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Confirming regional 1 cm differential geoid accuracy from airborne gravimetry: the Geoid Slope Validation Survey of 2011

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Abstract

A terrestrial survey, called the Geoid Slope Validation Survey of 2011 (GSVS11), encompassing leveling, GPS, astrogeodetic deflections of the vertical (DOV) and surface gravity was performed in the United States. The general purpose of that survey was to evaluate the current accuracy of gravimetric geoid models, and also to determine the impact of introducing new airborne gravity data from the ‘Gravity for the Redefinition of the American Vertical Datum’ (GRAV-D) project. More specifically, the GSVS11 survey was performed to determine whether or not the GRAV-D airborne gravimetry, flown at 11 km altitude, can reduce differential geoid error to below 1 cm in a low, flat gravimetrically uncomplicated region. GSVS11 comprises a 325 km traverse from Austin to Rockport in Southern Texas, and includes 218 GPS stations (\(\sigma _{\Delta h }= 0.4\) cm over any distance from 0.4 to 325 km) co-located with first-order spirit leveled orthometric heights (\(\sigma _{\Delta H }= 1.3\) cm end-to-end), including new surface gravimetry, and 216 astronomically determined vertical deflections \((\sigma _{\mathrm{DOV}}= 0.1^{\prime \prime })\). The terrestrial survey data were compared in various ways to specific geoid models, including analysis of RMS residuals between all pairs of points on the line, direct comparison of DOVs to geoid slopes, and a harmonic analysis of the differences between the terrestrial data and various geoid models. These comparisons of the terrestrial survey data with specific geoid models showed conclusively that, in this type of region (low, flat) the geoid models computed using existing terrestrial gravity, combined with digital elevation models (DEMs) and GRACE and GOCE data, differential geoid accuracy of 1 to 3 cm (1 \(\sigma )\) over distances from 0.4 to 325 km were currently being achieved. However, the addition of a contemporaneous airborne gravity data set, flown at 11 km altitude, brought the estimated differential geoid accuracy down to 1 cm over nearly all distances from 0.4 to 325 km.

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Notes

  1. The authors here distinguish between “EGM2008”, which is a global geopotential model of Earth’s external gravitational potential [also called an “Earth Gravitational Model” (EGM)] and “The EGM2008 geoid model”, which is one of an infinite number of geoid models which can be created from EGM2008 by choosing numerous variables, such as a reference \({W}_{0}\) value, a reference ellipsoid, a permanent tide system and the method of computing values “in the masses”. In this paper, “the EGM2008 geoid model” means the official model chosen by the EGM2008 team and distributed with EGM2008. For further information on this subtle distinction see Smith (1998).

  2. A similar approach can be used to reduce the spherical approximation error from Eqs (9) and (10). Here, progressively improved estimates of \(\overline{\delta C}_{\mathrm{nm}}^{\mathrm{s}}\) can be used to generate values \((N_{\mathrm{s}}=2{{,}}190)\) for the correction term \(\frac{\partial T}{\partial r}-\frac{\partial T}{\partial h}\), which can be added to the \(\delta g_{\mathrm{resid}}\) data to move this closer to \(\frac{\partial T}{\partial r}\).

  3. Future attempts at this type of analysis will seek to iterate this process, whereby initial estimates of the geopotential spectra could be used to generate improved estimates of the ‘\(\zeta _{\mathrm{e}}\)-to-\(N\)’ values which are specific to each geoid model.

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Acknowledgments

Above all others, the authors wish to thank the 96 separate individuals from within the National Geodetic Survey who participated to some extent in this survey, including 46 who actually performed the field work. The authors also wish to thank the Texas governor’s office, the Texas State Troopers, the Texas Department of Transportation, the Texas Maritime Museum, the Texas General Lands Office, the Texas Natural Resources Information System, the University of Texas, Bowie High School and all other representatives of the Lone Star State for welcoming us, and allowing us to work safely and unhindered on the grounds of the capital building, along the highways of Texas as well inside the Stephen F. Austin Building, the Texas Maritime Museum, Pickle Research Campus and Bowie High School.

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Appendices

Appendix A: DOV error budget estimation and field procedures

1.1 Field procedures

The DIADEM system was mounted on a trailer and pulled by an all-terrain vehicle (ATV) to the survey station at which the DOVs were to be determined. The instrument was then leveled, and data acquisition initiated. This consisted of recording images of the sky in the zenith direction, continuous monitoring of 6 high-precision tiltmeters and precisely timing the image acquisition by means of a dedicated low-cost L1-GPS receiver. Each such measurement was carried out in two opposite orientations of the system and then combined into one DOV “initial estimate” \((\eta _{i},\xi _{i})\). For the current campaign, approximately 50 such “initial estimates” were observed at each station. Finally, for the data post-processing, inertial accelerations were carefully filtered from the tiltmeter data so as to isolate the real signal of the tilt of the optical axis with respect to the gravity acceleration vector (plumb line). All tilt values were then computed at the epochs of the image acquisition and then combined with celestial calibration parameters which had been determined every other night. Astrometric processing utilized the Tycho-2 star catalogue and Earth rotation parameters from the International Earth Rotation Service (IERS). Finally, all DOV “initial estimates” \((\eta _{i},\xi _{i})\) for a single station were combined in a robust adjustment resulting in adjusted final surface of DOVs \((\eta ,\xi )\) and an empirical standard deviation \((\sigma _{\eta },\sigma _{\xi })\) for the error estimate.

