An Accuracy Assessment of the GEOID96 Geoid Height Model for the State of Ohio

Dennis G. Milbert, Ph.D.

National Geodetic Survey, NOAA

Abstract. Reliable estimates of accuracy have been difficult to produce. The most effective approaches have involved checks by Global Positioning System (GPS) ellipsoid heights measured on points with orthometric heights from spirit leveling. The accuracy of the GEOID96 model in the state of Ohio is assessed using GPS data from the Ohio High Accuracy Reference Network (HARN) survey. The GEOID96 geoid height model in the state of Ohio performs within its specified error assessment of 2.5 cm (one-sigma) over spacings of 50 km or more. There is evidence that GEOID96 may be even more accurate. A companion model, the G96SSS gravimetric geoid, contains a systematic North-South tilt of about 0.3 ppm relative to the GPS benchmarks in Ohio. This trend is well-modeled by GEOID96. Statistical tests do not show correlation with gravity coverage. Accuracy estimates of the adjusted ellipsoid height range from an optimistic 1.6 cm (one-sigma) to a pessimistic 4.7 cm (one-sigma). The accuracy and the spatial distribution of the GPS benchmarks do not support creation of a reliable map of geoid error by collocation. Recommendations are made for future surveys and data analysis. Additional gravity measurements are not recommended at this time.

Introduction

The impact of the Global Positioning System (GPS) can hardly be overstated. It has revolutionized the fields of surveying, mapping, navigation, and Geographic Information Systems (GIS). In particular, GPS has now provided surveyors a capability of making geodetic measurements with an accuracy that, in the past, could only be obtained by Federal agencies. Further, this new accuracy capability is achieved at a greater efficiency and economy than was ever possible before GPS.

The GPS is three-dimensional, so it provides heights as well as horizontal positions. The GPS heights are computed relative to a simple ellipsoidal model of the Earth; hence, they are called ellipsoidal heights. However, heights found on topographic maps, for example, are related to the gravity field of the Earth. These are called orthometric heights. The geoid height is the difference between the ellipsoidal and the orthometric heights. Geoid heights allow one to accurately relate the heights from GPS surveys to heights above mean sea level. One could say that geoid height models help supply the third dimension to GPS.

It was recognized at an early stage that the impressive efficiency of GPS surveying could be further enhanced, if it could simultaneously meet horizontal and vertical geodetic control network requirements. In one of the earliest studies, Soehngen et al. (1991), found elevations derived from GPS and the GEOID90 geoid model checking with geodetic leveling at the 5 mm level. While exceptional, this result certainly highlighted the possibility of establishing orthometric height control without the expense of leveling.

Establishing orthometric heights from GPS surveys requires careful control of the error sources. The planning of such surveys requires some knowledge of the expected accuracy of the geoid model, as well as the accuracy levels for the GPS measurements, and the existing geodetic control. Towards this end, a cooperative agreement was reached between the National Geodetic Survey (NGS) of the National Oceanic and Atmospheric Administration (NOAA), and the Ohio Department of Transportation (OHDOT). The central objective of this agreement was to establish a High Accuracy Reference Network (HARN) for the state of Ohio through a GPS survey. However, one element of the agreement was an accuracy evaluation of the geoid model for the state of Ohio. It was realized that such a study, as was performed for the Commonwealth of Virginia (Milbert 1991), would not only assess the geoid, but would also provide diagnostic information on the GPS network and level network accuracies, and indicate future survey requirements. In this way, the Ohio HARN would serve both horizontal and vertical control needs, and the maximum economic benefit could be obtained from that investment.

This paper begins with a review of the characteristics and computational procedures employed for the G96SSS and GEOID96 geoid height models. This review is particularly important, since a similar trend modeling technique is used in the Ohio data set analysis. Recent results on formal and empirical geoid accuracy estimates are then discussed, followed by remarks on the origin of the GEOID96 accuracy estimate for the United States. Then, the geoid model accuracy is analyzed for the state Ohio. This is performed with data sets from the NGS data base, as well as from a special adjustment of the Ohio HARN. The results are discussed along with rationale for the various recommendations and possibilities for network improvement. The paper ends with conclusions and recommendations for future activities.

The GEOID96 and G96SSS Geoid Height Models

The GEOID96 and G96SSS geoid height models were published by the National Geodetic Survey on October 10, 1996. These two models were released in a strategy to accommodate the disparate requirements of the surveying and mapping community and the scientific community. The G96SSS model is a purely gravimetric model. It is geocentric, relative to the GRS80 ellipsoid, and is referenced to the ITRF94(1996.0) frame. G96SSS is suited for various investigations, such as relating GPS receivers mounted in buoys to ocean circulation models.

GEOID96, on the other hand, was developed to support direct conversion between ellipsoid heights expressed in the NAD 83 (86) reference frame and orthometric heights expressed in the NAVD 88 vertical datum. Due to the non-geocentricity of the NAD 83 (86), coupled with a probable constant bias in the definition of NAVD 88, one will find discrepancies of a meter or more between a geocentric geoid model, such as G96SSS or GEOID93, and GPS on leveled benchmark values. By means of known reference frame relationships, and 2951 GPS benchmark points, a smooth conversion surface was developed to relate the G96SSS model and the GEOID96 model. Therefore, G96SSS and GEOID96 share identical high precision, high-resolution characteristics, and differ only in broader characteristics. Details on the motivation and computational techniques underlying this pair of geoid models can be found in Milbert (1995), Milbert and Smith (1996), Smith and Milbert (1997a), and Smith and Milbert (1997b).

To briefly review the computation technique, almost 1.8 million terrestrial and marine gravity data are used to obtain a high-frequency corrector to an underlying global geoid model, EGM96, in a remove-compute-restore procedure. To begin, four data grids are prepared. In the conterminous United States, these data sets are computed on a 2' by 2' regular grid extending from 24 N to 53 N and from 230 E to 294 E (66 W to 130 W). Each grid contains 871 rows and 1921 columns. The grids of EGM96 gravity anomalies and EGM96 geoid height are obtained by evaluating the EGM96 geopotential coefficients using the Clenshaw summation technique in Tscherning et al. (1983). Details about the EGM96 coefficients can be found in Lemoine et al. (1996). The grid of mean elevations is prepared by averaging 30" by 30" point elevations from the TOPO30 data set (Row and Kozleski 1991), supplemented by elevation data supplied by the Geodetic Survey Division, Geomatics Canada (GSD) and the National Imagery and Mapping Agency (NIMA, formerly DMA). This mean elevation grid is used in computing the indirect effect and in preparation of the input gravity anomaly grid.

The gravity anomaly grid contains atmospherically corrected Helmert (or Faye) anomalies referred to the GRS80 normal gravity model. Helmert anomalies are free air anomalies which have the terrain correction applied. These anomalies result from Helmert's second method of condensation, and have the attractive property that the indirect effect is small (Wichiencharoen 1982, pp. 4-7). The Helmert anomaly grid is developed in a two-step procedure. First, a grid of irregularly spaced, terrain corrected, atmospherically corrected Bouguer anomalies are processed into a Bouguer anomaly grid. Then, the Bouguer correction is restored through the mean elevation grid to yield the Helmert anomaly grid.

The gridding process, itself, uses a method of continuous curvature splines in tension (Smith and Wessel 1990) with tension parameter, T_B = 0.75. The method is one which honors the data and does not display large oscillations in areas without data coverage. However, any interpolation method will fail when data are spaced farther apart than the signal to be captured. To insulate oneself as much as possible from the influences of poor data coverage, one should attempt to remove the high frequency component of the signal. In terms of gravity data, terrain corrected, Bouguer anomalies have proven to be highly successful interpolants.

In keeping with the general computation strategy of Schwarz et al. (1990), a remove-compute-restore process is utilized. The grid of Helmert anomalies has low frequency effects removed by subtracting the EGM96 gravity anomalies. This grid of residual gravity anomalies is transformed into a grid of residual geoid heights by the "one-dimensional" FFT formulation of Haagmans et al. (1993). The final G96SSS grid is established by restoring the geoid heights computed from the EGM96 geopotential coefficients to the residual geoid heights, and adding appropriate corrections for the indirect effect.

The GEOID96 model is obtained by developing a conversion surface that incorporates the non-geocentricity of the NAD 83 (86) reference system and the NAVD 88 datum offset, as well as large scale systematic errors in the GPS height network, the level network, and the G96SSS model. First, the geocentric G96SSS model is transformed using the established relationships between ITRF94(1996.0) and NAD 83 (86) (Table 1). Next, residuals are formed between the transformed geoid model and the ellipsoid height minus orthometric height differences at 2951 points. A tilted plane (and offset) trend surface is fit to the residuals, and detrended residuals are formed. The detrended residuals are analyzed to establish an empirical covariance function, and an estimate of uncorrelated GPS ellipsoid height error. The detrended residual signal is predicted on a 30' by 30' grid by means of least-squares collocation with noise (Moritz 1980, p.102-106), and subsequently densified to 2' by 2' spacing. The conversion surface is generated by combining the predicted signal, the tilted plane and offset trend, and the reference system transformation. The GEOID96 geoid height model is computed by subtracting the smooth conversion surface from G96SSS grid.

Table 1 -- Transformation from NAD 83 (86) to ITRF94(1996.0)
X	-0.9738	± 0.0261	m
Y	+1.9453	± 0.0215	m
Z	+0.5486	± 0.0221	m
X	-0.02755	± 0.00087	arc sec
Y	-0.01005	± 0.00081	arc sec
Z	-0.01136	± 0.00066	arc sec
scale	-0.00778	± 0.00264	ppm

Figure 1 depicts a contour map of the GEOID96 geoid height model in the region of Ohio. One can see, for example, a geoid high of -33 m near Wooster (northeast Ohio) in contrast to a low of -35.5 m just southeast of Toledo. The most prominent feature in the entire area is the ridge in the southeast corner (West Virginia), which corresponds to the Appalachian mountains. The geoid structure in Ohio is seen to be more complicated in the western half of the state. To help visualize the geoid model, a mesh surface with a perspective view to the south is provided in Figure 2 .

Contour and surface plots are not provided for the G96SSS model, since the images appear to be virtually identical to those of GEOID96. Instead, a contour plot of the differences between G96SSS and the GEOID96 model is supplied in Figure 3 . The geoid difference portrayed in Figure 3 is a portion of the smooth conversion surface used to generate the GEOID96 model from the G96SSS model. The northwest tilt is due to the effect of the reference frame differences shown in Table 1 on ellipsoid height. Variations about that tilt are induced by the GPS benchmarks in the region.

Formal and Empirical Accuracy Estimates

At this point it is useful to review the results of error propagation studies of geoid height computation. One study which relates directly to this assessment is that of Kearsley (1986). Kearsley concludes that it is possible to compute geoid height differences to better than 5 cm over 100 km when using 6' by 6' mean gravity anomalies with no more than 3-mGal random error when systematic gravity and leveling errors are kept at less than a 0.3-mGal level. In fact, inspection of Kearsley's Table 3 indicates the random error component of relative geoid accuracy to be less than 3 cm at 100 km and only 1.5 cm at 30 km spacing. These results are directly proportional to the assumed random error of the mean gravity anomalies. If 1-mGal gravity anomalies were assumed instead (which is a common accuracy assessment), then the relative geoid error would be 3 times smaller.

