossRefGoogle Scholar
 Haagmans RRN, de Min E, van Gelderen M (1993) Fast evaluation of convolution integrals on the sphere using 1D FFT, and a comparison with existing methods for Stokes’ integral. Manuscr Geod 18(3):227–241Google Scholar
 Hipkin RG (2004) Ellipsoidal geoid computation. J Geod 78(3):167–179. doi: 10.1007/s001900040389y CrossRefGoogle Scholar
 Hirt C (2012) Efficient and accurate highdegree spherical harmonic synthesis of gravity field functionals at the Earth’s surface using the gradient approach. J Geod 86(9):729–744. doi: 10.1007/s001900120550y CrossRefGoogle Scholar
 Hirt C, Claessens SJ (2011) Ellipsoidal area mean gravity anomalies—precise computation of gravity anomaly reference fields for remove–compute–restore geoid computation. Stud Geophys Geod 55(4):598–607. doi: 10.1007/s1120001000702 CrossRefGoogle Scholar
 Hirt C, Featherstone WE, Claessens SJ (2011) On the accurate numerical evaluation of geodetic convolution integrals. J Geod 85(8):519–538. doi: 10.1007/s0019001104515 CrossRefGoogle Scholar
 Holmes SA, Featherstone WE (2002) A unified approach to the Clenshaw summation and the recursive computation of veryhigh degree and order normalised associated Legendre functions. J Geod 76(5):279–299. doi: 10.1007/s0019000202162 CrossRefGoogle Scholar
 Huang J, Véronneau M (2013) Canadian gravimetric geoid model 2010. J Geod 87(8):771–790. doi: 10.1007/s0019001306450 CrossRefGoogle Scholar
 Huang J, Fotopoulos G, Cheng MK, Véronneau M, Sideris MG (2007) On the estimation of the regional geoid error in Canada. IAG Symp 130:272–279. doi: 10.1007/9783540493501_41 Google Scholar
 Jekeli C (1981) The downward continuation to the Earth’s surface of truncated spherical and ellipsoidal harmonic series of the gravity and height anomalies, Report 323, Ohio State UniversityGoogle Scholar
 McCubbine JC, Featherstone WE, Kirby JF (2017) Fast Fourierbased error propagation for the gravimetric terrain correction. Geophysics. doi: 10.1190/GEO20160627.1 Google Scholar
 Milbert DG (1995) Improvement of a high resolution geoid height model in the United States by GPS height on NAVD 88 benchmarks. Int Geoid Serv Bull 4:13–36Google Scholar
 Moritz H (1980) Geodetic reference system 1980. Bull Géodésique 54(3):395–405. doi: 10.1007/BF02521480 CrossRefGoogle Scholar
 Paul MK (1978) Recurrence relations for integrals of associated Legendre functions. Bull Géodésique 52(3):177–190CrossRefGoogle Scholar
 Pavlis NK, Saleh J (2005) Error propagation with geographic specificity for very high degree geopotential models. IAG Symp 129:149–154. doi: 10.1007/3540269320_26 Google Scholar
 Pavlis NK, Holmes SA, Kenyon SC, Factor JK (2012) The development and evaluation of the Earth Gravitational Model 2008 (EGM2008). J Geophys Res Solid Earth 117(B4):B04406. doi: 10.1029/2011JB008916 CrossRefGoogle Scholar
 Pavlis NK, Holmes SA, Kenyon SC, Factor JK (2013) Correction to The development and evaluation of the Earth Gravitational Model 2008 (EGM2008). J Geophys Res Solid Earth 118(5):2633. doi: 10.1029/jgrb.50167 CrossRefGoogle Scholar
 Rapp RH (1961) The orthometric height, MS Dissertation, The Ohio State University, Columbus, USAGoogle Scholar
 Rexer M, Hirt C, Pail R, Claessens SJ (2013) Evaluation of the third and fourthgeneration GOCE Earth gravity field models with Australian terrestrial gravity data in spherical harmonics. J Geod 88(4):319–333. doi: 10.1007/s001900130680x