The theoretical error budget of a single DOV “initial estimate” was \(\sigma _{\eta i},\sigma _{\xi i} = 0.25^{\prime \prime }\) (random) + \(0.1^{\prime \prime }\) (systematic) (or 1.2 + 0.5 mm/km). The empirically determined error budget for the final DOV (adjusted from the approximately 50 successively observed “initial estimates”) was \(\sigma _{\eta },\sigma _{\xi } = 0.05^{\prime \prime }\) (random) + \(0.05^{\prime \prime }\) (systematic) (or 0.2 + 0.2 mm/km). As a final step, surface DOVs were corrected for curvature of the plumb line (cf. Section 5–6 of Heiskanen and Moritz 1967), to arrive at DOVs on the geoid, equivalent to ‘instantaneous’ (or “zero distance”) geoid slopes, in the N/S and E/W direction.

1.2 Error budget

In theory, the total errors for single DOV determination can be subdivided as follows. First, ‘astrometric’ errors (variation of the apparent places of stars caused by the propagation properties of the atmosphere) are the major contributors to the random error budget, but also have a systematic component. Second, the ‘scintillation’ part of the error can be seen as Gaussian random noise, and is so mitigated by observing up to 50 stars simultaneously and then computing the mean from the 50 individual DOV “initial estimates”. Third, the ‘systematic’ part of the error results from unmodeled atmospheric refraction, and is most problematic since it can only be assessed and reduced by making several DOV determinations with sufficient time spacing to reduce the time correlation. This last issue is currently being researched in astrogeodesy (e.g., Hirt 2006). For the GSVS11 campaign, nine stations homogeneously spread along the profile were re-observed a second time to assess the order of magnitude of random and systematic effects at spatial dimensions up to 300 km and time differences of up to 2 weeks.

1.3 Computation of empirical quality indicators

For each of the 216 DOV stations observed, robust means and empirical standard deviations \((\sigma _{\eta },\sigma _{\xi })\) were computed using all of the approximately 50 single DOV “initial estimates” \((\eta _{i},\xi _{i})\) obtained for that station. From Table 5, which summarizes the values for these standard deviations \((\sigma _{\eta },\sigma _{\xi })\) across 216 stations, we note a very high degree of internal precision (\(\sim \)0.02 arcsec) and reliability. This can be evaluated with the large number of stars which were identified and the quotient (resulting from the robust computation) between the number of DOV ‘initial estimates’ determined versus the number which were not rejected by the adjustment process. This quotient is better than 99 % (Table 6).

Table 5 Random and systematic error sources in the determination of a single DOV
Table 6 Empirical standard deviations and some statistical indicators representing the internal homogeneity of DOV determination at one location

A second very important quality check can be performed using the nine re-observed DOVs along the GSVS11 profile. The time differences between successive determinations at a single repeat station range from 9.7 to 35 days. The data and the results of the comparison are summarized in Table 7. Here the variables with the index 1 represent the first determination (proceeding south). Those with index 2 represent the second determination (proceeding north). The variables without index represent the arithmetic mean of the same variable with an index. The results show that the standard deviation of a determination of a DOV at one location can be estimated as 0.04 and 0.06 arcsec for \(\eta \) and \(\xi \) component, respectively. Moreover, these data show no significant systematic bias between observations taken at two different days.

Table 7 Comparisons of the DOVs determined on two different days

Appendix B: Collection, processing and accuracy of airborne gravity

The GRAV-D project collected airborne gravity data over southeastern Texas in 2009 (TX09). Flights were done with the National Oceanic and Atmospheric Administration’s (NOAA) Cessna Citation II, a small jet, at a nominal altitude of 35,000 feet (10,668 meters) and an actual mean height of \(\sim \)11,000m. This flight altitude is significantly higher than that used for other airborne gravity data collected for inclusion in geoid computations, since only a jet aircraft was available as a platform at the start of GRAV-D. Ground speed of the aircraft was nominally 280 knots. North-South oriented data lines are 600 km long with 10 km spacing and East-West oriented cross lines are 400 km long with 40 km spacing.