Figure 4 displays a schematic of the coverage of the gravity data in the National Geodetic Survey (NGS) data base and the gravity data supplied by the National Imagery and Mapping Agency (NIMA, formerly DMA), for Ohio and the surrounding region. One immediately notes the coverage in the western part of Ohio to be denser than in the eastern part. This is particularly fortunate, since the geoid is seen to be more complicated in western Ohio. To relate the coverage to the gravity gridding and geoid computation procedures used in GEOID96, the gravity is organized into 2' by 2' cells. Figure 5 shows the cell coverage, where a point on the figure indicates one or more gravity values for the cell in question. This portrayal again shows the sparser coverage in the western part of Ohio.

To relate the gravity coverage to the Kearsley study, Figure 6 was generated. This shows the gravity organized into 6' by 6' cells. In this portrayal, the gravity coverage looks more uniform. Even so, there are a few cells in Ohio where there is no gravity measurement. The coverage over the entire region is 92.4%.

The question to be asked is how well can the gravity anomalies be interpolated in the eastern part of Ohio? There is no easy answer from a theoretical standpoint. The measured free air anomalies in the gravity data sets have typical accuracy assessments of 1-mGal. One should expect the interpolants to display comparable random error. However, there can be localized terrain variations which could possibly cause an interpolated gravity anomaly to exceed 1-mGal random error. Recall that terrain corrections computed from a 30" by 30" elevation grid were used to generate the complete Bouguer anomalies that were input to the gridding procedure. This suggests that the gravity gridding procedure may be very well behaved.

Two recent studies of gravity densification and its impact on geoid computation provide some empirical evidence to the gravity accuracy question. Parks and Milbert (1995) report on a gravity densification survey in San Diego County, California. The additional gravity data lead to peak gravity improvements of 4 to 10 mGal in the centers of the void areas, and localized geoid changes of 2 to 3 cm within those voids. Given the rugged terrain in San Diego County, with mountain peaks of 1 to 2 km of elevation in fairly close proximity to terrain near sea level, this study is not completely applicable to the situation in eastern Ohio. However, it does provide an approximate "worst case" scenario for gravity gridding performance.

In contrast, a similar study was performed by Balde (1995) at the Forth Berthold Sioux Indian Reservation in west central North Dakota. In this work, additional gravity data supported data grids with three regions of 1' by 1' resolution. This gravity grid was then used to compute a local geoid grid. No new geoid structure seems evident in the three regions of 1' by 1' resolution. Balde reports no improvement over the previous GEOID93 model. Moreover, he does link this finding to the subdued topographic relief of North Dakota. Given the rather benign topography of Ohio, this study appears more relevant than that of San Diego.

Accuracy Estimates from the GEOID96 Geoid Height Model Computation

One product of the GEOID96 geoid height model computation is an estimate of the fit of the GEOID96 model to the GPS benchmarks. As reported in Milbert (1995), Milbert and Smith (1996), and Smith and Milbert (1997b), covariance statistics as a function of distance are computed for geoid model residuals to GPS benchmark values, and empirical covariance functions are developed. The results for GEOID96 and 2951 GPS benchmarks are displayed in Figure 7 . Covariances are statistics of errors common to two different variables; in this case, residuals at a given fixed separation. Covariances have units of squared error, and can take both positive and negative values, indicating direct or inverse relationships. To provide a somewhat more intuitive measure, the square root of the absolute value of the covariance (with the sign of the square root set to the sign of the original covariance) is plotted against distance. This allows the use of linear measures in both axes of Figure 7 . The plus symbols indicate the empirical statistics, and the solid line is a Gaussian function with a peak of 2.5 cm and a characteristic length of 40 km.

At zero distance, the empirical ("plus") statistic is simply the 5.5 cm RMS of the GPS benchmarks differences with GEOID96. The empirical statistics are seen to sharply drop from 5.5 cm at d = 0 km, down to 2.5 cm at d = 5 km. This reduction is evidence of an uncorrelated (white-noise) process. The source of the 5.5 cm of uncorrelated error is felt to be random error in the GPS ellipsoidal heights. The sources of the 2.5 cm of correlated error are not obvious, but geoid error is certainly one component. It is conceivable that adjusted ellipsoidal heights can also contribute to the correlated error.

In closing this section two remarks can be made. First, the GPS benchmark data set used to compute these empirical statistics was also the data set used to develop the conversion surface that lead to GEOID96. For this reason, the statistics are not absolute guides to GEOID96 accuracy. Smith and Milbert (1997b) show instances in the Pacific Northwest where the conversion surface absorbs long-wavelength systematic error from the old GPS networks, biasing GEOID96 relative to the Canadian GSD95 geoid model. However, GEOID96 does relate the biased ellipsoid heights in the region to the NAVD 88 orthometric heights at these empirical accuracy levels. Because the conversion surface is so smooth, the empirical statistics accurately reflect error processes that are significantly shorter than 400 km.

Second, the GEOID96 empirical statistics are averages over the entire conterminous United States. Therefore, one should be prepared to see better or worse results in any specific region of the country. As related in the introduction, this issue forms one motivation for this study.

Accuracy Estimates from the Ohio High Accuracy Reference Network

In this section, comparisons are made between geoid models and different sets of GPS ellipsoid heights co-located with orthometric heights computed from differential leveling. It must be realized that in any such comparison, GPS networks are less accurate in determining ellipsoidal height than horizontal position. And, one must be prepared for the possibilities of errors in the level network, particularly if the leveling is not of recent vintage. Isolation of problems in a level network is often hampered by the lack of redundancy, even though the component level lines are double run. For these reasons, the approach is one of outlier detection within a given geoid/GPS benchmark misclosure data set. Outliers are detected in part by excessive magnitude, and in part by excessive relative magnitudes when compared to their closest neighbors.

Ohio GPS Benchmark Data Set

Initially, a GPS on leveled benchmark data set was prepared from a retrieval from the main NGS data base. Such a retrieval is keyed on points having adjusted NAD 83 ellipsoid heights measured by First-Order (or higher) GPS specifications, and having adjusted NAVD 88 orthometric heights that are felt "reliable" (i.e. not posted, not single spur, etc...). This first data set is denoted OHIO62, since it contains 62 GPS benchmarks in and near Ohio.

Appendix 1 contains the output from an evaluation program (EVAL2) which processed the OHIO62 evaluation file using the GEOID96 model. In the first pass, geoid height estimates are obtained by differencing the GPS ellipsoid heights and the bench mark orthometric heights. Geoid height model values are computed through biquadratic interpolation from the specified geoid height model grid. In addition, differences between the measured geoid heights and the model geoid heights are accumulated in this first pass. Because of the possibility of datum offsets in ellipsoidal, orthometric, and/or geoid heights, the second pass of the program computes and removes the average discrepancy between the measured and modeled geoid heights. It is reasonable to interpret the results as relative height errors. As discussed above, these discrepancies will contain error contributions from GPS and leveling, as well as from the geoid model, itself.

The columns in the printout display from program EVAL2 shown in Appendix 1 are:

1. Geodetic latitude, decimal degrees
2. Geodetic longitude, decimal degrees, positive east
3. Ellipsoidal height, meters above GRS80 ellipsoid
4. Orthometric height, meters (NAVD88 in this report)
5. "measured" geoid height with the average discrepancy removed ("diff"), meters
6. Geoid height interpolated from designated model, meters
7. Interpolated geoid height - GPS benchmark value for geoid height, meters
8. Station serial number (arbitrary for this study) and the station name

Inspection of Appendix 1 shows that the average discrepancy (labeled "geoid offset") is less than 1 mm. While the extremely low value is somewhat accidental, it does reflect one of the major benefits in using the GEOID96 model; NAD 83 non-geocentricity and the NAVD 88 datum offset are incorporated into the geoid height model (Milbert and Smith 1996). The RMS of the 62 points is 5.97 cm, which is higher than the 5.5 cm RMS achieved for the entire United States data set. Closer inspection shows an obvious outlier at point number 61, designated W 118, with a misfit of -33.6 cm. For a graphic portrayal, the discrepancies (column 7) for the OHIO62 data set are plotted in units of cm in Figure 8 . The outlier is in the north, just across the Ohio-Michigan border.

The outlier at W 118 characterizes a typical situation encountered in comparison of GPS and level networks with high resolution geoid height models. The source of the error is not clear, but it is evident that the error is not due to the geoid height model. Based in part on experiences in Parks and Milbert (1995), a local gravity error of 100 mGal or more (a 100 sigma outlier) would be needed to induce a 33 cm geoid error. Inspection of the gravity coverage in Figure 4 shows extremely dense coverage. This gravity error would have to have occurred repeatedly. Further, such a hypothetical gravity error would have been detected in a quality control checks involving comparisons with the EGM96 gravity field, in automatic gravity spike detection, and in visual inspection of digitally enhanced images of the gravity grids.

A pair of outliers is seen to the south, just across the Ohio-Kentucky border. The stations, I71 T 34 and L 34 have discrepancies of -10.5 cm and -7.4 cm respectively. It is felt that these discrepancies are excessive when compared to the +4.0 cm discrepancy just to the north at 7008. The relative discrepancy is 14.5 cm in magnitude over a spacing of less than 30 km. By contrast, the GEOID96 empirical covariance function for the entire United States indicates one can expect a relative discrepancy of a little less than 2.5 cm (one sigma) over such a spacing (before inclusion of random GPS ellipsoid height error).

Of particular note is the 10.1 cm outlier at DAY G. Since this point lies in the GPS benchmark cluster near Dayton, these points are plotted at a larger scale in Figure 9 . It is seen that DAY G lies only 1 km from a cluster of 4 other GPS benchmarks. At this scale, each division represents only 0.01 degree (about 1 km) of latitude and/or longitude. Also, recall that the GEOID96 model grid spacing is only 2' by 2' (0.0333 degree). The relative discrepancy of 16.5 cm between DAY G and DAYT A is not due to the GEOID96 model, since the model in such a local area is only a flat plane.

The orthometric height at DAY G is of First-Order Class II accuracy, adjusted in January 1994. It is stamped with the year 1990, and was recovered in 1992 and 1993. The ellipsoidal height is derived from A-Order GPS measurements, adjusted in April 1993 as part of the New England HARN survey (Maralyn Vorhauer, 1997, private communication). It should be emphasized that the station, DAY G, was not a part of the Ohio HARN 1995 survey and adjustment, GPS882 (Edwards 1996). As such, DAY G derives its ellipsoid height from older, long baseline GPS measurements that predate the availability of CORS stations, GPS antenna phase center variation models, or NGS precise orbits.

Because of the problematic characteristics of the OHIO62 GPS benchmark data set, no further analysis is performed with these data. It is anticipated that when the statewide readjustment of Ohio is performed, that the discrepancies between the existing, older, GPS ellipsoid heights, and those of the newer HARN adjustment will be resolved.

CORS-Fixed GPS Benchmark Data Set

In order to achieve a more homogenous GPS benchmark data set, the A-Order, B-Order, and First-Order GPS surveys constituting the Ohio HARN were combined in a special adjustment. In this computation, performed by Ms. Gloria Edwards, NGS, the NAD 83 (1996.0) coordinates of 3 CORS stations were held fixed.

