National Geoid Height Models for the United States: USGG2009 and GEOID09

Daniel R. Roman, Yan Ming Wang, Jarir Saleh, Xiaopeng Li

NOAA/National Geodetic Survey

Abstract. New geoid height models for the United States have been determined. The United States Gravimetric Geoid of 2009 (USGG2009) serves as the basis for scientific and engineering applications. It is a one arc-minute model (2 km by 2 km nodal spacing) based on the EGM08 reference model and improved surface gravity and terrain data. GEOID09 was developed from USGG2009 in conjunction with thousands of control points where GPS-derived ellipsoid heights are available on leveled bench marks (GPSBM's). Models have been developed for all regions of the United States and its territories: Conterminous United States (CONUS), Alaska, Hawaii, Guam, the Commonwealth of the Northern Marianas Islands, American Samoa, Puerto Rico and (eventually) the Virgin Islands. GEOID09 is also a one arc-minute model and serves in CONUS as the recognized standard for transforming between NAD 83 and NAVD 88 for most geodetic and surveying applications. The final CONUS GEOID09 model resulted in sub-cm accuracy, having less than a centimeter of correlated signal with 1.5 cm random GPS observation errors. The GEOID09 models for the other states & territories serve a similar function with respect to the local vertical datums and have similar quality. This workshop will cover the development of these models, their limitations, and utility as well as the future of the U.S. vertical datum.

1 Introduction

The difference between the orthometric height and the GPS observed ellipsoidal height is called the geoid height or undulation. The geoid surface is said to undulate about the ellipsoid surface. If the geoid undulation is known everywhere, then the GPS height can be simply converted into orthometric height by subtracting the geoid height. NGS has been developing and experimenting with these models since the start of the GPS era in the early 1990’s (e.g. GEOID90). Geoid modeling has progressively gained in significance as a part of the Height Modernization program. It directly supports efforts at improving the National Spatial Reference System, a part of the National Spatial Data Infrastructure mandated by Executive Order 12906 (Clinton, 1994).

With the continued loss of monumented marks, geoid height models have steadily served as a ready mechanism for deriving orthometric heights (cf. GPS-derived orthometric heights). As defined in the NGS Ten Year Plan (ref), geoid height models will eventually define the vertical datum by 2017 instead of serving as a bridge between the ellipsoidal and vertical datums. The work described here is one step in that direction.

New geoid height models were developed for all regions of the United States and its territories. Two types of models were created: gravimetric geoids defined from gravity and terrain data, and hybrid geoids defined from the gravimetric geoids and GPS-derived ellipsoid heights on leveled bench marks (GPSBM’s). These models were developed for all of the United States and its territories including the Conterminous United States (CONUS), Alaska, Hawaii, Guam, the Commonwealth of the Northern Marianas Islands (CNMI), American Samoa, and Puerto Rico and the Virgin Islands.

The gravimetric geoid models were based on the Earth Gravity Model of 2008, EGM08 (Pavlis et al. 2008) and have significantly reduced long wavelength errors. The gravimetric geoid USGG2009 is more accurate than previous models mainly contributed from the new computation method, new satellite gravity model and improved altimetric gravity anomalies. In comparison with GPSBM implied geoid undulations, the improvement goes from 9.1 cm to 7.3 cm for USGG2003 and USGG2009, respectively.

In turn, the hybrid models are significantly different because of the National Readjustment of 2007, which modified most GPS-derived coordinates (including heights) at the cm- to dm-level nationwide.  The changes in vertical direction are mostly clustered around 2 cm, with few changes exceed 10 cm or higher. The hybrid models transform between ellipsoidal heights above the NAD 83 datum and orthometric heights above the respective vertical datum around the country. For the Conterminous United States and Alaska, this is the North American Vertical Datum of 1988 (NAVD 88). The focus of this paper will be on GEOID09 in CONUS with some discussion if the development of the GEOID09 files for other regions.

The CONUS GEOID09 is produced by tailoring USGG2009 to fit the 12,715 GPSBM’s. The root mean square (RMS) of difference between the GEOID09 and the NAVD 88 that is represented by the GPSBM’s at the benchmarks is ±2m nationwide. The RMS values of the differences (formal error) between the geoid heights from GEOID09 and the GPSBM implied geoid heights at the benchmarks are given in state by state basis. For most areas, the errors are around ±2cm or better relative to NAVD 88. But there are few spots where the errors are larger, due to the errors in GPSBM’s, in gravity data and due to poor data distribution and other errors.