Three geodetic GPS receivers on the aircraft (two NovAtel DL4 Plus and one Trimble) and three on the ground (all Ashtech Z-extremes) provided 1 Hz data for differential kinematic GPS positioning. Aircraft attitude data were collected at 20 Hz with an Applanix POS-AV 510 inertial navigation system (INS). The GPS and INS data were processed in Applanix’s POSPac v.8 software in a loosely coupled solution. The final average position accuracies for the survey were 0.022 m in the horizontal and 0.043 m in the vertical.

The gravimeter used was a Micro-g LaCoste Turnkey Airborne Gravity System (TAGS), a beam-style gravity sensor mounted on a gyro-stabilized platform. The manufacturer reported accuracy is 1 mGal. The airborne gravity data were processed with NGS in-house software which builds upon the processing methods presented primarily by Childers (1996) and secondarily by Diehl et al. (2008). This software calculates a full-field gravity value at altitude by correcting the raw TAGS gravity values for: vertical accelerations from GPS, the Eötvös correction, sensor offlevel, sensor drift, and a tie to absolute gravity. After application of these corrections and an along-track, time-domain 120 s Gaussian filter, the airborne gravity data’s smallest recoverable feature is approximately 17 km half-wavelength. This is slightly larger than the 13.8 km half-wavelength feature recovery estimated solely from the altitude of the survey (Childers et al. 1999). Airborne data error can be represented by statistical analysis of all intersecting final data lines and cross lines, called crossover analysis, if a reasonable population of crossing points exists. For the TX09 survey, the 267 crossing points yield an RMS error of 3.40 and 0.90 mGal mean. No adjustments were used for this crossover analysis or applied to the final gravity data, to maintain integrity of the airborne gravity prior to inclusion in geoid models.

Appendix C: Harmonic analysis of GPS/leveling

For Fig. 7, the associated ‘hybrid’ (GPS/L/EGM) geoid is constructed by first averaging the spherical spectra \(\overline{C}_{\mathrm{nm}}^{\mathrm{s}} \)of EGM2008 and GOCO02S, up to the maximum degree provided for GOCO02S, which is \(N_{\mathrm{s}}=250.\) This constitutes the spherical spectrum of the hybrid geoid out to \(n=250\). The remaining spherical coefficients \(\overline{C}_{\mathrm{nm}}^{\mathrm{s}}\) for the hybrid geoid \((251\le n \le 2{{,}}190)\) are copied from EGM2008. This spherical spectrum \(\overline{C}_{\mathrm{nm}}^{\mathrm{s}}\) is then used to create a global \(1^{\prime }\times 1^{\prime }\) ‘reference’ geoid model, designated ‘\(N_{\mathrm{ref}}\)’, using exactly the same procedures as those which were used to compute the EGM2008 geoid model (Pavlis et al. 2012). Values for ‘\(N_{\mathrm{ref}}\)’ are also computed for, and removed from, the geoid heights implied by the co-located GPS/L data ‘\(N_{GPS/L}\)’, yielding residual geoid heights ‘\(N_{\mathrm{resid}}\)’ .

$$\begin{aligned} N_{\mathrm{resid}} =N_{GPSL} -N_{\mathrm{ref}} \end{aligned}$$
(18)

Figure 22a shows ‘\(N_{\mathrm{resid}}\)’ for the CONUS GPS/L stations. Prominent is the North-West to South-East tilt, highlighting again the significant distortions in the CONUS GPS/L, primarily the NAVD 88 orthometric heights (H). This tilt accounts for the majority of the 59 mGal standard deviation of \(N_{\mathrm{resid}}\), and so must be removed prior to gridding and harmonic analysis. This is achieved by fitting a plane through the \(N_{\mathrm{resid}}\) in Fig. 22a, for which the geoid values described by this plane are \(N_{\mathrm{plane}}\), and removing the plane to yield the ‘de-trended’ residual geoid values ‘\(N_{\mathrm{trend}}\)’:

$$\begin{aligned} N_{\mathrm{trend}} =N_{\mathrm{resid}} -N_{\mathrm{plane}} \end{aligned}$$
(19)
Fig. 22
figure 22

a Top Residuals (cm) of the NGS GPS/leveling with respect to (minus) a gravimetric geoid computed from arithmetic average of GOCO02S and EGM2008 for \(2 \le n \le 250\), and EGM2008 alone \((251 \le n \le 2{,}190)\) (\({ N} = 23{,}961\), \(\hbox {Min}=-158.2\), \(\hbox {Max}=152.2\), \(\hbox {Mean}=12.7\), \(\hbox {SD}=59.0\)); b Below Residuals (cm) in a from which a planar trend has been removed (\({ N}=23{,}961\), \(\hbox {Min}= -48.1\), \(\hbox {Max}= 46.3\), \(\hbox {Mean}=0.0\), \(\hbox {SD}=10.9\))