Name                          Latitude          Longitude          Ellipsoid Ht
DETROIT 1                     42 17 50.42501 N  083 05 43.04993 W  146.277 m
GAITHERSBURG CORS-L1 PHS CTR  39 08 02.34060 N  077 13 15.51927 W  109.047 m
ST LOUIS 2                    38 36 40.70827 N  089 45 32.02618 W  166.777 m

Appendix 2 reproduces the summary statistics page generated by program ADJUST (Milbert and Kass 1987). In this data set, 487 GPS vectors (with inter-vector correlations induced by common session reductions) were adjusted to establish the coordinates for 244 points. While this was not a minimally constrained adjustment, the CORS coordinates were felt to be so accurate that they would not bias the adjustment. The ratio of GPS vectors to established points reflects the dual occupation requirement for such surveys. The large variance of unit weight is consistent with those obtained by other GPS network adjustments, and is generally indicative of overly optimistic assessments of GPS vector accuracies. Of interest, the variance of unit weight obtained in this adjustment is smaller than that obtained in the minimally constrained adjustment of the B-Order vectors in GPS882 (Edwards 1996). With this adjustment we now have a consistent set of coordinates in an NAD 83 (1996.0) reference system, established with recent GPS measurements and data reduction procedures.

Estimates of ellipsoidal height accuracy relative to the CORS stations are obtained by formal error propagation in program ADJUST. Although not reproduced here, the a priori standard deviations of adjusted ellipsoid heights range from about 3 to 4 mm for the 244 points. However, since assigned GPS vector accuracies are much too optimistic, it is more appropriate to consider the a posteriori statistics. When scaling by the a posteriori standard deviation of unit weight, 11.81, it is seen the adjusted ellipsoid height accuracy ranges around 3.5 to 4.7 cm (one-sigma).

However, one should be somewhat cautious in the use of a posteriori statistics from GPS adjustments. This is because the statistics are typically obtained by simple scaling of the vector covariance matrix, which is expressed in an X,Y,Z system. It is known that GPS measurements are less accurate in height, than in latitude or longitude; usually a factor of 2 to 3 times less accurate. But, the a priori variances and covariances assigned to vectors by GPS reduction software might not accurately reflect this ratio. In such circumstances, it is not appropriate to scale the horizontal and the vertical components of GPS error by the same amount. Instead, separate variance components for scaling of horizontal and vertical error should be estimated. For this reason, the a posteriori height accuracy of 3.5 to 4.7 cm is not considered completely definitive, at this point. It is useful to consider the GPS vector adjustment residuals.

GPS vector measurements are typically expressed as X,Y,Z Cartesian coordinate triples. Program ADJUST computes observation residuals in X,Y,Z, but also rotates each residual vector into a local horizon system (North, East, and Up). A scatter plot of the 487 height residuals is shown in Figure 10 . The residual cluster reflects an RMS of 1.6 cm, also shown in Appendix 2 . As has been seen in many other GPS adjustments, the measurement scatter in the vertical is roughly triple that found in either the North or East components. To assist in the possibility of future survey projects, the largest ellipsoid height residuals are reproduced in Appendix 3 .

It must be emphasized that residuals from a least-squares adjustment underestimate the true measurement error. Since the GPS stations were established in a dual occupation scheme; in general, each ellipsoidal height is determined from only two measurements. A more realistic assessment of the actual error of individual vectors would be roughly 40% larger than the amount indicated by the residual statistics, or 2.3 cm (one-sigma) of random error in ellipsoid height determinations. The situation is alleviated somewhat by the fact that we are interested in adjusted ellipsoidal heights. If one assumes the absence of systematic errors, the adjusted values should be more accurate than the individual determinations by a compensating 40%. Although this figure is probably optimistic, the adjusted ellipsoid heights might be considered to have 1.6 cm (one-sigma) accuracy with respect to the CORS stations.

Now, having established a consistent set of ellipsoidal heights, with a range for the error statistics, a new evaluation file of GPS benchmarks was generated. This file is designated CORS51, since it contains 51 GPS benchmarks and is obtained from the CORS-fixed special adjustment. The names and positions of the 11 stations which are in the OHIO62 file, but not in the CORS51 file, are reproduced in Table 3. (Note: Station M 176 was inadvertently omitted in preparing the CORS51 file.) Appendix 1 shows that, in general, these points have larger than average discrepancies. This is due to ellipsoidal height inconsistency, rather than geoid error. As mentioned earlier, it is anticipated that these inconsistencies will be resolved when the statewide readjustment of Ohio is performed.

Name            Adjust Date   Latitude          Longitude         Ellipsoid Ht
ARP              Dec. 1995    39 20 44.51901 N  81 26 15.86971 W  221.403 m
DAY G            Apr. 1993    39 54 51.85283 N  84 12 45.65097 W  269.329 m
DAYT A           May  1995    39 54 18.27972 N  84 12 16.81024 W  270.445 m
DAYT B           May  1995    39 54 14.31844 N  84 12 06.70869 W  270.018 m
I71 T 34         Sep. 1994    38 52 03.98977 N  84 37 28.17301 W  243.308 m
L 34             Sep. 1994    38 40 58.46624 N  84 04 01.40429 W  256.908 m
M 176            Aug. 1996    41 03 13.21094 N  81 59 11.33577 W  311.471 m
M 319            Jun. 1995    41 27 48.60139 N  82 11 33.80166 W  147.252 m
VANDALIA AZ MK   May  1995    39 54 03.81041 N  84 11 45.50825 W  267.192 m
W 118            Sep. 1995    41 45 16.66798 N  84 09 02.71374 W  208.033 m
ZOB A            Jun. 1995    41 17 44.60621 N  82 12 20.46681 W  211.243 m

Appendix 4 contains the output from an evaluation program (EVAL2) which processed the CORS51 evaluation file using the GEOID96 model; as was done earlier with the OHIO62 data set. The average discrepancy is larger, but still only 1.2 cm for the entire state. The RMS discrepancy of the 51 points has dropped significantly, now down to 3.3 cm. Note that this reduction was achieved solely by computing a consistent set of ellipsoidal heights. Closer inspection shows only one outlier at point number 43, designated E 338, with a misfit of 9.2 cm. This point was also an outlier in the earlier OHIO62 data set. For a graphic portrayal, the discrepancies (column 7) for the CORS51 data set are plotted in units of cm in Figure 11 . The outlier is in the south central part of the state, near Chillicothe.

The outlier at E 338 has no obvious cause. It has nearby gravity measurements. It is not found in the list of large ellipsoid height residuals in Appendix 3 . Detailed investigations by Dave Zilkoski and Bruce Ward, NGS, have not uncovered any obvious problems with the leveling. However, the leveling is older, and it is possible that a mark can be disturbed in the interval between the leveling and the GPS occupations (Dave Zilkoski, 1997, private communication).

Because of the magnitude of the discrepancy at E 338, this point is deleted from further analysis. The resulting evaluation file is denoted as CORS50, since it now contains 50 GPS benchmark points. The output from the EVAL2 program relative to GEOID96 is reproduced in Appendix 5 , and is plotted in Figure 12 for completeness. The deletion of the outlier changed the average discrepancy to 1.0 cm, and the RMS discrepancy to 3.1 cm.

When considering the average discrepancy of 1.0 cm, it must be realized that the ellipsoid heights from the special adjustment, as a group, are subject to any systematic errors on the baselines connecting the Ohio HARN to the CORS points in Table 2. When one inspects the height residual list in Appendix 3 , one sees large height residuals involving both ST LOUIS 2 and DETROIT 1. It is possible that a portion of the 1.0 cm average discrepancy is due to these residuals slightly shifting the set of ellipsoidal heights relative to the CORS stations.

The RMS discrepancy of 3.1 cm is a combination of GPS, leveling, and geoid error. Recall, accuracy assessment of the adjusted ellipsoidal heights ranged from a 3.5 to 4.7 cm value from a posteriori error propagation to a 1.6 cm value from study of GPS residuals. If one accepts the former accuracies, then the RMS discrepancy of 3.1 cm is completely within the expected noise of the GPS heights. No geoid or leveling error is discernable. If the latter accuracy of 1.6 cm for ellipsoid heights is assumed (which implies uncorrelated error processes in the GPS ellipsoidal heights), this would indicate 2.6 cm (one sigma) error for the geoid and the leveling in Ohio. This quantity agrees with the 2.6 cm and the 2.5 cm values obtained from the empirical covariance functions for the entire United States in Milbert and Smith (1996) and Smith and Milbert (1997b), respectively.

Accuracy Analysis Related to Gravity Distribution

The foregoing results suggest that, even in the worst case, the GEOID96 model in Ohio is performing at an accuracy consistent with the United States as a whole. And, it can not be ruled out that the GEOID96 model in Ohio is more accurate. However, the unique gravity distribution in Ohio, with denser gravity coverage in the west ( Figures 4 and 5 ), invites additional analysis. Is the GEOID96 model in west Ohio significantly more accurate than the GEOID96 model in east Ohio?

To perform this test, the CORS50 GPS benchmark evaluation file is split into two subsets. A CORSW file was created which contains the points west of 277 E (83 W). The CORSE file contains the remaining points. The demarcation was chosen to generally conform to the boundary of gravity coverage density seen in Figures 4 and 5 . It was found that this demarcation lead to 25 points in each of the subset files. Separate EVAL2 program runs were made for the subset files and GEOID96, with the key elements summarized in Table 4.

Data Set	Number of Points, N	Average Discrepancy, x	RMS About Average, s
CORS-East	25	1.49 cm	3.13 cm
CORS-West	25	0.57 cm	3.03 cm

Table 4. Summary Statistics from Evaluation File Subsets

The first test is to see if the difference in average discrepancies is significant. By Dixon and Massey (1969, pp. 109-119), formulate a null hypothesis:

H₀: x_e = x_w (average discrepancies are equal)

and perform a two-sided test against the alternative hypothesis that the discrepancies are not equal. We compute the z statistic (ibid, pg. 114) equal to 1.05. For large sample size, z is approximately normal even if the population, itself, is not normally distributed. Allowing an alpha=0.10, the critical value to be exceeded is ±1.645. The null hypothesis can not be rejected.

The second test is to see if the difference in variances, s², is significant. Formulate a null hypothesis:

H₀: s_e² = s_w² (variances are equal)

and perform a two-sided test against the alternative hypothesis that the variances are not equal. We compute the F statistic (ibid, pg. 110) equal to 1.067. This is compared against F(24,24), where the numbers in parenthesis indicate the number of degrees of freedom in the numerator and demoninator of the F statistic. Allowing an alpha=0.10, the critical value to be exceeded is 1.98 Once again, the null hypothesis can not be rejected.

The results of the first test indicate that there is no detectable systematic offset in the GEOID96 model between the eastern half and western half of Ohio. The results of the second test indicate the GEOID96 model is not noticeably noisier in the eastern or western halves. The tests support an assertion that the discrepancies computed from two subsets, CORSE and CORSW, share a common distribution. These results are consistent with the numbers by Kearsley (1986) regarding adequacy of 6' by 6' gravity coverage, and are consistent with the lack of geoid improvement found by Balde (1995) when densifying gravity to 1' by 1' in North Dakota. If one accepts the a posteriori estimates of ellipsoid height accuracy, then the tests were largely a comparision of GPS ellipsoid height errors. Even under that assumption, the previous statements are still valid.