GEOID09 are given in the same format as its predecessors. The software used in interpolation of the geoid heights can be used in the same way as before by only change the dataset name. For more information, please visit the National Geodetic Survey (NGS)’ web site http://www.ngs.noaa.gov/geoid/.

 2 Theoretical Improvements for Gravimetric Geoid Models

During recent years, satellite gravity missions, such as the Gravity Recovery and Climate Experiment (GRACE), have improved the accuracy of the satellite gravity models significantly in comparison with previous models. The accuracy of the satellite models has been reported to be better than 1 cm to degree and order 70 (Tapley et al., 2005). Based on those satellite gravity models, a high degree and order spherical harmonic coefficient gravity model, the Earth Gravity Model for 2008 (EGM08) was published by the U.S. National Geospatial-Intelligence Agency (NGA) (Pavlis et al, 2008). This model was developed combining the GRACE observations with the surface gravity and altimetric gravity (gravity implied by altimeter observations over the oceans). The maximum degree and order of the model is 2160, which corresponds to a resolution of five arc-minutes (5’) or about a ten kilometer nodal spacing. 

Historically, the NGS gravimetric geoid models have been computed using Faye anomalies in the Stokes integral (a function that relates gravity to geoid heights) as a simplification of Helmert anomalies. Faye anomalies are easier to compute, however, they involve several approximations. The accuracy of these approximations depend on assumptions that may not work in some places, particularly in the mountains. At the same time, the coefficients models have been used in a simple remove-restore fashion to take the advantage of much more accurate long and medium wavelength of satellite model. Knowing that the coefficient models are developed under the theory of harmonic continuation, while the Faye anomalies are a result of the Helmert’s 2^nd condensation, there is an inconsistency problem between them.  The effect of the inconsistency has been ignored, since other errors in gravimetric geoid are believed to be much larger. This situation changed dramatically after publication of the EGM08. The accuracy of the EGM08 has been improved to such a level that the error due to the inconsistency becomes obvious, thus it has to be taken into account.

There are two ways to solve the inconsistency problem: (a). Modifying the coefficient model to make it compatible with the Faye anomalies. In other words, to add the indirect effect of the Helmert’s 2^nd condensation to the coefficient model; (b). Abandon the method of Helmert’s 2^nd condensation, instead using the harmonic continuation method for geoid computation (e. g., Wang 1990; Sjöbeger, 2006).  There is no topographic correction to the gravity observations (direct effect). The topographic effect only appears as a correction to the disturbing potential or to the geoid.

In comparisons of two methods, the harmonic continuation has its advantage on simplicity and computation efficiency. There is no topographic correction for gravity data. In addition, the gravity data are treated in the same way the coefficient models are developed, thus the coefficient models can be used in a simple remove-restore fashion. Based on these reasons, the harmonic continuation method becomes a nature choice for our geoid computations.

3 USGG2009

There are 1.3 million terrestrial and ship-borne gravity data points in NGS database that are a result of decades of data collection effort from various sources. Another 8000 points were downloaded recently from NGA to add to this total. The altimetric data have been improved in recent years and become available (Anderson, 2008). The most noticeable improvement is the altimeter retracking for the data near shore which directly impacts the geoid in the states bordering the oceans. The altimetric gravity data fill in the oceans surrounding the terrestrial data and provide continuity across the sea-shore boundary. It is very important for the geoid models over islands, such as Puerto Rico, the Virgin Islands, Guam and the Northern Mariana Islands.

Many systematic and random errors exist in these data, particularly in the terrestrial data. Hence, editing the data is critical for accurate geoid computation. While the satellite models provide the long wavelengths of the gravity field very accurately, it is not very useful for data editing as the errors in the point gravity data are mainly medium and high frequency. EGM08 contains this satellite data plus the medium and short wavelength up to resolution of 5’, thus it is used for data editing. The reference gravity is computed at the location of the observation on the Earth’s surface (latitude, longitude and height), then subtracted from the observed gravity. Nationwide, the RMS value of the residual gravity anomaly is 5.3 mGal with a range of 300 mGals. If a 3 sigma outlier (16 mGal) filter is applied, the RMS value of the residual is reduced to 2.9 mGal. Fig. 1 shows the location of 117,649 points that were edited out. 

Figure 1 Location of outliers of gravity anomaly data. These data were made available by NGA and NGS but were rejected from the USGG2009 model due to their poor comparison.

