The CARIB97 high resolution geoid height model for the Caribbean Sea
Dru A. Smith1 and Howard J. Small2
1National Oceanic and Atmospheric
Administration/National Geodetic Survey,
1315 East-West Highway, Silver Spring Maryland, 20910
phone: 001- (301) 713-3202
fax: 001 - (301) 713-4172
2National Imagery and Mapping Agency,
3200 South Second Street, St. Louis, Missouri, 63118
Correspondence to: D. Smith
(This paper has been published in Journal of Geodesy, Vol. 73, No. 1, p. 1-9, 1999)
A 2'x2' resolution geoid model, CARIB97, has been computed covering the Caribbean Sea. The geoid undulations refer to the GRS-80 ellipsoid, centered at the ITRF94 (1996.0) origin. The geoid level is defined by adopting the gravity potential on the geoid as W0=62636856.88 m2/s2 and a gravity-mass constant of GM=3.986004418 x 1014 m3/s2. The geoid model was computed by applying high-frequency corrections to the EGM96 global geopotential model in a remove-compute-restore procedure. The permanent tide system of CARIB97 is non-tidal.
Comparison of CARIB97 geoid heights to 31 GPS/tidal (ITRF94/local) benchmarks show an average offset (h-H-N) of 51 cm, with an RMS of 62 cm about the average. This represents an improvement over the use of a global geoid model for the region. However, because the measured orthometric heights (H) refer to many differing tidal datums these comparisons are biased by localized permanent ocean dynamic topography (PODT). Therefore, we interpret the 51 cm as partially an estimate of the average PODT in the vicinity of the 31 island benchmarks.
On an island-by-island basis, CARIB97 now offers the ability to analyze local datum problems which were previously unrecognized due to a lack of high resolution geoid information in the area.
Keywords: geoid, mean sea level, ocean dynamic topography, GPS, reference systems
In 1996, NOAA's National Geodetic Survey (NGS) was contracted by the Federal Aviation Administration (FAA) to provide position control at various Caribbean airports. Using the Global Positioning System (GPS), geodetic latitude, longitude and ellipsoid height were established at airport control points. Following the survey, local geoid undulations were computed at 31 benchmarks where a GPS (ellipsoidal) height and a tidal (local mean sea level) height were both available. These local geoid undulations were compared with the global geoid undulation model generated from Earth Gravity Model 1996 (EGM96) geopotential coefficients and global Bouguer anomalies (National Imagery and Mapping Agency 1996). Discrepancies of over 1 meter (caused mostly by geoid omission error, local height definition problems, and permanent ocean dynamic topography -- the separation between the geoid and local mean sea level) were noticed in that comparison. Without a high resolution geoid model, the amount of geoid error in those discrepancies prevented an accurate determination of the internal consistency of the local benchmarks. Therefore, CARIB97 was proposed as a method both to minimize geoid error in the Caribbean Sea, as well as to provide a unified vertical reference surface for the entire area.
The gravity coverage which is sufficient for geoid modeling depends on the accuracy of geoid undulations required. Kearsley (1986) shows that it is possible to compute geoid undulation 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. Although the level of random errors is well within the stated 1-mGal error bars of NGS gravity points, our gravity coverage does not approach 6'x6' in some areas of the Caribbean. Therefore, NGS proposed a partnership with the National Imagery and Mapping Agency (NIMA), whose gravity holdings are extensive. Both agencies had interests in a Caribbean geoid, and CARIB97 is the product of that partnership.
Although we expected to reduce geoid error, we knew that our ability to check the geoid model against local GPS/tidal benchmarks would be hindered by two things: 1) a lack of knowledge of the local permanent ocean dynamic topography (PODT), and 2) a lack of information regarding how, or if, the local heights are tied to local mean sea level. Still, with an accurate geoid model, we should be able to identify the areas where potential height definition and/or surveying errors are causing internal inconsistencies in an island's vertical datum.
Section 2 briefly examines geoid undulations from the EGM96 global spherical harmonic geopotential model. Section 3 will discuss the data and theory that produced CARIB97. Section 4 discusses the GPS/tidal benchmarks in the Caribbean Sea, with section 5 showing the investigation of local height problems on those benchmarks. A comparison with EGM96 is found in section 6, and conclusions follow in section 7. Section 8 contains some new, experimental results, which were computed during the writing of this paper and which represent future directions for research.
2. Geoid Undulations calculated with EGM96
CARIB97, like G96SSS (the gravimetric geoid which served as the foundation for GEOID96; Smith and Milbert 1998) was calculated using an approximation to Helmert's second method of condensation, combined with the "remove-compute-restore" Fast Fourier Transform procedure, and the EGM96 geopotential model as a foundation. The gravity anomalies ("remove") and the geoid undulations ("restore") were simultaneously computed by evaluating a spherical harmonic series, using the EGM96 potential coefficients. The evaluation points are on the geoid, and therefore "in the masses" for most terrestrial points. As pointed out by Rapp (1997), such "geoid undulations" can be erroneous (as much as 3.5 meters in the Himalayas). Therefore one must be careful when speaking of "EGM96 geoid undulations". Throughout this paper, the term "EGM96 geoid undulations" will refer to geoid undulations computed from the EGM96 coefficients alone. Geoid undulations which are computed through a combination of geopotential coefficients and Bouguer anomalies (ibid) will be called "EGM96/Bouguer geoid undulations". In the Caribbean Sea (9 to 28 North latitude and 274 to 302 East longitude), the differences between these two types of geoid undulations average -2.2 cm, with an RMS of 1.8 cm. The maximum difference is -45 cm (Colombia) which is far from negligible.