GEOID96 Conversion Surface Behavior in Ohio

The preceding results have been developed from the GEOID96 model. Recall, however, that GEOID96 was computed from the gravimetric geoid model, G96SSS, by means of a conversion surface ( Figure 3 ). One component of the conversion was known reference frame relationships between NAD 83 (86) and ITRF94(1996.0). Another component was empirical, developed from the discrepancies between the transformed GPS on benchmarks. For the entire United States, the empirical component of the conversion surface was found to be very smooth, and well-modeled by a Gaussian covariance function of characteristic length L = 400 km. It is possible that a different covariance function, developed locally for Ohio, might have different properties. If so, this could lead to a slightly different conversion surface, and a variant of GEOID96. It might identify potential geoid errors not resolved from the L = 400 km function.

To begin, the geodetic latitudes, longitudes, and ellipsoidal heights of the CORS50 data set were transformed to the ITRF94(1996.0) frame (Table 1). This data set is denoted ITRF50. Note that the scale difference was not applied since the scale of the GPS baselines is felt to be accurate. Next, the evaluation program (EVAL2) is executed to generate misclosures between the G96SSS geoid model and the ITRF50 file. The output is reproduced in Appendix 6 . The average discrepancy is seen to be 50.0 cm. This sizable number is to be expected. The major component of that average discrepancy is the bias in the NAVD 88 datum. Another component is a 12.3 cm offset of G96SSS from an ideal tide-free global geopotential (Smith and Milbert 1997b). Also of interest is the RMS discrepancy, which is 4.1 cm. This figure is in contrast to the 3.1 cm RMS obtained for GEOID96 and the CORS51 data set.

The next step is to identify and remove any trend before computing a local empirical covariance function. A plane is fit to the discrepancies of Appendix 6 . The results are summarized in Table 5. Since the trend is almost exactly North-South, the discrepancies are shown in a scatter plot against latitude in Figure 13 . The trend is clear despite the noise.

          Number of Points              50
          Tilt of Plane                 0.28 PPM
          Azimuth of Maximum Tilt       351 degrees
          Average Discrepancy           50.06 cm
          Pre-fit RMS                   4.09 cm (about average discrepancy)
          Post-fit RMS                  3.12 cm

It is seen that by merely removing a tilted plane from GPS benchmark discrepancies relative to G96SSS, one obtains an RMS of fit equivalent to that from GEOID96. This RMS indicates that for a state the size of Ohio, the original L = 400 km conversion surface has much in common with a tilted plane. This is also evident when inspecting Figure 3 . Of more interest, the tilt relative to the G96SSS gravimetric geoid is essentially North-South at 0.3 PPM. This behavior is also seen in a figure of detrended residuals in Smith and Milbert (1997b), which is reproduced here as Figure 14 . The trend seems to form a transition between a circular feature centered in western Virginia, and an elongated feature over northern Wisconsin. If the trend is due to geoid error, then it seems that the error source(s) lies beyond the borders of Ohio. It is also remarkable that the direction of this trend is at a right angle to the East-West difference in gravity coverage ( Figures 4 and 5 ). This rules out gravity coverage variations within Ohio as a cause.

The detrended discrepancies are then used to compute a local empirical covariance function for Ohio. Note that we can only proceed with this step if we provisionally accept the 1.6 cm figure for adjusted ellipsoid height accuracy. If one accepts the a posteriori estimates of 3.5 to 4.7 cm of error, then one would only be trying to model GPS ellipsoid height error.

However, it is at this stage that difficulties began to surface. Slightly different estimates of the characteristic length of the covariance function were obtained depending upon the size of the bins used to accumulate the covariance statistics. Figure 15a illustrates the empirical statistics (plus symbols) for radial bin sizes of 25 km. The solid line is the covariance function with a length, L = 15, and a peak at 2.5 cm. However, there are only 30 samples in the bin from 12.5 km to 37.5 km. Figure 15b portrays the empirical statistics (plus symbols) for bin sizes of 30 km. The solid line is the empirical covariance function with a length, L = 20, and a peak at 2.5 cm. In this case, there are 45 samples in the bin from 15 km to 45 km. For both functions, the RMS is 3.1 cm; and we have adopted, under an assumption of uncorrelated ellipsoid height determinations, ellipsoid height error to be 1.6 cm. Therefore, the peaks of the local empirical functions were set to 2.5 cm in both cases. Since the sampling is larger for the 30 km radial bins, the L = 20 km empirical covariance function ( Figure 15b ) is selected.

Based upon the set of detrended discrepancies, the local empirical covariance function selected above, and a random discrepancy noise component of 1.6 cm, least-squares collocation was used to predict a local discrepancy surface. This follows the general approach performed in computing the conversion surface for the GEOID96 model. The output listing from the collocation program, CGRID, is provided in Appendix 7 , and the detail statistics for the 50 data points in Appendix 8 . It is seen that the RMS of fit is only 1.1 cm. This quantity is too small when compared to the (optimistic) assigned noise component of 1.6 cm. This difference is a symptom that error in adjusted ellipsoid height is being absorbed into predicted signal, the local discrepancy surface. At the GPS benchmarks, the predicted discrepancies (signal) range from +4.7 to -3.9 cm. A contour plot of the surface is shown in Figure 16 , and a perspective view of the surface is provided in Figure 17 .

Figures 16 and 17 illustrate why an abnormally small RMS of fit was obtained in the collocation solution. The GPS benchmark station spacing of the ITRF50 data set is generally much greater than the 20 km characteristic length obtained from the empirical covariance function statistics. Therefore, significant areas have zero predicted discrepancy, due to the absence of GPS benchmark data and the rapid decay of the covariance function. Because of the rapid decay, the discrepancy surface is able to undulate quickly enough to absorb most of the measured discrepancies, whatever their sources, leaving only a 1.1 cm random part. The predicted signal range of -3.9 to +4.7 cm is significantly larger than the 2 to 3 cm geoid variation in the gravity densification project reported by Parks and Milbert (1995). Recall that those were peak values in centers of void areas in a mountainous region. The predicted signals are excessive for gravity-induced geoid height errors.

Based on these results, it is not felt that a reliable map of residual geoid error can be developed from the data at hand. This does not imply that GEOID96 and its conversion surface are perfect. Rather, more accurate GPS on leveled benchmark data at a much finer (~20 km or less) spacing would be required to resolve any such hypothetical fine-scale geoid error.

Discussion

The general result that can be gleaned from this analysis is that the GEOID96 geoid height model in the state of Ohio is performing within its specified error assessment of 2.5 cm (one-sigma) over spacings of 50 km or more. It is quite possible that the relative geoid accuracy in Ohio is better, perhaps 1 cm, based upon Kearsley (1986) and 1-mGal gravity anomaly interpolants. This suspicion is also based, in part, on the much smaller local empirical covariance function characteristic lengths of L = 15 km to L = 20 km ( Figures 15a and 15b ), compared to that of GEOID96 with L = 40 km ( Figure 7 ). However, the CORS50 data set is not accurate enough, nor dense enough, to prove or disprove such a hypothesis. This is particularly true if one adopts the larger a posteriori error statistics instead of the RMS-derived error statistics.

A second, important, result from this study is to highlight particular problem locations that could profit from more detailed analysis, data reduction, and/or resurvey. These are now listed:

• The largest problem, the -33.6 cm discrepancy at W 118, has undergone some preliminary research. It seems to be due to a combination of an abnormally large distribution correction in the leveling and possible frost heave disturbing the mark between the last leveling and the GPS occupation.





• A number of height discrepancies that were unique to the OHIO62 may be resolved when the statewide readjustment is performed. Such a readjustment could allow the GPS benchmarks in question to get new ellipsoid heights. After such analysis, specific recommendations could then be made about the points in Table 3. However, it can be stated that the readjustment will not correct the height discrepancy at DAY G. This is because DAY G does not possess a direct connection to the Ohio HARN survey. It is recommended that DAY G (or some nearby point, accurately connected to DAY G) be connected to the HARN by GPS measurements.





• Another large discrepancy, 9.2 cm at E 338, could be checked. E 338 does not appear in the list of large residuals of Appendix 3 . It is likely that the ellipsoidal height is sound, although an additional GPS determination could be considered. The gravity anomalies in the vicinity of E 338 only differ by ±2 mGal, indicating sub-centimeter geoid precision. In-house review of the level line shows no abnormalities. However, the leveling is older, and the mark could have been disturbed (Dave Zilkoski, 1997, private communication).





• Detailed analysis of the CORS-fixed GPS adjustment, which generated the ellipsoidal heights for the data sets of CORS51 and CORS50, should be performed. Of particular concern are the large residuals in Appendix 3 . Some of the residuals involve connections to the CORS stations. Re-reduction or resurvey of those lines might be appropriate. One can also see repeated instances of stations (such as DEF 66 or K 323) appearing in Appendix 3 . It is possible that one or two errors of significant size are being adjusted into other nearby measurements, causing those measurements to take larger residuals. More detailed analysis of the GPS adjustment and results from the GPS reductions might isolate the problems.

It is also possible that study of the patterns of residuals may not isolate problem GPS vectors. Recall that the basic structure of the survey was two occupations per survey point. While this amount of redundancy can discern a problem (as opposed to "no-check" geometries), it will be difficult to pinpoint specific problem vectors. Unfortunately, when there is less redundancy, the problem can be sizable without generating large residuals. This situation is related to the fact that residuals underestimate measurement error. The degree to which this occurs is related to the amount of network redundancy. And, of course, if a large error is adjusted into nearby measurements, then the residuals on such nearby measurements will overestimate the true errors. And, in lower redundancy situations, the residuals will become more correlated with one another.

This lower redundancy issue, with its associated impact on the a posteriori error assessments, may have lead to the situation encountered when developing a local empirical covariance function and discrepancy surface by collocation with noise. In Figures 16 and 17 , some structure is seen in central Ohio and western Ohio. Yet, western Ohio is exceptionally well covered with gravity ( Figures 4 and 5 ). And, the features in central Ohio are along the edge of very dense gravity coverage, rather than being more distant from the dense gravity. It is not likely that the surface is a map of geoid error. While it could be a map of level network error, it is likely that the larger features are mostly due to GPS ellipsoid height error. The magnitudes are consistent with the 3.5 to 4.7 cm range of the a posteriori error.

To continue this thought, consider the empirical covariance function computed using the GEOID96 model with 2951 GPS benchmarks displayed in Figure 7 . There is no explanation for why the function peaks at 2.5 cm (one-sigma), nor why the characteristic length is L = 40 km. To be conservative, accuracy statements were assigned to GEOID96 assuming that the entire content of correlated error is due to geoid model error. It is unknown how much of the correlated error could be due to GPS. It is felt that the leveling contribution would not exceed 1 cm (one-sigma), although it could decorrelate over 40 km, about one side of a level loop (Dave Zilkoski, 1997, private communication). It can be stated that the discrepancy RMS, now at 5.5 cm, has steadily decreased since experiments began in 1993. This is because the RMS is dominated by random GPS ellipsoid height error, and the GPS surveys have gotten more accurate over the intervening years.