 Ridgway KR, Dunn JR, Wilkin JL (2002) Ocean interpolation by fourdimensional weighted least squaresapplication to the waters around Australasia. J Atmos Oceanogr Technol 19(9):1357–1375. doi: 10.1175/15200426(2002) 019<1357:OIBFDW>2.0.CO;2 CrossRefGoogle Scholar
 Rodríguez E, Morris CS, Belz JE (2006) A global assessment of the SRTM performance. Photogramm Eng Remote Sensing 72(3):249–260. doi: 10.14358/PERS.72.3.249 CrossRefGoogle Scholar
 Roelse A, Granger HW, Graham JW (1971 2nd, ed. 1975) he adjustment of the Australian levelling survey 1970–1971, Technical Report 12. Division of National Mapping, Canberra, AustraliaGoogle Scholar
 Rothacher M (2002) Estimation of station heights with GPS. IAG Symp 124:81–90. doi: 10.1007/9783662046838_17 Google Scholar
 Sandwell DT, Smith WHF (2009) Global marine gravity from retracked Geosat and ERS1 altimetry: ridge segmentation versus spreading rate. J Geophys Res Solid Earth 114(B1):B01411. doi: 10.1029/2008JB006008 CrossRefGoogle Scholar
 Sandwell DT, Garcia E, Soofi K, Wessel P, Smith WHF (2013) Towards 1 mGal global marine gravity from CryoSat2, Envisat, and Jason1. Lead Edge 32(8):892–899. doi: 10.1190/tle32080892.1 CrossRefGoogle Scholar
 Sandwell DT, Müller RD, Smith WHF, Garcia E, Francis R (2014) New global marine gravity model from CryoSat2 and Jason1 reveals buried tectonic structure. Science 346(6205):65–67. doi: 10.1126/science.1258213 CrossRefGoogle Scholar
 Sansò F, Rummel R (1997) Geodetic boundary value problems in view of the one centimeter geoid, Lecture Notes in Earth Sciences 65, vol 65. Springer, HeidelbergCrossRefGoogle Scholar
 Sideris MG, She BB (1995) A new highresolution geoid for Canada and part of the US by the 1D FFT method. Bull Géodésique 69(1):92–108. doi: 10.1007/BF00819555 CrossRefGoogle Scholar
 Smith DA (1998) There is no such thing as The EGM96 geoid: Subtle points on the use of a global geopotential model. Int Geoid Serv Bulletin 8:17–28Google Scholar
 Smith DA, Roman DR (2001) GEOID99 and G99SSS: 1arcminute geoid models for the United States. J Geod 75(9–10):469–490. doi: 10.1007/s001900100200 CrossRefGoogle Scholar
 Smith WHF, Wessel P (1990) Gridding with a continuous curvature surface in tension. Geophysics 55(3):293–305. doi: 10.1190/1.1442837 CrossRefGoogle Scholar
 Sussanna V, Janssen V, Gibbings P (2016) Relative performance of AUSGeoid09 in mountainous terrain. J Geod Sci 6(1):34–42. doi: 10.1515/jogs20160002 Google Scholar
 Tapley BD, Bettadpur S, Watkins M, Reigber C (2004) The gravity recovery and climate experiment: mission overview and early results. Geophys Res Lett 31(9):L09607. doi: 10.1029/2004GL019920 CrossRefGoogle Scholar
 Tscherning CC, Arabelos D, Strykowski G (2001) The 1cm geoid after GOCE. IAG Symp 123:267–270. doi: 10.1007/9783662048276_45 Google Scholar
 Vaníček P, Featherstone WE (1998) Performance of three types of Stokes’s kernel in the combined solution for the geoid. J Geod 72(12):684–697. doi: 10.1007/s001900050209 CrossRefGoogle Scholar
 Volkov DL, Larnicol G, Dorandeu J (2007) Improving the quality of satellite altimetry data over continental shelves. J Geophys Res Oceans 112:C06020. doi: 10.1029/2006JC003765 CrossRefGoogle Scholar
 Wessel P, Smith WHF (1996) A global, selfconsistent, hierarchical, highresolution shoreline database. J Geophys Res Solid Earth 101(B4):8741–8743. doi: 10.1029/96JB00104 CrossRefGoogle Scholar
 Wessel P, Smith WHF, Scharroo R, Luis JF, Wobbe F (2013) Generic mapping tools: improved version released. EOS Trans AGU 94(45):409–410. doi: 10.1002/2013EO450001 CrossRefGoogle Scholar
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