Both types of geoid undulation are useful in this study. The EGM96 geoid undulations are mathematically consistent with the EGM96 gravity anomalies (Smith and Milbert, 1998), a requirement in the "remove-compute-restore" procedure. They will be used as approximations to Helmert gravity anomalies in the remove phase, and co-geoid undulations in the restore phase (Smith and Milbert, 1998). The EGM96/Bouguer geoid undulations are our best estimates of true geoid undulations available from a global model; therefore they are the geoid undulations we will use in comparison with either CARIB97 or the GPS/tidal benchmarks.
When computing the EGM96 geoid undulations (and gravity anomalies), it was necessary to define the absolute level of the geoid through an adopted value of W0. We used the value put forth by Burša (1995) of W0 = 62636856.88 m2/s2. We also adopt the GRS-80 ellipsoid (Moritz, 1988) as our reference ellipsoid. We also adopt a best global value of GM from Burša (1995), GM=3.986004418 x 1014 m3/s2. These parameters, in concert with the EGM96 geopotential coefficients allow us to calculate a grid of geoid undulations and gravity anomalies, relative to the GRS-80 ellipsoid. (As a clarifying note, the latest estimates of the true values of W0 and GM must be chosen to accurately compute the location of the geoid. If we were to use the GRS-80 GM and U0 values as our approximations to the true GM and W0 values, we would induce an error in the geoid undulation of -93 cm and -41 cm, respectively). Finally, as with all geoid computations, we must define our treatment of the permanent tide system. For the sake of consistency with EGM96, as well as consistency with NGS GPS and leveling data, we will work in the non-tidal (or tide-free) system. The exact equations used to compute the gravity anomalies and geoid undulations from EGM96 geopotential coefficients are (Heiskanen and Moritz, 1967, chapters 1 and 2):
Because EGM96 serves as the long-wavelength foundation for CARIB97, we may say that (nominally) CARIB97 is in the tide-free system. We say "nominally" because the terrestrial and ship gravity measurements are reduced to the International Gravity Standardization Network of 1971 (IGSN 71) which is in the mean tide system, due to the use of the Honkasalo correction term (Morelli, 1971, p. 117). This discrepancy should be studied and ultimately resolved in future geoid models.
3. Computing CARIB97
CARIB97 was computed using the remove-compute-restore technique. This is the method most commonly used at NGS for geoid computations. Complete details about the general gridding of gravity data and remove-compute-restore FFT geoid computation procedures can be found in (Smith and Milbert, 1998). This section will summarize the procedure and highlight specific issues pertaining to CARIB97.
The NGS gravity data base contains approximately 120,000 terrestrial and marine gravity measurements in the Caribbean Sea. These points are mostly found near Florida and Puerto Rico, to support previous geoid models in those areas. In order to model the geoid for the entire Caribbean Sea, NGS invited NIMA to join the project and to contribute their extensive releasable gravity holdings in the area. To the NGS holdings, NIMA added 545,000 gravity measurements. When duplications were removed, the final gravity data set contained 621,000 points. To this, were added 153,000 satellite altimeter derived gravity anomalies. These non-gridded gravity points served as the base data set for CARIB97.
In addition, we compiled a 30 arcsecond gridded topography model based on two different grids. The first is GTOPO30, distributed by the U.S. Geological Survey (USGS, 1997), and was the grid used for all areas of the Caribbean except Florida. The second data set is TOPO30, distributed by the National Geophysical Data Center (Row and Kozleski, 1991), and used for Florida. This mixing of topography data was done because attempts were made to clean TOPO30 of systematic errors and blunders at NGS (D. Milbert, NGS, personal communication), but its coverage doesn't extend past Florida. Therefore GTOPO30 was used to fill in the rest of the Caribbean Sea.
Terrain corrected Bouguer anomalies are gridded first, using splines in tension (Smith and Wessel, 1990) with a tension parameter TB=0.50. This is a looser tension than used in G96SSS (Smith and Milbert, 1998). Due to the impact of tension parameters in areas of sparse data (Cuba, Colombia, Venezuela), differing tensions were tried, until the value of TB=0.50 was found to yield the best overall results. The grid of gravity anomalies covers an area of 7N to 30N and 272E to 304E (56W to 88W).
