Contribution of GOCE RL05 models to Height System Unification in North America B. Amjadiparvar(1), E. Rangelova(1), M. G. Sideris(1) , C. Gerlach(2) (1) Department of Geomatics Engineering, University of Calgary (2) Bavarian Academy of Sciences and Humanities, Munich, Germany

Outline Overview of classical levelling-based vertical datums in North America Computation of datum offsets Factors affecting the datum offset estimation Indirect bias term Systematic levelling errors and distortions GOCE geoid omission error Sensitivity of the offsets and their errors to data inaccuracies Conclusions 1

Classical levelling-based vertical datums + Gravimetry Levelling network over a region with respect to the MSL Tide Gauge (datum origin) Mean Sea Level Classical levelling-based vertical datums 2 5

Overview of Classical levelling-based vertical datums in NA CGVD28 and NAVD88 Levelling data collected over the time span of many decades CGVD28 (until Nov 2013) constrained to the Atlantic and Pacific MSL Normal-orthometric heights Poor absolute accuracy and local distortions NAVD88 constrained to Rimouski Orthometric heights Large NW-SE long-wavelength tilt Poor absolute accuracy CONUS and Alaska NAVD88 Mexico NAVD88 CGVD28 Objectives of the GOCE+ HSU project Investigate the North American height datums unification by means of GOCE using the GBVP approach Compute high accuracy local datum offsets with respect to the regional W0 level Improve gravity anomaly data and height information for the computation of the continental geoid modelin NA by 2022 3

Computation of datum offset (1/2) datum zone j Datum i Geoid Ellipsoid datum j datum zone i Connecting two datums by means of a GOCE-based geoid. ITRF height of a BM or of the mean sea level (MSL) at a TG; Orthometric height in the national datum of the BM or of the MSL; Geoid height determined from a GOCE-based gravity model; Residual geoid height determined from local gravity data 4

Computation of datum offset (2/2) Basic relationships Multiple Vertical Datum Problem If the indirect bias term is negligible (less than 1 cm), the observation equation reduces to Mean datum offset, i.e., the offset of a mean equipotential surface from the geoid or a reference level defined by a conventional Wo value Omission error Indirect bias term Data variances known and uncorrelated h, H and N 5

GNSS-levelling and TG Stations 6

Indirect bias term (1/3) Procedure based on Gerlach and Rummel (Journal of Geodesy 2013) Mean offsets computed with GOCE DIR-R5 D/O 210 (Canada and USA), 250 (Alaska) and 280 (Mexico) using GNSS-levelling stations Reference level Wo = 62636856.0 m2s-2 7

Indirect bias term (2/3) Stokes’s kernel 8

Indirect bias term (3/3) Residual Stokes’s kernel nmax =70 9

Indirect bias term (3/3) Residual Stokes’s kernel nmax = 120 9

Indirect bias term (3/3) Residual Stokes’s kernel nmax = 150 9

Indirect bias term (3/3) Residual Stokes’s kernel nmax = 180 9

Indirect bias term (3/3) Residual Stokes’s kernel nmax = 200 9

Indirect bias term (3/3) Residual Stokes’s kernel nmax = 210 9

Indirect bias term (3/3) Residual Stokes’s kernel nmax = 250 9

Indirect bias term (3/3) Residual Stokes’s kernel nmax = 280 9

Systematic levelling errors Differences between geoid heights computed by extended DIR-R5 model and the GNSS-levelling geoid heights Canada USA Alaska Mexico 10

GNSS-levelling and TG Stations TG and GNSS-levelling stations with heights in NAVD88 and CGVD28 Region TG GNSS-lev Atlantic Canada  7 77 Pacific Canada 5 21 Atlantic USA I 10 799 Atlantic USA II 1677 Atlantic USA III 8 3280 Pacific USA I 201 Pacific USA II 9 476 Gulf of Mexico 13 1867 11