Clearly, there are some problems in the data in the mountainous regions. The differences may be a combination of errors in the gravity observation, location error and error in EGM08. Of all the error sources, the errors in recorded height for the gravity measurements are likely the major contributor. Excluding at the bench marks, the elevations of the gravity survey are mostly read from a topography map, a common practice before the advent of GPS. This practice may introduce height errors in the range of meters to hundreds of meters. If the height is off by 10 meters, the error in the computed gravity anomaly is 3 mGals, which is one to two orders larger than the errors in the gravity observations themselves. The assertion of inaccurate heights in the gravity database is confirmed by the observation that the most outliers happened in mountainous region where the height could not be accurate determined by reading a topo-map.

The data that passed this editing procedure is plotted in the Figure 2. Note that there were 5,420,245 points passed, but most of these are from the offshore altimetric data. Very few of these were rejected as seen in Figure 1.

Figure 2 Edited gravity anomalies distribution over CONUS

EGM08 actually creates height anomalies instead of geoid heights – these are related functions of the gravity field but not identical. However, EGM08 software comes with its own topographic effect correction and it converts the height anomaly directly into the geoid height. So the model is used in slightly different remove-restore fashion than in the past (Smith and Roman 2001). The reference gravity from coefficient model is removed from the surface gravity data at the observation location (latitude, longitude and height), then residual gravity anomalies are inserting into Stokes integral to compute the residual geoid. Since the residual anomaly is so small, that the downward continuation effect is in mm level and ignored at this time.

In the Stokes integral, a modified kernel is used to eliminate any disagreement between the NGS gravity data and the EGM08 model through degree and order 360. This effectively adopts the EGM08 model for signals corresponding to 111 km and longer (large to very large features).  The surface gravity anomalies provide higher frequencies of the gravity field that occur in spherical harmonics higher than degree 360 (361 to infinity).

The following figure shows the residual geoid, which is the value-added signal contributed by NGS to the NGA model. Note that most of the signal is very short wavelength (the dimpling) draped onto a long wavelength feature (the slow shift in colors from red to purple).

Figure 3 Residual geoid signal in USGG2009. This is the signal added to the EGM08 model to create USGG2009. Dimpling reflects the short wavelength signal omitted by EGM08. The color shifts indicate longer wavelength differences due to exclusion of poorer surface gravity data.

It is also useful to plot the changes from USGG2003 to USGG2009 Fig. 4 shows these differences.

Figure 4 Differences between USGG2003 and USGG2009 mainly highlight the improvements to the underlying reference models. EGM96 was used for USGG2003, while EGM08 was used for USGG2009.

The changes from USGG2003 to USGG2009 are quite significant. The long wavelength changes are due to the differences between the EGM96 and EGM08, and the improved altimetric gravity anomalies. Over the oceans, there are clear features that are associated with the mean ocean dynamic topography of the Gulf Stream and other ocean currents. Improvement on the oceans is mostly due to better satellite gravity model determined from the GRACE satellite mission and incorporated into EGM08. Over land areas, the long wavelength changes are due to the same thing plus the inconsistency caused between the methods of Helmert 2^nd condensation and harmonic continuation. The short wavelength changes are the contributions of the edited surface gravity data, and the methodology inconsistency. Along the shorelines, the altimetric data retracking (Anderson, 2008) improves the gravity recovery from altimetry, and indirectly, improves the geoid for the shoreline statistics. In previous models, altimetric data in shoreline area edited out because of their poor quality.  Typical distances offshore for exclusion were about 20-100 km corresponding to the 500 fathom curve where ocean models become more rigorous. Now, gravity data derived from re-tracked altimetry can approach the shoreline.

One final note on the USGG2009 model: it does not include any of the recently acquired airborne gravity data collected as a part of the Gravity for the Re-definition of the American vertical Datum (GRAV-D) project. This model is intended as a baseline for determining the impact of GRAV-D; see http://www.ngs.noaa.gov/GRAV-D/ for more details. As GRAV-D moves forward, updated gravimetric geoid models will reflect the improvements. Those models will be posted to the web under a separate location. As more data and improved techniques are applied, this series of models will eventually lead to the selection of the optimal gravimetric geoid for replacing the North American Vertical Datum of 1988.