Further research is possible. One approach would be to perform an analysis of variance, where discrepancy subsets could be gathered for combinations of older and newer GPS and for mountainous and flat terrain. This could help separate the influence of geoid error, which is correlated with mountains, from the influence of ellipsoid height error, which is correlated with age of GPS surveys. In addition, one could also perform analysis of variance with the Ohio HARN GPS adjustment, where distinct horizontal and vertical scale factors for the covariance matrix would be estimated. This would lead to more realistic estimates of the adjusted GPS ellipsoid height error in Ohio.

Recommendations and Conclusions

It is recommended that the orthometric height at W 118 be recomputed by means of GPS and GEOID96.

It is recommended that the height discrepancies unique to OHIO62 be resolved as much as possible when the statewide readjustment is performed. It is also recommended that DAY G (or some nearby point, accurately connected to DAY G) be connected to the HARN by GPS measurements.

It is recommended that the ellipsoidal height at E 338 be remeasured by means of GPS, and the mark be checked for possible disturbances since the last leveling measurements.

It is recommended that additional investigation be performed on the large residuals associated with the CORS50 GPS adjustment. Such investigation may identify problem vectors that should be re-reduced or reobserved.

It is not recommended that gravity coverage in Ohio be densified at this time. While a complete uniform coverage of 2' by 2' spacing is ideal, these results and those of prior studies indicate that addditional gravity measurements in Ohio would not have a significant effect on a new geoid model.

It can be safely concluded that the GEOID96 geoid height model in the state of Ohio is performing within its specified error assessment of 2.5 cm (one-sigma) over spacings of 50 km or more. It is also possible that the relative geoid accuracy in Ohio is better.

It was seen that the G96SSS gravimetric geoid model contains a systematic North-South tilt of about 0.3 ppm relative to the GPS benchmarks in Ohio. This trend is probably due to G96SSS error, but the origin is external to the state. The trend is well-modeled by the conversion surface incorporated into GEOID96.

The estimated accuracy of the adjusted ellipsoid heights from the CORS-fixed GPS adjustment ranged from a 3.5 to 4.7 cm value from a posteriori error propagation to a 1.6 cm value from study of GPS residuals. The smaller quantity implied a local empirical covariance function that was somewhat consistent with that of the United States as a whole. However, the difficulties in developing a reliable map of geoid error by collocation show that the smaller ellipsoid height error estimate is too optimistic. This does not imply that GEOID96 and its conversion surface are perfect. More accurate GPS ellipsoid heights on leveled benchmark data at a much finer (~20 km or less) spacing would be required to resolve any such hypothetical fine-scale geoid error.

It is concluded that additional study involving analysis of variance of GPS benchmark/geoid model discrepancies be performed. Such research would involve the national data set as well as the subset for Ohio.

Acknowledgments

This study incorporates the contributions of numerous NGS employees involved in the creation and evaluation of the gravity, NAVD 88, and GPS data sets. In particular, Dr. Dru Smith is the co-author of the G96SSS and GEOID96 models. Ms. Gloria Edwards, Network Analysis Branch, NGS, was instrumental in computing the special GPS network adjustments used in this report. Mr. Dave Zilkoski and Mr. Bruce Ward, Spatial Reference System Division, NGS, analyzed the leveling networks of problem points in and around Ohio. The National Imagery and Mapping Agency (NIMA, formerly DMA) provided a major portion of the NGS land gravity data, and was instrumental in the creation of various 3" and 30" digital elevation data grids in use today. NIMA was also a partner in a joint project with Goddard Space Flight Center, that developed the EGM96 global geopotential model. Dr. Walter Smith, NOAA, provided altimeter-derived gravity anomalies used in the G96SSS and GEOID96 models. This work was funded in part by a cooperative agreement between the National Oceanic and Atmospheric Administration and the Ohio Department of Transportation.

References

Balde, M., 1995: Vertical control by GPS satellite surveying rapid methods and high resolution local geoid computation. Applications to high accuracy mapping and geophysical data acquision. Ph.D. Dissertation, Center for Lithospheric Studies, University of Texas at Dallas, Richardson, TX., 236 pp.

Dixon, W.J. and F.J. Massey, Jr., 1969: Introduction to Statistical Analysis. Third Edition. McGraw-Hill, Inc., New York, 109-119.

Edwards, G., 1996: Computation of horizontal control, Ohio HARN 1995. NGS internal report No. GPS882, Information Services Branch, National Geodetic Survey, NOAA, Silver Spring, MD.

Haagmans, R., E. de Min, and M. van Gelderen, 1993: Fast evaluation of convolution integrals on the sphere using 1D FFT, and a comparison with existing methods for Stokes' integral. Manus. Geod., 18(5), 227-241.

Kearsley, A.H.W., 1986: Data requirements for determining precise relative geoid heights from gravimetry. Journal of Geophysical Research, 91(B9), 9193-9201.

Lemoine, F.G., et al, 1996: EGM96, The NASA GSFC and NIMA Joint Geopotential Model. Proceedings of the International Symposium on Gravity, Geoid, and Marine Geodesy, Tokyo, Japan, September 30 -October 4, 1996.

Milbert, D.G., 1995: Improvement of a high resolution geoid height model in the U.S. by GPS height on NAVD88 benchmarks. IGeS Bulletin N.4, "New Geoids in the World", International Geoid Service, Milan, 1995, 13-36.

Milbert, D.G., 1991: An accuracy assessment of the GEOID90 geoid height model for the Commonwealth of Virginia. Unnumbered Report, June 1991, Information Services Branch, National Geodetic Survey, NOAA, Silver Spring, MD., 52pp.

Milbert, D.G., W.G. Kass, 1987: ADJUST: the horizontal observation adjustment program. NOAA Technical Memorandum NOS NGS-47, Information Services Branch, National Geodetic Survey, NOAA, Silver Spring, MD., 53pp.

Milbert, D.G. and D.A. Smith, 1996: Converting GPS Height into NAVD 88 Elevation with the GEOID96 Geoid Height Model . Proceedings of GIS/LIS '96 Annual Conference and Exposition, Denver, November 19-21, 1996, American Congress on Surveying and Mapping, Washington, D.C., pp. 681-692.

Mortiz, H., 1980: Advanced Physical Geodesy. Herbert Wichmann Verlag, Karlsruhe, 500 pp.

Parks, W. and D.G. Milbert, 1995: A geoid height model for San Diego County, California, to test the effect of densifying gravity measurements on accuracy of GPS-derived orthometric heights. Surv. and Land Info. Sys., 55(1), 21-38.

Row III, L.W. and M.W. Kozleski, 1991: A microcomputer 30-second point topography database for the conterminous United States, User Manual, Version 1.0, National Geophysical Data Center, Boulder, CO, 40 pp.

Schwarz, K.P., M.G. Sideris, and R. Forsberg, 1990: The use of FFT techniques in physical geodesy. Geophysical Journal International, 100, 485-514.

Soehngen, H.F., Z.A. Blackman, and E. Lange, 1991: Geoid modeling and effect on computed orthometric elevations for a county wide GPS project, Nassau County, New York. Technical Papers of ACSM-ASPRS Fall Convention, Atlanta, October 28-November 1, 1991, American Congress on Surveying and Mapping, Washington, D.C., A261-A278.

Smith, D.A. and D.G. Milbert, 1997a: Evaluation of Preliminary Models of the Geopotential in the United States , IGeS Bulletin, International Geoid Service, Milan, Italy, in press.

Smith, D.A. and D.G. Milbert, 1997b: The GEOID96 high resolution geoid height model for the United States. In preparation for submission to Journal of Geodesy.

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Figure Captions

Figure 1 : GEOID96 geoid height referred to the GRS80 ellipsoid (contour interval 0.1 m)

Figure 2 : GEOID96 geoid height referred to the GRS80 ellipsoid (perspective view to the south).

Figure 3 : G96SSS - GEOID96 geoid height differences (contour interval 0.05 m).

Figure 4 : Gravity coverage in the vicinity of Ohio.

Figure 5 : Gravity coverage (2' by 2' cells).

Figure 6 : Gravity coverage (6' by 6' cells).

Figure 7 : Empirical error statistics, differences from 2951 GPS benchmarks and GEOID96.

Figure 8 : GEOID96 model vs. OHIO62 GPS benchmarks, discrepencies about mean.

Figure 9 : GEOID96 model vs. OHIO62 GPS benchmarks, discrepencies about mean, detail.

Figure 10 : Ellipsoidal height difference residuals, CORS-fixed GPS adjustment, Ohio.

Figure 11 : GEOID96 model vs. CORS51 GPS benchmarks, discrepencies about mean.

Figure 12 : GEOID96 model vs. CORS50 GPS benchmarks, discrepencies about mean.

Figure 13 : G96SSS model vs. ITRF50 GPS benchmarks, discrepencies about mean, plotted against latitude.

Figure 14 : G96SSS model vs. 2951 GPS benchmarks, detrended discrepencies (contour interval 0.05 m).

Figure 15a : Local empirical covariance function, G96SSS model vs. ITRF50 GPS benchmarks, detrended discrepencies with 25 km bin size and Gaussian function fit of C₀ = 2.5²cm², L=15 km

Figure 15b : Local empirical covariance function, G96SSS model vs. ITRF50 GPS benchmarks, detrended discrepencies with 30 km bin size and Gaussian function fit of C₀ = 2.5²cm², L=20 km

Figure 16 : Predicted detrended discrepencies, G96SSS model vs. ITRF50 GPS benchmarks (contour interval = 0.01 m).

Figure 17 : Predicted detrended discrepencies, G96SSS model vs. ITRF50 GPS benchmarks (perspective view to northeast).

Appendicies

Appendix 1 -- GEOID96 model vs. OHIO62 GPS benchmarks, evaluation program output.