Omission error –RL 04 and RL 05 Region DIR-R5 DIR-R4 TG Stations GNSS-lev Stations Atlantic Canada 5.4 -4.3 2.9 -4.1 Pacific Canada -17.6 -39.3 -13.5 -39.7 Atlantic USA I 11.0 -3.7 20.2 2.2 Atlantic USA II -14.9 7.2 -15.4 9.2 Atlantic USA III -2.2 10.2 -1.3 Pacific USA I -17.3 -13.1 -14.1 -12.9 Pacific USA II -6.6 -10.6 -15.5 Gulf of Mexico -2.1 -8.7 -1.8 -15.6 Alaska --- -20.7 -22.2 Mexico -0.8 -0.3 Atlantic USA 0.6 0.3 4.6 2.3 Pacific USA -11.6 -11.3 -13.3 Alaska and Mexico, for which only GNSS-BMs are available are added in the table for completeness. The mean of the omission error has a direct effect on the mean datum NAVD88 and CGVD28 offsets. When averaged over the tide gauges, the omission error tend to cancel out. Effect on the mean offsets is 17 cm at most on the west coast and only 2 cm in Gulf of Mexico. The number of GNSS-lev BMs varies from one coastal region to another. In the best case, the mean of the omission error is below 1 cm in Mexico (relatively small number of BMs bur well distributed over the territory even though the terrain is rough) and 2 cm in southern Atlantic USA (very large number of BMs and evenly distributed but over moderately flat terrain). The worst scenario is a very small number of BMs on rough terrain and clustered (see Pacific Canada): 40 cm. A large decrease in the GOCE omission error compared to the Release 4 direct approach model GO_CONS_GCF_2_DIR_R4 is observed for the Atlantic US coast, where the large number of tide gauges allows a fair comparison between the two releases. The mean of the omission error, which directly affects the height datum offsets, decreases from 4.6 cm to 0.6 cm, and the standard deviation decreases from 28.7 cm to 25.8 cm. 12

Sensitivity of the offsets and their errors to data inaccuracies (1/3) Motivation in coastal areas, errors of computed heights of mean sea level are usually unknown CGVD28 heights do not come with errors; even for more recent adjustments like NAVD88 error VCM are not available. Objectives: vary the error of the geometric geoid height (h-H) in a simulation study to estimate the variation of the offsets and their errors. Data sets DIR-R5 (210) geoid heights NGOCE and geoid error VCM (block-diagonal) Residual geoid heights Nres from EGM2008 and residual geoid errors NGeom= hMSL- HMSL and NGeom = hBM - HBM . Errors are unknown. 13

Sensitivity of the offsets and their errors to data inaccuracies (2/3) Test on the a-posteriori estimate of the variance Test statistic with Reject null hypothesis if is the degree of freedom, is the level of significance Least-squares adjustment model block-diagonal VCM error propagation from a map of propagated errors of gravity anomalies (scaled by 2); is varied from 1 cm to 40 cm. 14