The de-trended geoid values are shown in Fig. 22b. Note the reduction in the standard deviation from 59 to 11 mGal. These detrended geoid values are then interpolated into a \(1^{\prime }\times 1^{\prime }\) grid using least squares collocation (Moritz 1980). Since the intention is that it is the GPS/L, rather than EGM2008 and GOCO02S, which should define the ‘hybrid’ geoid inside CONUS, no error model is adopted for the GPS/L in the gridding (Moritz 1980, Eq. 9–28). The global \(1^{\prime }\times 1^{\prime }\) reference geoid \(N_{\mathrm{ref}}\) is then restored to the \(1^{\prime }\times 1^{\prime }\) gridded values for \(N_{\mathrm{trend}}\), yielding the final global hybrid geoid ‘\(N_{\mathrm{hybrid}}\)’ :

$$\begin{aligned} N_{\mathrm{hybrid}} =N_{\mathrm{trend}} +N_{\mathrm{ref}} \end{aligned}$$
(20)

which can be directly compared to the EGM2008 geoid model (Pavlis et al. 2012), denoted by ‘\(N_\mathrm{EGM2008}\)’. For GOCO02S, the spherical spectrum from EGM2008 \((251\le {n} \le 2{,}190)\) is added to the GOCO02S spectrum, and this combined \((N_{\mathrm{s}}=2{,}190)\) EGM is used to compute the GOCO02S geoid model ‘\(N_\mathrm{GOCO02S}\)’, again following the EGM2008 recipe.

Each of the three geoid models, \(N_{\mathrm{hybrid}},N_\mathrm{EGM2008}\) and \(N_\mathrm{GOCO02S}\), is then analyzed into its component surface ellipsoidal harmonic spectrum using a least squares formulation in which the full normal matrix is approximated with the “Type 1” Block-Diagonal form (Pavlis 1998; Pavlis et al. 2012). For each geoid model, this returns the surface harmonic spectrum \(\overline{N}_{\mathrm{nm}}^e\) which best reproduces the geoid \(N(\delta ,\lambda )\) in the least-squares sense, according to the mathematical model in Eq. (12). For a degree-wise comparison of GOCO02S and EGM2008 against the CONUS GPS/L data, the harmonic spectrum for \(N_{\mathrm{hybrid}}\) is used in a harmonic synthesis of Eq. (12) for \(n=2\) only (i.e, \(\bar{N}_{2,m}^e)\) to generate geoid values \(N_{\mathrm{hybrid}}^{n=2}\) which contain only the harmonics for \(n=2\). This is performed only at GPS/L stations which are proximal to at least four other stations at distances greater than 0.5’ and less than \(2^{\circ }\) spherical distance, and with at least one of these proximal stations in each of the four azimuthal quadrants. This is repeated for \(N_\mathrm{EGM2008}^{n=2}\) and \(N_\mathrm{GOCO02S}^{n=2}\). At each qualifying GPS/L station, the difference \(N_{\mathrm{hybrid}}^{n=2} -N_\mathrm{EGM2008}^{n=2}\) is computed, and the standard deviation of all the differences over CONUS is obtained. Each \(N_{\mathrm{hybrid}}^{n=2} -N_\mathrm{EGM2008}^{n=2}\) is weighted so that, for any given State, the total weight of all the stations in that State towards the CONUS-wide standard deviation is proportional to the land area of that State. This CONUS-wide standard deviation is then also computed for the \(N_{\mathrm{hybrid}}^{n=2} -N_\mathrm{GOCO02S}^{n=2}\) differences.

This entire procedure is performed for all ellipsoidal harmonic degrees \(2 \le n \le 2{,}159\). The CONUS-wide standard deviation for each set of \(N_{\mathrm{hybrid}}^n -N_\mathrm{EGM2008}^n\) differences, and the \(N_{\mathrm{hybrid}}^n -N_\mathrm{GOCO02S}^n\) differences, are plotted graphically by degree, in a manner similar to a typical degree-variance presentation, as shown in Fig. 7. Note that GOCO03S and xEGM-G were also included in this analysis, although they did not contribute to the hybrid EGM/GPS/L geoid.

Finally, it should be noted here that this recipe for the harmonic analysis of GPS/L data is being refined and improved. Future implementations will benefit from a systematic cleaning of the GPS/L data, as well as some appropriate down-weighting in the statistics of GPS/L station clusters that are geographically isolated, straddling state boundaries, or otherwise suspect in quality.

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Smith, D.A., Holmes, S.A., Li, X. et al. Confirming regional 1 cm differential geoid accuracy from airborne gravimetry: the Geoid Slope Validation Survey of 2011. J Geod 87, 885–907 (2013). https://doi.org/10.1007/s00190-013-0653-0

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