Evaluating geoid grid:  
geoid96.b                                                   
Evaluation data file :  
oh88                                                        
  62  unrejected points

geoid offset=   -7.3929079117157D-04

    lat       lon      ellip      msl       diff     interp     int-diff    ssn name
  38.55320 277.21162  133.657   166.502   -32.8457  -32.8464   -0.0006      4968V 353                         
  38.68291 275.93294  256.908   290.642   -33.7347  -33.8082   -0.0735      3147L 34                          
  38.83375 277.83777  137.871   171.604   -33.7337  -33.7395   -0.0057      3096KITTY                         
  38.83986 277.14868  167.500   200.770   -33.2707  -33.2374    0.0334      5309Y 214                         
  38.86777 275.37551  243.308   277.108   -33.8007  -33.9053   -0.1046      2735I71 T 34                      
  39.01797 276.97407  142.944   176.28    -33.3367  -33.2699    0.0668      4047PIKETON                       
  39.04514 276.97718  138.306   171.666   -33.3607  -33.3030    0.0578      4045PIK                           
  39.10957 275.46801  119.692   153.773   -34.0817  -34.0415    0.0402      04837008                          
  39.21157 277.77404  200.320   234.415   -34.0957  -34.0623    0.0334      0750ALBANY                        
  39.32960 276.33368  332.429   365.017   -32.5887  -32.5615    0.0272      0961B 340                         
  39.33163 277.01492  158.113   191.876   -33.7637  -33.6591    0.1047      1763E 338                         
  39.33362 278.57488  208.893   243.090   -34.1977  -34.1594    0.0384      3016K 323                         
  39.34570 278.56226  221.403   255.608   -34.2057  -34.1775    0.0282      0858ARP                           
  39.44285 275.66648  169.013   202.619   -33.6067  -33.5682    0.0385      4675T 347                         
  39.53003 275.22516  283.099   316.373   -33.2747  -33.3035   -0.0288      3884OXFORD                        
  39.58985 275.77276  253.514   286.866   -33.3527  -33.3048    0.0480      4562SOUTHPORT AZ MK               
  39.75944 275.79248  194.917   228.051   -33.1347  -33.1265    0.0082      2142G 346                         
  39.79332 277.06186  189.253   223.287   -34.0347  -33.9596    0.0752      4546SMITH                         
  39.80043 275.93407  210.870   243.955   -33.0857  -33.0635    0.0222      4003PAT AZ MK                     
  39.84652 277.47041  242.230   276.397   -34.1677  -34.1091    0.0586      4158Q 190                         
  39.89482 276.39526  326.165   359.078   -32.9137  -32.9137    0.0001      2396H 34                          
  39.89819 275.79975  271.050   304.057   -33.0077  -33.0397   -0.0319      5004VANDALIA                      
  39.90106 275.80403  267.192   300.170   -32.9787  -33.0394   -0.0607      5005VANDALIA AZ MK                
  39.90398 275.79814  270.018   303.000   -32.9827  -33.0373   -0.0545      1642DAYT B                        
  39.90508 275.79533  270.445   303.417   -32.9727  -33.0364   -0.0636      1641DAYT A                        
  39.91440 275.78732  269.329   302.461   -33.1327  -33.0321    0.1006      1640DAY G                         
  39.94091 278.10717  239.804   273.635   -33.8317  -33.8089    0.0229      2846IVORY                         
  39.94164 279.24586  173.443   207.220   -33.7777  -33.7786   -0.0009      0953B 316                         
  39.94728 278.10836  234.083   267.855   -33.7727  -33.8033   -0.0306      5049W 183                         
  40.03943 278.35310  314.283   348.089   -33.8067  -33.7655    0.0413      2068FORD                          
  40.05075 275.80188  222.950   255.951   -33.0017  -33.0092   -0.0075      3123L 164                         
  40.09943 275.39083  276.064   309.331   -33.2677  -33.2759   -0.0081      1630DAR GAR                       
  40.29846 275.83652  279.959   313.041   -33.0827  -33.1114   -0.0286      1914F 348                         
  40.35594 277.00956  266.641   300.671   -34.0307  -34.0132    0.0175      3231LEONARD                       
  40.47022 278.58140  237.468   270.882   -33.4147  -33.4267   -0.0120      4033PHILPORT                      
  40.51860 277.51643  302.510   336.171   -33.6617  -33.6064    0.0553      1344CAUSEWAY                      
  40.59469 276.78264  244.955   279.495   -34.5407  -34.5361    0.0047      4659T 23                          
  40.62261 279.43516  195.868   230.135   -34.2677  -34.2863   -0.0186      1265C 338                         
  40.70449 275.97325  260.176   294.730   -34.5547  -34.5674   -0.0127      0643A 290                         
  40.76946 277.25560  323.900   358.058   -34.1587  -34.1204    0.0383      5111WACHS                         
  40.77353 275.90710  230.381   265.054   -34.6737  -34.6501    0.0236      2399H 348                         
  40.83723 278.73587  302.043   335.389   -33.3467  -33.3698   -0.0230      0927B 10                          
  40.88490 276.12196  223.692   258.946   -35.2547  -35.2597   -0.0050      4966V 349                         
  40.88770 275.19742  214.740   248.131   -33.3917  -33.3840    0.0078      1023BAXTER                        
  40.89278 276.17647  225.444   260.760   -35.3167  -35.3140    0.0028      1054BLACK                         
  41.01265 276.31581  205.425   240.808   -35.3837  -35.4151   -0.0314      5072W 350                         
  41.03347 278.52747  284.965   318.297   -33.3327  -33.3642   -0.0314      0744AIRPORT C OF A                
  41.05367 278.01352  311.471   344.920   -33.4497  -33.4436    0.0062      3410M 176                         
  41.06830 277.31360  255.166   289.787   -34.6217  -34.5819    0.0399      5192WILLARD                       
  41.12726 276.76247  198.486   233.486   -35.0007  -35.0063   -0.0056      4249R 344                         
  41.29572 277.79431  211.243   245.568   -34.3257  -34.3580   -0.0323      5491ZOB A                         
  41.29753 277.79441  211.045   245.467   -34.4227  -34.3620    0.0607      5492ZOB B                         
  41.34299 277.82172  205.731   240.190   -34.4597  -34.4308    0.0289      3355LORPORT                       
  41.38570 276.35399  170.451   205.739   -35.2887  -35.3265   -0.0377      1269C 351                         
  41.38726 276.36389  169.348   204.633   -35.2857  -35.3229   -0.0371      1111BOWLPORT                      
  41.46350 277.80728  147.252   181.899   -34.6477  -34.7076   -0.0598      3433M 319                         
  41.53998 278.36585  143.451   177.800   -34.3497  -34.3885   -0.0387      2134G 321                         
  41.54293 277.26815  141.851   177.309   -35.4587  -35.4601   -0.0014      5415Z 317                         
  41.54381 277.26851  140.820   176.281   -35.4617  -35.4606    0.0011      3490MARBLEHEAD                    
  41.75284 278.71168  141.458   176.025   -34.5677  -34.5623    0.0055      5269X 323                         
  41.75463 275.84925  208.033   242.426   -34.3937  -34.7299   -0.3361      5036W 118                         
  41.82332 276.54720  143.098   178.441   -35.3437  -35.2992    0.0446      4802TT 4 L                        

  62 points --    mean =     2.0628660070455D-15
  62 points --     rms =     5.9701024120379D-02

Appendix 2 -- CORS-fixed GPS adjustment summary statistics, adjustment program output.

                                                    NATIONAL GEODETIC SURVEY
 PROGRAM ADJUST                                         ADJUSTMENT PROGRAM                                               PAGE 274
                                                          VERSION   4.00  


 RESIDUAL STATISTICS

 OBSERVATION NUMBERS OF 20 GREATEST STANDARDIZED RESIDUALS (V/SD)

    142    70    76    61   241  1192   143  1195  1288    85   991  1219    97   934   187     2   712  1273     7   727

  TOTAL=  1476        NO-CHECK=    0
  MAX V=  3.0D+02    MAX V/SD=  117.150
  MIN V= -9.1D-02    MIN V/SD=  -51.996
 MEAN V=  6.0D-01   MEAN V/SD=    0.899


               N      VTPV      RMS       RN        VTPV/RN      MEAN ABS
                               VTPV                              RESIDUAL
 DELTA X     489   42601.1     9.33   245.96         173.20      0.579 (METERS)
 DELTA Y     489   40783.3     9.13   245.96         165.81      0.614 (METERS)
 DELTA Z     489   19433.1     6.30   245.92          79.02      0.625 (METERS)
 DOPPLER X     0       0.0     0.00     0.00           0.00      0.000 (METERS)
 DOPPLER Y     0       0.0     0.00     0.00           0.00      0.000 (METERS)
 DOPPLER Z     0       0.0     0.00     0.00           0.00      0.000 (METERS)
 DIRECTION     0       0.0     0.00     0.00           0.00      0.000 (SECONDS)
 H ANGLE       0       0.0     0.00     0.00           0.00      0.000 (SECONDS)
 ZEN DIST      0       0.0     0.00     0.00           0.00      0.000 (SECONDS)
 DISTANCE      0       0.0     0.00     0.00           0.00      0.000 (METERS)
 AZIMUTH       0       0.0     0.00     0.00           0.00      0.000 (SECONDS)
 OTHER         9      44.2     2.22     0.16         273.63      0.000
 TOTAL      1476  102861.7     8.35   738.00         139.38

               N          RMS                                    MEAN ABS
          CONTRIB.      RESIDUAL                                 RESIDUAL
 NORTH       487          0.005                                  0.003 (METERS)
 EAST        487          0.007                                  0.004 (METERS)
 UP          487          0.016                                  0.012 (METERS)

0DEGREES OF FREEDOM      =              738
 VARIANCE SUM            =           102861.7
 STD.DEV.OF UNIT WEIGHT  =               11.81
 VARIANCE OF UNIT WEIGHT =              139.38

Appendix 3 -- Large ellipsoid height difference residuals, CORS-fixed GPS adjustment.

 Ellipsoid Height Residuals Exceeding 3.2 cm in Magnitude

  Length    Residual    Standpoint       Forepoint
    (m)        (m)   
---------------------------------------------------------
   73022    -0.0847     BLACK            17G A                     
   60968    -0.0606     BLACK            T 23                      
   27587    -0.0564     DEF 66           NAPPORT                   
  587213    -0.0525     BOLTON CBL 0     ST LOUIS 2                
   46342    -0.0440     MILLPORT         PHILPORT                  
   37407    -0.0407     CMH A            DEL GAR                   
   35113    -0.0384     DEF 66           MILFORD 2 RM A            
   58268    -0.0383     DEF 66           V 349                     
   34662    -0.0382     BLACK            I95 A                     
   27118    -0.0354     MILLPORT         I40 A                     
   40954    -0.0347     DEF 66           OOTTPORT                  
   42926    -0.0343     BLACK            56D A                     
   24782    -0.0332     DEF 66           RESERVOIR                 
   35666    -0.0329     BLACK            RND HEAD 

  109225     0.0330     K 323            DIST 5 HQ                 
   27849     0.0345     OAK GAR          TDZ B   
  110655     0.0347     K 323            Q 190                     
   80989     0.0352     K 323            PER GAR                   
   89243     0.0403     K 323            ALDERMAN AZ MK 2          
   79807     0.0406     AUBURN           B 340                     
   78262     0.0407     BOLTON CBL 0     T 23                      
   37433     0.0423     AUBURN           AP STA A2 LUK             
   39291     0.0428     WILLARD          17G A                     
   54711     0.0475     K 323            MRG GAR                   
  130543     0.1013     NAPPORT          DETROIT 1 
         

Appendix 4 -- GEOID96 model vs. CORS51 GPS benchmarks, evaluation program output.