Sensitivity of the offsets and their errors to data inaccuracies (3/3) Region Mean offset range [cm] Mean offset error range Threshold of the geometric geoid height error [cm] TG GNSS-lev Atlantic Canada [-58.0, -59.9] 1.9 [-56.9, -58.4] 1.5 [1.2, 11.8] 10.6 [1.0, 2.1] 1.1 31.0 15.5 Pacific Canada [2.3, 3.8] 1.5 [11.0, 11.2] 0.2 [2.6, 14.1] 11.5 [1.5, 1.6] 0.1 30.9 1.4 Atlantic USA I [-34.6, -35.1] 0.5 -34.7 0 [1.2, 2.0] 0.8 [1.0, 1.4] 0.4 4.9 2.7 Atlantic USA II [-34.0, -34.7] 0.7 -33.2 0 [1.2, 2.7] 1.5 7.6 4.7 Atlantic USA III [-3.5, -10.1] 6.6 [-5.2 to -6.6] 1.4 [1.3, 10.0] 8.7 1.0 0 28.1 14.3 Pacific USA I [-93.0, -94.1] 1.1 [-101.3, -101.7] 0.4 [1.6, 10.6] 9.0 [1.0, 1.2] 0.2 29.6 7.0 Pacific USA II [-69.4, -71.2] 1.8 [-70.1, -73.3] 3.2 [1.7, 7.7] 6.0 [1.0, 1.1] 0.1 22.4 8.7 Gulf of Mexico [-10.6, -11.9] 1.3 [-1.0, -2.3] 1.3 [1.2, 5.7] 4.5 20.1 11.6 Alaska --- [-149.2, -156.0] 6.8 [1.1, 2.6] 1.5 31.1 Mexico [-5.0, -12.6] 7.6 [1.1, 1.6] 0.5 The test on the variance is performed for all 8 regions with TGs and GNSS-lev stations. For Mexico and Alaska, we have only GNSS-lev stations. The error limit of the geometric geoid is actually the maximum error of N_geom, for which the estimated variance is tested 1.0 with the given parametric and stochastic models. As we see in the TGs case, except for the US Atlantic, the N_geom large errors are responsible for larger variations in the accuracy of the estimated offsets. Best case: Atlantic USA I and II – Practically, identical offsets computed with TGs and GNSS-lev BMs. The abundance of GNSS-lev data leads to a very small 1-1.4 cm error of the offset. The offset error computed with TGs is 2 -3 cm. A sub-cm variation of the offset and a 1 cm variation of its error when computed with the tide gauges. Either TGs or GNSS-lev BMs can be used in HSU. Mixed case: Pacific US I: Stable offset with a 1 cm error if computed with GNSS-lev BMs. Large offset and error variation when computed with TGs. This is a consequence of the large omission error at the tide gauges, which does not average out. Worst case GNSS-lev data - Alaska: Large offset variation, large error variation. Shows the effect of the sparse GNSS-lev stations with poor configuration and coverage, and as a result, large mean omission error. Also, very large systematic levelling errors contribute to the poor results. Worst case TGs – Pacific Canada: Large offset and error variations, a consequence of the combination few TGs and very large mean omission error. The Atlantic US III offset and error with TGs is correct. We could not find the reason for this odd results. There are no blunders among the TGs. 15

Conclusions The indirect bias term can be omitted for a GGM of DO 180 in North America (below 1 cm); therefore the LVD offset is a (weighted) mean of the discrepancies between the geometrical and gravimetric geoid heights computed at TG or GNSS-levelling stations. The MEAN of the GOCE omission error of the release 5 models are generally decreased compared to release 4 at the GNS-levelling and TG stations where a large number of stations are available to allow a fair comparison between the two releases. With a limited number of tide gauges available, the datum offset error can be very sensitive to height data errors. With a larger number of GNSS-levelling stations, a more accurate datum offset can be estimated which is not very sensitive to height data error. 16

Thank you for your attention "This work was a contribution to the ESA STSE – GOCE+ Height System Unification with GOCE project." 17

DIR-R5 omission error Omission error in the vicinity of tide gauges 13 A small number of tide gauges combined with rugged terrain as in Pacific Canada results in a large mean omission error even for large radii of the spherical cap. Tide gauges used for HSU should be al least 0.6 degree apart in order to mimimize the effect of the omission error on the mean datum offset. Another way to assess the omisssion error is to compute areal averages around each tide gauge station. The residual geoid height from EGM2008 above D/O 210 is averaged in a spherical cap that is centered at a tide gauge and has a varying radius from 0.1 to 2.0 degrees. The standard deviation of this average omission error at the tide gauges as well as the mean are plotted. The standard deviation gradually decreases with the spherical distance to below 5 cm for all five regions at around 1.0 degree (~70 km for 50N). Evidently, the combination of a small number of tide gauges with a rugged terrain as in the Pacific Canada results in a large mean omission error even for large radii of the spherical cap. For all other regions, the mean of the average omission error quickly decreases with the distance and from 0.7 degree varies within 1 cm for all regions except the Canadian Pacific. An important observation is that the mean of the omission error computed by averaging in a spherical cap around the tide gauges may differ largely from the mean of the omission error computed direcly at the tide gauges in rugged terrain (see Pacific Canada). Another implication of the graphs is that tide gauges used for HSU should be al least 0.6 degree apart in order to mimimize the effect of the omission error on the mean datum offset. 13