4 GPSBM2009

Another essential data set for geoid modeling is the GPSBM’s derived from the differences between the NAD 83 and NAVD 88 datums on the bench marks. As GPS became operational in the early 1990’s, NGS quickly adopted the new technology and established High Accuracy Reference Networks (HARN’s) in each state. Ellipsoidal coordinates were readjusted then based on the other points in a state’s HARN through about 1998. At about that time, a nationwide network of Continuously Operating Reference Stations (CORS) was established.

This network provides not only the 3-D coordinates of stations, but also monitored the coordinates changes (velocities). HARN’s were now adjusted based on the CORS stations, to which new stations were constantly being added. This series of adjustments in conjunction with older baselines and other datum errors, led to the National Readjustment of 2007 (NRA2007). Prior to the NRA2007, approximately 18,000 points met the criteria for being used to make a hybrid geoid model. After that, the number dropped to closer 13,000. While the number of points used here represent approximately the same number as were available for GEOID03, the distribution and quality of these points has significantly changed.

The coordinate changes after NRA2007 directly impact the hybrid modeling, because the implied geoid heights are derived from the difference between NAVD 88 and NAD 83 on a bench mark – changing only the ellipsoid height on the bench mark creates a “new” geoid height. To a lesser extent, this has been the basis for rolling out new hybrid geoid models in the past. New models were published to reflect the current state of the database. It was necessary to hold off on publishing this model for this reason. We originally intended a GEOID06, but that became GEOID07, which became GEOID08, and finally became GEOID09. Holding off on this was necessary because of the magnitude of the changes. Certainly all coordinates changed, but the most important for geoid modeling is the ellipsoidal height change. Figure 5 shows the absolute changes in height.

Figure 5 Absolute height changes after the National Readjustment of 2007 (NRA2007). Note states like California with a highly noisy character and states like Minnesota with a significant trend.

The noticeable height changes happened in California, Alabama, Georgia, Minnesota and Alaska. The changes are a combination of the Earth’s dynamics (crustal motion, subsidence, glacial rebound) as well as the steady changes in coordinate adjustments (e.g., HARNS, CORS, NRA). Figure 5 highlighted the point differences, while Figure 6 shows their impact in the locales.

Figure 6 Height changes after the National Readjustment of 2007. Values are from Figure 5 and gridded to highlight regions where gaps in coverage create dm-level artifacts.

The GPSBM’s are relatively much sparser than the point gravity data by a couple orders of magnitude. Additionally, they tend to be clustered in states with active geodetic surveying programs. Hence, a given GPSBM control value may influence a large region around it. Gridding the data highlights the long wavelength artifacts that can result across gaps in the coverage. These gaps take the form of large features of significant (dm-level) magnitude and do not represent real signal. Hence, error estimates of hybrid geoid models are always overly optimistic as they assume a sufficiency of coverage throughout the country. Use of any hybrid geoid model should also rely on examining the coverage of the data used to make it.

Table 1. Comparisons between GPSBM2009 and USGG2009 & GEOID09. All values in cm. SD represents 1σ. Double that to get the 95% confidence level. The σ_h column represents the state average for all points for a given state derived from the NRA2007. AVE and SD are provided for both USGG2009 and GEOID09.