Evaluating geoid grid:  
geoid96.b                                                   
Evaluation data file :  
cors88                                                      
  51  unrejected points

geoid offset=    1.2107492484300D-02

    lat       lon      ellip      msl       diff     interp     int-diff    ssn name
  41.82332 276.54720  143.098   178.441   -35.3309  -35.2992    0.0317      0002TT 4 L                        
  39.94164 279.24586  173.443   207.220   -33.7649  -33.7786   -0.0137      0006B 316                         
  39.80043 275.93407  210.870   243.955   -33.0729  -33.0635    0.0094      0012PAT AZ MK                     
  39.94091 278.10717  239.804   273.635   -33.8189  -33.8089    0.0100      0036IVORY                         
  39.94728 278.10836  234.083   267.855   -33.7599  -33.8033   -0.0434      0037W 183                         
  41.03347 278.52747  284.965   318.297   -33.3199  -33.3642   -0.0443      0067AIRPORT C OF A                
  40.47022 278.58140  237.468   270.882   -33.4019  -33.4267   -0.0248      0079PHILPORT                      
  41.34299 277.82172  205.731   240.190   -34.4469  -34.4308    0.0161      0115LORPORT                       
  39.58985 275.77276  253.514   286.866   -33.3399  -33.3048    0.0351      0183SOUTHPORT AZ MK               
  39.89819 275.79975  271.050   304.057   -32.9949  -33.0397   -0.0448      0209VANDALIA                      
  40.70449 275.97325  260.176   294.730   -34.5419  -34.5674   -0.0255      0218A 290                         
  40.88490 276.12196  223.692   258.946   -35.2419  -35.2597   -0.0178      0226V 349                         
  41.01265 276.31581  205.425   240.808   -35.3709  -35.4151   -0.0443      0229W 350                         
  39.53003 275.22516  283.099   316.373   -33.2619  -33.3035   -0.0416      0999OXFORD                        
  39.89482 276.39526  326.165   359.078   -32.9009  -32.9137   -0.0128      1299H 34                          
  40.62261 279.43516  195.868   230.135   -34.2549  -34.2863   -0.0314      1599C 338                         
  40.76946 277.25560  323.900   358.058   -34.1459  -34.1204    0.0255      1701WACHS                         
  41.53998 278.36585  143.451   177.801   -34.3379  -34.3885   -0.0506      1802G 321                         
  40.09943 275.39083  276.064   309.331   -33.2549  -33.2759   -0.0210      1902DAR GAR                       
  39.21157 277.77404  200.320   234.415   -34.0829  -34.0623    0.0206      2061ALBANY                        
  38.83375 277.83777  137.871   171.604   -33.7209  -33.7395   -0.0186      2064KITTY                         
  38.83986 277.14868  167.500   200.770   -33.2579  -33.2374    0.0205      2079Y 214                         
  40.35594 277.00956  266.641   300.671   -34.0179  -34.0132    0.0047      2103LEONARD                       
  41.38726 276.36389  169.348   204.633   -35.2729  -35.3229   -0.0500      2144BOWLPORT                      
  39.84652 277.47041  242.230   276.398   -34.1559  -34.1091    0.0468      2303Q 190                         
  40.03943 278.35310  314.283   348.089   -33.7939  -33.7655    0.0284      3099FORD                          
  39.10957 275.46801  119.692   153.773   -34.0689  -34.0415    0.0274      31997008                          
  40.89278 276.17647  225.444   260.761   -35.3049  -35.3140   -0.0091      3205BLACK                         
  39.32960 276.33368  332.429   365.018   -32.5769  -32.5615    0.0153      3699B 340                         
  41.06830 277.31360  255.166   289.787   -34.6089  -34.5819    0.0270      3901WILLARD                       
  40.51860 277.51643  302.510   336.172   -33.6499  -33.6064    0.0435      4203CAUSEWAY                      
  41.75284 278.71168  141.458   176.025   -34.5549  -34.5623   -0.0074      4305X 323                         
  40.59469 276.78264  244.955   279.495   -34.5279  -34.5361   -0.0082      5101T 23                          
  41.29753 277.79441  211.045   245.467   -34.4099  -34.3620    0.0479      5106ZOB B                         
  40.77353 275.90710  230.381   265.054   -34.6609  -34.6501    0.0108      5108H 348                         
  40.05075 275.80188  222.950   255.951   -32.9889  -33.0092   -0.0203      5504L 164                         
  39.75944 275.79248  194.917   228.051   -33.1219  -33.1265   -0.0046      5799G 346                         
  41.54381 277.26851  140.820   176.281   -35.4489  -35.4606   -0.0117      6205MARBLEHEAD                    
  41.54293 277.26815  141.851   177.309   -35.4459  -35.4601   -0.0142      6206Z 317                         
  39.79332 277.06186  189.253   223.287   -34.0219  -33.9596    0.0623      6502SMITH                         
  39.04514 276.97718  138.306   171.666   -33.3479  -33.3030    0.0449      6602PIK 95                        
  39.01797 276.97407  142.944   176.280   -33.3239  -33.2699    0.0540      6699PIKETON                       
  39.33163 277.01492  158.113   191.876   -33.7509  -33.6591    0.0918      7199E 338                         
  41.12726 276.76247  198.486   233.486   -34.9879  -35.0063   -0.0184      7406R 344                         
  40.29846 275.83652  279.959   313.041   -33.0699  -33.1114   -0.0415      7501F 348                         
  40.83723 278.73587  302.043   335.389   -33.3339  -33.3698   -0.0359      7601B 10                          
  39.33362 278.57488  208.893   243.090   -34.1849  -34.1594    0.0255      8099K 323                         
  40.88770 275.19742  214.740   248.131   -33.3789  -33.3840   -0.0051      8104BAXTER                        
  39.44285 275.66648  169.013   202.619   -33.5939  -33.5682    0.0257      8399T 347                         
  41.38570 276.35399  170.451   205.739   -35.2759  -35.3265   -0.0506      8704C 351                         
  38.55320 277.21162  133.657   166.502   -32.8329  -32.8464   -0.0135      9994V 353                         

  51 points --    mean =     3.2044084161730D-15
  51 points --     rms =     3.3478820501037D-02

Appendix 5 -- GEOID96 model vs. CORS50 GPS benchmarks, evaluation program output.

Evaluating geoid grid:  
geoid96.b                                                   
Evaluation data file :  
cors50                                                      
  50  unrejected points

geoid offset=    1.0270870971682D-02

    lat       lon      ellip      msl       diff     interp     int-diff    ssn name
  41.82332 276.54720  143.098   178.441   -35.3327  -35.2992    0.0336      0002TT 4 L                        
  39.94164 279.24586  173.443   207.220   -33.7667  -33.7786   -0.0119      0006B 316                         
  39.80043 275.93407  210.870   243.955   -33.0747  -33.0635    0.0112      0012PAT AZ MK                     
  39.94091 278.10717  239.804   273.635   -33.8207  -33.8089    0.0119      0036IVORY                         
  39.94728 278.10836  234.083   267.855   -33.7617  -33.8033   -0.0416      0037W 183                         
  41.03347 278.52747  284.965   318.297   -33.3217  -33.3642   -0.0424      0067AIRPORT C OF A                
  40.47022 278.58140  237.468   270.882   -33.4037  -33.4267   -0.0230      0079PHILPORT                      
  41.34299 277.82172  205.731   240.190   -34.4487  -34.4308    0.0179      0115LORPORT                       
  39.58985 275.77276  253.514   286.866   -33.3417  -33.3048    0.0369      0183SOUTHPORT AZ MK               
  39.89819 275.79975  271.050   304.057   -32.9967  -33.0397   -0.0429      0209VANDALIA                      
  40.70449 275.97325  260.176   294.730   -34.5437  -34.5674   -0.0237      0218A 290                         
  40.88490 276.12196  223.692   258.946   -35.2437  -35.2597   -0.0160      0226V 349                         
  41.01265 276.31581  205.425   240.808   -35.3727  -35.4151   -0.0424      0229W 350                         
  39.53003 275.22516  283.099   316.373   -33.2637  -33.3035   -0.0398      0999OXFORD                        
  39.89482 276.39526  326.165   359.078   -32.9027  -32.9137   -0.0109      1299H 34                          
  40.62261 279.43516  195.868   230.135   -34.2567  -34.2863   -0.0296      1599C 338                         
  40.76946 277.25560  323.900   358.058   -34.1477  -34.1204    0.0273      1701WACHS                         
  41.53998 278.36585  143.451   177.801   -34.3397  -34.3885   -0.0487      1802G 321                         
  40.09943 275.39083  276.064   309.331   -33.2567  -33.2759   -0.0191      1902DAR GAR                       
  39.21157 277.77404  200.320   234.415   -34.0847  -34.0623    0.0224      2061ALBANY                        
  38.83375 277.83777  137.871   171.604   -33.7227  -33.7395   -0.0168      2064KITTY                         
  38.83986 277.14868  167.500   200.770   -33.2597  -33.2374    0.0224      2079Y 214                         
  40.35594 277.00956  266.641   300.671   -34.0197  -34.0132    0.0065      2103LEONARD                       
  41.38726 276.36389  169.348   204.633   -35.2747  -35.3229   -0.0482      2144BOWLPORT                      
  39.84652 277.47041  242.230   276.398   -34.1577  -34.1091    0.0486      2303Q 190                         
  40.03943 278.35310  314.283   348.089   -33.7957  -33.7655    0.0303      3099FORD                          
  39.10957 275.46801  119.692   153.773   -34.0707  -34.0415    0.0292      31997008                          
  40.89278 276.17647  225.444   260.761   -35.3067  -35.3140   -0.0073      3205BLACK                         
  39.32960 276.33368  332.429   365.018   -32.5787  -32.5615    0.0172      3699B 340                         
  41.06830 277.31360  255.166   289.787   -34.6107  -34.5819    0.0289      3901WILLARD                       
  40.51860 277.51643  302.510   336.172   -33.6517  -33.6064    0.0453      4203CAUSEWAY                      
  41.75284 278.71168  141.458   176.025   -34.5567  -34.5623   -0.0056      4305X 323                         
  40.59469 276.78264  244.955   279.495   -34.5297  -34.5361   -0.0063      5101T 23                          
  41.29753 277.79441  211.045   245.467   -34.4117  -34.3620    0.0497      5106ZOB B                         
  40.77353 275.90710  230.381   265.054   -34.6627  -34.6501    0.0126      5108H 348                         
  40.05075 275.80188  222.950   255.951   -32.9907  -33.0092   -0.0185      5504L 164                         
  39.75944 275.79248  194.917   228.051   -33.1237  -33.1265   -0.0028      5799G 346                         
  41.54381 277.26851  140.820   176.281   -35.4507  -35.4606   -0.0099      6205MARBLEHEAD                    
  41.54293 277.26815  141.851   177.309   -35.4477  -35.4601   -0.0124      6206Z 317                         
  39.79332 277.06186  189.253   223.287   -34.0237  -33.9596    0.0641      6502SMITH                         
  39.04514 276.97718  138.306   171.666   -33.3497  -33.3030    0.0468      6602PIK 95                        
  39.01797 276.97407  142.944   176.280   -33.3257  -33.2699    0.0558      6699PIKETON                       
  41.12726 276.76247  198.486   233.486   -34.9897  -35.0063   -0.0166      7406R 344                         
  40.29846 275.83652  279.959   313.041   -33.0717  -33.1114   -0.0396      7501F 348                         
  40.83723 278.73587  302.043   335.389   -33.3357  -33.3698   -0.0340      7601B 10                          
  39.33362 278.57488  208.893   243.090   -34.1867  -34.1594    0.0274      8099K 323                         
  40.88770 275.19742  214.740   248.131   -33.3807  -33.3840   -0.0033      8104BAXTER                        
  39.44285 275.66648  169.013   202.619   -33.5957  -33.5682    0.0275      8399T 347                         
  41.38570 276.35399  170.451   205.739   -35.2777  -35.3265   -0.0487      8704C 351                         
  38.55320 277.21162  133.657   166.502   -32.8347  -32.8464   -0.0117      9994V 353                         

  50 points --    mean =     8.5265128291212D-16
  50 points --     rms =     3.1164337679022D-02

Appendix 6 -- G96SSS model vs. ITRF50 GPS benchmarks, evaluation program output.