Required Data Sets GNSS ellipsoidal height Transformed to a common ITRFxx and epoch Unknown in Mexico’s data set USA: NAD83  ITRF2005 MSL in the national datum ideally tide gauges with a 19-year long records and no data gaps inclusion of shorter tide gauge series to improve the network configuration Geoid height Long and medium wavelengths from a GOCE GGM with the highest possible spectral resolution (d/o 210 for North America) defined by the improvement over EGM2008

DIR-R5 omission error Region Mean TG Stations GNSS-lev Stations Atlantic Canada 5.4 -4.3 Pacific Canada -17.6 -39.3 Atlantic USA I 11.0 -3.7 Atlantic USA II -14.9 7.2 Atlantic USA III -2.2 Pacific USA I -17.3 -13.1 Pacific USA II -6.6 -10.6 Gulf of Mexico -2.1 -8.7 Alaska --- -20.7 Mexico -0.8 Atlantic USA 0.6 0.33 Pacific USA -11.6 -11.3 A large decrease in the GOCE omission error compared to the DIR-R4 observed for the Atlantic US coast 4.6 cm to 0.6 cm Alaska and Mexico, for which only GNSS-BMs are available are added in the table for completeness. The mean of the omission error has a direct effect on the mean datum NAVD88 and CGVD28 offsets. When averaged over the tide gauges, the omission error tend to cancel out. Effect on the mean offsets is 17 cm at most on the west coast and only 2 cm in Gulf of Mexico. The number of GNSS-lev BMs varies from one coastal region to another. In the best case, the mean of the omission error is below 1 cm in Mexico (relatively small number of BMs bur well distributed over the territory even though the terrain is rough) and 2 cm in southern Atlantic USA (very large number of BMs and evenly distributed but over moderately flat terrain). The worst scenario is a very small number of BMs on rough terrain and clustered (see Pacific Canada): 40 cm. A large decrease in the GOCE omission error compared to the Release 4 direct approach model GO_CONS_GCF_2_DIR_R4 is observed for the Atlantic US coast, where the large number of tide gauges allows a fair comparison between the two releases. The mean of the omission error, which directly affects the height datum offsets, decreases from 4.6 cm to 0.6 cm, and the standard deviation decreases from 28.7 cm to 25.8 cm.

Statistics of the DIR5 commission error in different regions (up to DO 210 in Canada and USA, 250 in Alaska and 280 in Mexico) [cm] Region Min Max Mean TG GNSS-lev Canada Atlantic 0.9 Canada Pacific US Atlantic I US Atlantic II US Atlantic III US Pacific I US Pacific II Gulf of Mexico Alaska --- 2.9 3.2 3.0 Mexico 4.0 4.5 4.2

Statistics of the residual geoid error in different regions [cm] Min Max Mean TG GNSS-lev Canada Atlantic 0.7 0.6 2.0 2.7 0.1 1.2 Canada Pacific 2.1 1.6 3.5 10.2 2.8 3.8 US Atlantic I 1.9 6.3 1.0 1.5 US Atlantic II 0.8 1.7 3.0 US Atlantic III 0.5 2.4 3.9 0.9 US Pacific I 1.4 2.5 7.9 1.8 US Pacific II 3.1 12.8 2.2 Gulf of Mexico 1.3 Alaska --- 8.5 3.2 Mexico 1.1 12.5 7.8