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, TB = 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:
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: xe = xw (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: se2 = sw2 (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.
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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.
Appendix 2 -- CORS-fixed GPS adjustment
summary statistics, adjustment program output.
Appendix 3 -- Large ellipsoid height
difference residuals, CORS-fixed GPS adjustment.
Appendix 4 -- GEOID96 model vs. CORS51 GPS
benchmarks, evaluation program output.
Appendix 5 -- GEOID96 model vs. CORS50 GPS
benchmarks, evaluation program output.
Appendix 6 -- G96SSS model vs. ITRF50 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
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
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
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
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
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