ST	No. Pts.	σh cm	USGG2009		GEOID09		ST	No. Pts.	σh cm	USGG2009		GEOID09
			Ave.	SD	Ave.	SD				Ave.	SD	Ave.	SD
AL	53	1.3	-40.0	5.5	0.1	2.3	NE	119	0.6	-75.8	4.4	-0.1	0.8
AZ	153	0.8	-62.0	7.4	0.0	1.6	NV	64	1.0	-82.8	8.0	0.1	1.4
AR	58	1.2	-49.2	3.2	0.0	0.8	NH	13	1.5	-41.6	2.9	-0.2	1.3
CA	557	2.7	-83.2	12.2	0.0	2.3	NJ	198	0.4	-42.9	2.7	0.0	1.1
CO	459	1.1	-69.2	8.1	0.0	2.8	NM	89	1.0	-51.3	8.9	0.0	1.4
CT	8	1.7	-41.3	4.0	-0.1	1.6	NY	105	1.2	-46.4	6.9	0.0	1.4
DE	20	2.5	-40.3	4.2	0.2	2.5	NC	1231	0.8	-35.8	5.2	0.0	1.7
DC	16	0.6	-44.6	2.0	-0.2	2.0	ND	41	0.7	-97.7	3.4	0.2	0.8
FL	1413	1.6	-5.8	7.7	0.0	1.5	OH	191	1.4	-60.4	5.0	0.0	2.0
GA	95	1.2	-33.9	6.4	0.0	1.0	OK	71	1.4	-51.8	5.4	0.0	1.0
ID	87	1.4	-101.7	7.9	0.0	1.5	OR	163	0.9	-108.7	8.5	0.0	1.4
IL	266	1.1	-68.7	8.3	0.1	1.2	PA	80	0.7	-49.8	4.1	0.0	1.1
IN	113	1.1	-60.7	6.3	-0.1	2.0	RI	15	0.9	-42.9	1.7	0.0	1.4
IA	73	1.0	-77.4	4.9	0.0	0.8	SC	663	1.0	-38.1	6.2	0.0	1.1
KS	94	1.0	-65.9	5.3	0.0	1.2	SD	172	1.1	-84.7	5.9	0.1	0.9
KY	97	1.6	-50.2	4.1	-0.1	1.7	TN	248	1.1	-49.9	3.7	0.0	1.9
LA	343	1.0	-22.2	8.9	0.0	0.6	TX	188	1.4	-36.8	8.1	0.0	2.4
ME	57	0.9	-39.8	4.6	0.0	0.9	UT	37	0.8	-79.8	9.1	0.1	2.1
MD	186	0.8	-43.2	2.9	0.0	1.9	VT	279	1.0	-42.1	4.3	0.0	1.3
MA	30	1.1	-40.7	3.5	0.0	1.1	VA	190	1.0	-41.6	4.5	0.0	1.8
MI	199	2.0	-64.5	7.2	0.1	1.7	WA	183	1.3	-116.4	8.2	0.0	1.9
MN	2943	0.4	-87.3	3.6	0.0	1.0	WV	41	1.2	-51.3	4.8	0.1	1.9
MS	104	1.7	-44.8	4.3	0.0	1.6	WI	551	0.3	-73.3	3.4	0.0	0.7
MO	96	0.9	-58.7	5.3	0.0	1.1	WY	89	2.2	-84.1	9.3	0.0	2.6
MT	150	2.2	-101.3	8.7	0.0	1.7	all	12961	1.2	-58.5	6.3	0.0	1.5

Table 1 shows the data that are available for the 2009 model. These data were cleaned in three steps. First, the error value determined from the NRA2007 was used as an initial filter. The data were broken up by state and a statewide standard deviation determined. Points that fell outside of two sigma (95% confidence level) were rejected. This process was repeated three times. This resulted in dropping 303 points. By performing this on a state-by-state basis, a relative parity was obtained by ensuring that those states with high quality data did not cause too many point to be unduly eliminated in states with poorer quality. Next, the points were compared to USGG2009 values interpolated to the bench mark locations. Raw differences were taken to see if any outliers existed. One was found at over 2 meters. Next, multi-matrix least squares collocation (MMLSC) was used in the initial development of GEOID09. In determining the statistical behavior of the model, two more points were found to lie outside the behavior of the others (double or more the residuals of any other points).

This is encouraging because it means that the use of the ellipsoidal height error derived as a part of the NRA 2007 provides a real diagnostic tool for locating problem bench marks. Note that 303 points were located this way and only 3 others were found using techniques employed in the development of GEOID03 and earlier models. The remaining 12,979 points with 579 added from Canada were now used to make GEOID09 and they’re distributed as shown in Figure 7.

Figure 7 GPS-derived ellipsoid heights on leveled bench marks for 2009 (GPSBM2009). These point values represent the control data used to make GEOID09. The color scale shows the error estimate of the ellipsoidal height (σ_h) determined as a part of the NRA2007.

5 GEOID09

Finally, we get to the development of GEOID09. Most of the work has been completed in preparing the underlying gravimetric geoid, USGG2009, and cleaning the available GPSBM’s (GPSBM2009). The remaining task is to determine the best means of melding the two into a cohesive whole.

The general approach taken remains to determine the difference between and geoid height values determined by interpolating a gravimetric geoid (USGG2009) and the GPSBM’s (GPSBM2009). These residual signals mainly represent the misfit between the gravimetric geoid and NAVD 88. By developing a mathematical model from these control data, a conversion surface may determined to fit to the USGG2009 model that creates the GEOID09 model. The best tool for performing this fit was determined to be Multi-Matrix Least Squares Collocation (MMLSC) during the development of GEOID03 (Roman et al 2004). This tool was necessary due to the multiple complex features that correlated at different powers and scales of distance.