Evaluating geoid grid:  
g96sss.b                                                    
Evaluation data file :  
itrf50                                                      
  50  unrejected points

geoid offset=   0.50059679077149

    lat       lon      ellip      msl       diff     interp     int-diff    ssn name
  41.82333 276.54720  141.938   178.441   -36.0024  -35.9072    0.0952      0002TT 4 L                        
  39.94165 279.24586  172.200   207.220   -34.5194  -34.5554   -0.0360      0006B 316                         
  39.80044 275.93407  209.654   243.955   -33.8004  -33.8039   -0.0035      0012PAT AZ MK                     
  39.94092 278.10717  238.571   273.635   -34.5634  -34.5617    0.0017      0036IVORY                         
  39.94729 278.10836  232.851   267.855   -34.5034  -34.5556   -0.0522      0037W 183                         
  41.03348 278.52747  283.762   318.297   -34.0344  -34.0500   -0.0156      0067AIRPORT C OF A                
  40.47023 278.58139  236.247   270.882   -34.1344  -34.1457   -0.0113      0079PHILPORT                      
  41.34300 277.82172  204.544   240.190   -35.1454  -35.0910    0.0544      0115LORPORT                       
  39.58986 275.77275  252.294   286.866   -34.0714  -34.0583    0.0131      0183SOUTHPORT AZ MK               
  39.89820 275.79975  269.839   304.057   -33.7174  -33.7715   -0.0541      0209VANDALIA                      
  40.70450 275.97324  258.987   294.730   -35.2424  -35.2432   -0.0008      0218A 290                         
  40.88490 276.12196  222.507   258.946   -35.9384  -35.9247    0.0137      0226V 349                         
  41.01266 276.31581  204.242   240.808   -36.0654  -36.0735   -0.0081      0229W 350                         
  39.53004 275.22515  281.883   316.373   -33.9894  -34.0522   -0.0628      0999OXFORD                        
  39.89483 276.39525  324.948   359.078   -33.6294  -33.6513   -0.0219      1299H 34                          
  40.62262 279.43515  194.645   230.135   -34.9894  -35.0135   -0.0241      1599C 338                         
  40.76946 277.25560  322.701   358.058   -34.8564  -34.8035    0.0529      1701WACHS                         
  41.53999 278.36585  142.265   177.801   -35.0354  -35.0509   -0.0155      1802G 321                         
  40.09944 275.39082  274.863   309.331   -33.9674  -33.9880   -0.0206      1902DAR GAR                       
  39.21158 277.77404  199.068   234.415   -34.8464  -34.8812   -0.0348      2061ALBANY                        
  38.83376 277.83777  136.607   171.604   -34.4964  -34.5999   -0.1035      2064KITTY                         
  38.83987 277.14868  166.243   200.770   -34.0264  -34.0767   -0.0503      2079Y 214                         
  40.35595 277.00956  265.431   300.671   -34.7394  -34.7213    0.0181      2103LEONARD                       
  41.38727 276.36388  168.176   204.633   -35.9564  -35.9565   -0.0001      2144BOWLPORT                      
  39.84653 277.47041  241.001   276.398   -34.8964  -34.8608    0.0356      2303Q 190                         
  40.03944 278.35309  313.051   348.089   -34.5374  -34.5140    0.0234      3099FORD                          
  39.10958 275.46801  118.461   153.773   -34.8114  -34.8219   -0.0105      31997008                          
  40.89279 276.17646  224.259   260.761   -36.0014  -35.9791    0.0223      3205BLACK                         
  39.32961 276.33368  331.195   365.018   -33.3224  -33.3419   -0.0195      3699B 340                         
  41.06830 277.31359  253.975   289.787   -35.3114  -35.2481    0.0633      3901WILLARD                       
  40.51861 277.51643  301.301   336.172   -34.3704  -34.3074    0.0630      4203CAUSEWAY                      
  41.75285 278.71168  140.276   176.025   -35.2484  -35.2256    0.0228      4305X 323                         
  40.59470 276.78263  243.755   279.495   -35.2394  -35.2262    0.0132      5101T 23                          
  41.29753 277.79440  209.857   245.467   -35.1094  -35.0239    0.0855      5106ZOB B                         
  40.77354 275.90710  229.195   265.054   -35.3584  -35.3203    0.0381      5108H 348                         
  40.05076 275.80188  221.743   255.951   -33.7074  -33.7302   -0.0228      5504L 164                         
  39.75945 275.79248  193.702   228.051   -33.8484  -33.8682   -0.0198      5799G 346                         
  41.54382 277.26851  139.644   176.281   -36.1364  -36.1015    0.0349      6205MARBLEHEAD                    
  41.54294 277.26815  140.675   177.309   -36.1334  -36.1010    0.0324      6206Z 317                         
  39.79333 277.06186  188.026   223.287   -34.7604  -34.7111    0.0493      6502SMITH                         
  39.04515 276.97718  137.057   171.666   -34.1084  -34.1194   -0.0110      6602PIK 95                        
  39.01798 276.97407  141.694   176.280   -34.0854  -34.0887   -0.0033      6699PIKETON                       
  41.12727 276.76246  197.302   233.486   -35.6834  -35.6626    0.0208      7406R 344                         
  40.29846 275.83652  278.759   313.041   -33.7814  -33.8149   -0.0335      7501F 348                         
  40.83724 278.73587  300.832   335.389   -34.0564  -34.0694   -0.0130      7601B 10                          
  39.33363 278.57487  207.638   243.090   -34.9514  -34.9841   -0.0327      8099K 323                         
  40.88771 275.19741  213.565   248.131   -34.0654  -34.0321    0.0333      8104BAXTER                        
  39.44286 275.66648  167.789   202.619   -34.3294  -34.3305   -0.0011      8399T 347                         
  41.38571 276.35399  169.279   205.739   -35.9594  -35.9600   -0.0006      8704C 351                         
  38.55321 277.21162  132.391   166.502   -33.6104  -33.7145   -0.1041      9994V 353                         

  50 points --    mean =    -1.1368683772162D-15
  50 points --     rms =     4.0923763017576D-02

Appendix 7 -- G96SSS model vs. ITRF50 GPS benchmarks, collocation program output.

program cgrid
version 0.7 -- alpha
Enter input file name : eval2.t
Enter stats file name : stats
Enter grid  file name : grid
Enter column for Z value  --> 7
Remove plane   (y/n)? : y
Enter trend file name : trend
Sparse solve   (y/n)? : n
  50 records in input file
az=351.   ppm=  0.28
Enter cov. model code : 1
Enter signal sigma    : 0.025
Enter correl.leng.(km): 20

Enter data sigma      : 0.016

Min/max lat  =     38.553210000000    41.823330000000
Min/max long =     275.19741000000    279.43515000000
Enter min lat: 38 
Enter max lat: 42
Enter # row: 41
Enter min lon: 274
Enter max lon: 281
Enter # col: 71
Enter code for data kind  --> 1
Enter covar.plot.dist.: 300
Enter covar.bin.width.: 30
Enter covar.file name : ecov
Beginning empirical statistics ....
          mean =     2.49245E-16
var about zero =     9.75492E-04    rms about zero =     3.12329E-02
Beginning covariances ....
Beginning collocation ....
Beginning prediction ....

n  =  50     rms=    1.07104E-02
vmn=   -3.00866E-02     vmx=    2.41950E-02

Appendix 8 -- G96SSS model vs. ITRF50 GPS benchmarks, collocation program fits to detrended discrepencies.

         Lat.     Long.     Discrep.  Predict.   Resid.
        (deg.)    (deg.)       (m)       (m)       (m)
     --------------------------------------------------
       41.82333 276.54720     0.046     0.032     0.014
       39.94165 279.24586    -0.016    -0.012    -0.005
       39.80044 275.93407     0.008    -0.003     0.012
       39.94092 278.10717     0.017    -0.005     0.022
       39.94729 278.10836    -0.037    -0.004    -0.032
       41.03348 278.52747    -0.033    -0.024    -0.008
       40.47023 278.58139    -0.011    -0.007    -0.003
       41.34300 277.82172     0.025     0.032    -0.007
       39.58986 275.77275     0.031     0.021     0.010
       39.89820 275.79975    -0.046    -0.024    -0.022
       40.70450 275.97324    -0.017    -0.003    -0.014
       40.88490 276.12196    -0.008    -0.004    -0.004
       41.01266 276.31581    -0.033    -0.022    -0.011
       39.53004 275.22515    -0.045    -0.031    -0.014
       39.89483 276.39525    -0.011    -0.007    -0.004
       40.62262 279.43515    -0.025    -0.018    -0.007
       40.76946 277.25560     0.039     0.033     0.007
       41.53999 278.36585    -0.049    -0.035    -0.014
       40.09944 275.39082    -0.020    -0.016    -0.004
       39.21158 277.77404     0.002     0.000     0.002
       38.83376 277.83777    -0.055    -0.039    -0.016
       38.83987 277.14868    -0.004    -0.002    -0.002
       40.35595 277.00956     0.017     0.013     0.003
       41.38727 276.36388    -0.036    -0.030    -0.006
       39.84653 277.47041     0.052     0.039     0.012
       40.03944 278.35309     0.037     0.022     0.015
       39.10958 275.46801     0.021     0.015     0.006
       40.89279 276.17646     0.001    -0.007     0.008
       39.32961 276.33368     0.008     0.006     0.002
       41.06830 277.31359     0.041     0.031     0.009
       40.51861 277.51643     0.058     0.043     0.015
       41.75285 278.71168    -0.016    -0.013    -0.002
       40.59470 276.78263     0.003     0.004     0.000
       41.29753 277.79440     0.058     0.036     0.021
       40.77354 275.90710     0.019     0.005     0.015
       40.05076 275.80188    -0.019    -0.027     0.007
       39.75945 275.79248    -0.007    -0.004    -0.003
       41.54382 277.26851    -0.003    -0.003     0.000
       41.54294 277.26815    -0.005    -0.003    -0.002
       39.79333 277.06186     0.065     0.049     0.017
       39.04515 276.97718     0.028     0.027     0.002
       39.01798 276.97407     0.037     0.026     0.011
       41.12727 276.76246    -0.006    -0.005     0.000
       40.29846 275.83652    -0.038    -0.027    -0.011
       40.83724 278.73587    -0.023    -0.019    -0.004
       39.33363 278.57487     0.003     0.002     0.001
       40.88771 275.19741     0.008     0.006     0.002
       39.44286 275.66648     0.021     0.018     0.003
       41.38571 276.35399    -0.037    -0.031    -0.006
       38.55321 277.21162    -0.049    -0.035    -0.014