Before you can fit the model using MMLSC, any biases and trends must be removed. The trend for GEOID03 was 0.14 ppm, indicating that the mismatch between the NAVD 88 datum and the gravimetric geoid trended up 14 cm for every 1,000 km. For GEOID09, this number increased significantly to 0.31 ppm. This is largely due to the underlying reference model, EGM08, and is actually good news from a modeling perspective. First order trends are easy to model, it’s the features that span 10’s to 1000’s of km that are harder to model correctly. After removing the 0.14 ppm trend in 2003, the power of the remaining signal was (14.8 cm)². In 2009, the power was reduced to (6.7 cm)². With the bias and trend removed, MMLSC is employed to model this remaining signal.

Classical LSC permits fitting one Gaussian function that rolls off smoothly with distance. Unfortunately, the residual data implied here do not behave in this manner. MMLSC stacks multiple matrices to better emulate the actual data than the classical approach. Even though the magnitude of the signal to be modeled has been reduced, MMLSC was employed more extensively in 2009 to over come an earlier problem found in GEOID03.

GEOID99 (Smith and Roman 2001) used a single Gaussian function at 400 km. In 2003, only two functions were stacked: one at 60 km and the other at 650 km. This was later determined to be deficient in some regions where the data were much sparser than 60-120 km. In these regions, the only effective model was then at 650 km, which produced a poorer local comparison than for GEOID99. Most regions saw a drastic improvement under GEOID03 but not all.

Figure 8 Comparisons of analytic (MMLSC) models with empirical data for USGG2003 (left) and USGG2009 (right). Note that the scale for USGG2003 MMLSC is four times larger than for USGG2009. This indicates that there is less systematic signal to model. Also, note that the behavior of the curve is much smoother and fits better – indicating a better model.

Hence in 2009, six functions were stacked: 30 km, 60 km, 90 km, 120 km, 260 km, and 600 km. Figure 8 shows the comparisons between the math models and the empirical data for GEOID03 and GEOID09. Note that the left graph in Figure 8 has a scale on the Y-axis that is four times larger than the right figure due to the magnitude of the residual signal left after the trends were removed. The blue line shown in both plots represents the analytic signal derived from the math model (MMLSC). The black diamonds represent the binned values of the empirical data derived from the residuals between the gravimetric geoid and the GPSBM2009.

The analytic line in 2003 does a poorer job of fitting than for 2009. This is because the character of the binned empirical data rolls off more regularly in 2009 than in 2003. Note how the diamonds form a series of humps in 2003. This is not so prominently the case in 2009. This is thought to be because several state-sized features were removed from the residual signal by cleaning up the GPS-derived ellipsoidal heights as a part of the NRA2007. Recall that there was an obvious first order trend seen in Figures 5 and 6 in Minnesota. This trend was present in the 2003 data but removed in the 2009 data. This then also simplifies the MMLSC model. The resulting analytic functions are then used to predict the conversion surfaces seen in Figure 9.

Figure 9 Collocation surface determined for GEOID03 (left) and GEOID09 (right). Color scales are identical. Note difference in magnitude of modeled signal. Many features are gone or significantly reduced.

These are both at the same scale to highlight the magnitude of the changes. Many features are no longer present. Examine in the left plot the state of Illinois. Note how you can see the state boundary in the 2003 model. It is nearly gone in the 2009 model. This likely represented a systematic difference between the Illinois ellipsoidal heights determined in their earlier HARN adjustment that has been reduced significantly after the NRA2007.

These models were then added to trend surfaces derived from the 0.14 and 0.31 ppm trends determined for the respective models. Biases of around 58 cm were restored (common to both models), which likely represents the selection of Father Point/Rimouski as the datum point for NAVD 88. Also, a surface was added to account for the shift between NAD 83 and ITRF00 on which the USGG2009 and USGG2003 models are based. The conversion surface is then the sum of the collocation surface (from MMLSC), trend, bias, and ellipsoidal datum shift (ITRF => NAD 83). When the conversion surface is added to the gravimetric geoids, the hybrid geoids are determined. These models are then compared to the GPSM data again. No significant signal should remain as this whole effort has been to model these differences into the hybrid geoid models. Figure 10 shows the comparisons between GEOID03 (left) and GEOID09 (right) to their respective GPSBM data sets.

Figure 10 Comparisons of analytic (MMLSC) models with empirical data for GEOID03 (left) and GEOID09 (right). There is actually no signal to be seen in the right figure. The red dashed line shows the modeled residual values for GEOID03. GEOID09 is effectively precise to below a cm-level. The spikes on the Y-axes for both show the uncorrelated signal due to GPS random errors of observation.

What is remarkable here is that both did well. GEOID03 had around 1 cm of correlated signal that fell off quickly (10 km). Note where the analytic signal (blue line) intercepts the Y-axis – this value represents the correlated signal. The difference between this point and the spike on the Y-axis represents the uncorrelated signal (around (2.1 cm)² in magnitude).

For GEOID09, this is even better. The red line is identical to the blue line shown in the left graph. This is necessary as there is no correlated signal remaining. Note how the empirical data (black diamonds) oscillate around zero almost up to auto-correlation, then there is a jump up to (1.5 cm)² on the Y-axis.

In the Table 1, statistical values are given for all states in CONUS and the District of Columbia with respect to GEOID09 in the rightmost columns.. Note that the national average is 0.000 m and the national standard deviation is 0.015 m. States with red values show those with worse averages than the national value.

6 Other Regions

While other regions are not discussed here, it should be noted that the scope of the models has been expanded. USGG2009 and GEOID09 models will be developed for Alaska, Hawaii, Guam, the Commonwealth of Marianas Islands, American Samoa, Puerto Rico and the U.S. Virgin Islands. Most of these are complete. For all regions, gravimetric geoid models are developed that predict heights between a GRS-80 ellipsoid centered on the ITRF00 reference frame.

For Alaska, Guam, CNMI, and American Samoa, hybrid models will tie between NAD 83 and the local established datum. Puerto Rico’s vertical datum is not yet robust enough to support a conversion surface. The 29 stations that exist are essentially in a straight line. Hawaii and the USVI have no established datum. For Hawaii, PR, and USVI the gravimetric geoid will be adjusted to use NAD 83 instead of ITRF00 and will serve as the hybrid model.

For the USVI in particular, a greater effort is being made. Their models will be out later this year and will reflect the likely first vertical datum based on a gravimetric geoid as a apart of the Gravity for the Redefinition of the American Vertical Datum (GRAV-D) project. High (35,000 ft) and low (5,000 ft) data will be combined with the existing data to create an improved gravimetric geoid model that will be used to define heights in the USVI. This exemplar will serve for how GRAV-D will be implemented eventually for the entire country.

Finally, DEFLEC09 models will be developed for all regions. These models will serve an expanding role in the GPS/INS community seeking continuity of navigation during times and places experiencing sporadic GNSS coverage.

7 Software and applications

The GEOID09 will be available on the NGS website through the Geoid web page and the NGS Geodetic Tool Kit:

http://www.ngs.noaa.gov/GEOID/

http://www.ngs.noaa.gov/TOOLS/

Descriptions and write-ups will be available on the respective pages in the Geoid respective web pages, while the data and tools to use it will be available through the Tool Kit.

8 Summary

USGG2009, GEOID09 and DEFLEC09 represent improved models for all users. USGG2009 incorporates the EGM08 reference field based on the GRACE data. It reflects improvements in terrain and processing and performs better than previous models. GEOID09 is based on this model and uses the most recent GPS-derived ellipsoidal heights on leveled bench marks (GPSBM’s) in the NGS database.

This is significant in view of the recent National Readjustment of 2007 that caused decimeters of change in ellipsoidal coordinates around the country. Use of earlier hybrid geoid models (e.g., GEOID03) will produce significant errors if used with the updated data available for NGS bench marks. Significant effort was made to ensure that GEOID09 better reflects the varying spatial distribution and quality of the GPSBM’s for each state. This effort has resulted in a hybrid model that will precisely provide the distance between NAD 83 and NAVD 88 based on the local spatial distribution. However, this does not address accuracy.

GRAV-D is being implemented in Puerto Rico and the U.S. Virgin Islands as a first step in defining a model that is both precise and accurate. The model will be available later this year and will be determined from dense, modern airborne gravity data.

Finally Deflection of the Vertical models (DEFLEC09) will be made available to assist in GPS/INS applications. These models will be developed for all regions as well.

Acknowledgments

Special thanks to Dr. Dru Smith for ideas and discussions centered on refining the Least Squares Collocation approach to better model local geoid height signals.

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