The extensive ocean areas of the Caribbean Sea contained ship gravity data with varying track spacings. In order to have more complete gravity coverage, we relied heavily on satellite altimeter-derived gravity anomalies. Unlike previous geoid models at NGS, the altimeter anomalies used for CARIB97 were acquired from the National Survey and Cadastre of Denmark (Knudsen and Andersen, 1997). These anomalies were computed from the Geodetic Mission (GM) of ERS-1 and GEOSAT altimeter data, using the EGM96 geopotential field as a base model in a remove-compute-restore fashion. For CARIB97, preliminary tests convinced us to assign a higher priority to altimetric, rather than ship, gravity anomalies. Also, the altimetric anomalies were used as close as 3 km from shore. Using the altimetry data as a check, some ship gravity data were edited out of the final gravity data set. The large mismatches between some ship tracks and altimetry (over 25 mGal offsets in some cases) may be indicative of no cross-over adjustments on the ship tracks. Whatever the cause, this is an excellent method for identifying questionable ship tracks and cleaning gravity data sets.
As per G96SSS (Smith and Milbert, 1998), the geoid undulation is computed through the generalized Stokes' integral, under the assumption of Helmert's second method of condensation. Faye anomalies are substituted for Helmert anomalies. A discussion of the problems with this assumption can be found in Smith and Milbert (1998) and with more detail in (Martinec et al., 1993). After gridding the terrain corrected Bouguer anomalies, we restore the Bouguer plate and remove EGM96 gravity anomalies, yielding residual Faye anomalies. We make the assumption that Faye anomalies are equivalent to Helmert anomalies, and proceed with Stokes' integral, in a 1-D FFT fashion of (Haagmans et al, 1993). This yields residual co-geoid undulations. We restore the EGM96 geoid undulations, and apply a first order indirect effect (dN = pGrH2/g; Wichiencharoen, 1982) to yield CARIB97.
3.1 Cuban Gravity Data Gaps
One serious problem with this procedure was the lack of gravity coverage on Cuba, the largest island in the Caribbean Sea. Only about 250 sparsely sampled gravity points were available, with most clustered in surveys near Guantanamo Bay and Santa Clara. Some of these points appear to be outliers relative to their few neighbors. Large gaps in the gravity coverage, over 100 km each, are common on Cuba. To fill these gaps during the gridding of the Bouguer anomalies, an interpolation procedure must be used. In addition to splines in tension (Smith and Wessel, 1990), we also experimented with using collocation over the island of Cuba itself.
Large gravity gaps cause serious problems in geoid modeling. Interpolating with splines across such a gap usually induces a large systematic error in the geoid model. As pointed out in Small (1992), least squares collocation is a reliable alternative to splines in tension. If one performs interpolations using collocation, assuming zero observational noise, the results should be compatible with splines in tension in areas where data are dense. However, in sparse areas collocation may offer an advantage by using a-priori knowledge of the covariance of the data. When splines in tension yielded questionable Bouguer anomalies on Cuba's interior, we decided to test collocation in the same region. We used program GEOGRID (Tscherning et al,. 1992) to grid terrain corrected Bouguer anomalies in the area 19- 24 N, 274 - 286 E. Although GEOGRID does not use empirically derived covariance functions, it allows the user to define the signal noise, and correlation length. The correlation length of refined Bouguer anomalies will depend on the roughness of the topography, resolution of terrain correction grid, and underlying geological features. In one study (Forsberg, 1984), residual terrain model (RTM) Bouguer anomalies (which differ from refined Bouguer anomalies at long wavelengths) have been shown to have correlation lengths of 20 km in Colorado and 33 km in California. Our own limited tests indicate a correlation length near Cuba around 15-30 km. Using a correlation length of 15 km, with 1 mGal of signal noise, refined Bouguer anomalies were gridded, and compared with the splined grid. The collocation-interpolated Bouguer anomalies were systematically larger than the splined values by 10 mGals (as large as +/- 100 mgals) in central Cuba. This yielded geoid differences at the 50 to 100 cm level. Other correlation lengths, and noise levels were tried; in each case, the results gave worse geoid agreement with EGM96 than either splines or the 15 km, 1 mgal noise test.
For each type of interpolation method tried (splines with varying tensions, collocation with varying correlation lengths and noise levels), a high-resolution geoid model was produced. However, there were always consistently large (2-8 meters) disagreements with the EGM96 geoid in and around Cuba. Because these large discrepancies tended to affect areas hundreds of kilometers away from them, it was necessary to force an agreement with EGM96 in the Cuba region. Splines in tension (TB=0.5) were used to grid the Bouguer anomalies. Then, after computing residual Faye anomalies, a central portion of Cuba was modified. To force an agreement with EGM96, the residual free-air anomalies in a (1.4 x 1.5) central part of Cuba were zeroed, with a 1 degree taper to avoid sharp delineations. This method yields significantly lower residual co-geoid undulations near Cuba, and thus yields better agreement between the high-resolution geoid and the EGM96 (and EGM96/Bouguer) geoid. However we must note that this method does not yield very detailed geoid undulations on Cuba; instead, general agreement with EGM96 is enforced. The only real solution to this problem is the acquisition of gravity data on Cuba. In addition to the Cuba gravity gap, a similar problem exists in the northeast coast of Venezuela. The solution was also to zero a small portion of the residual free-air gravity anomalies in that area as well.
In each of these gridding tests, the data being gridded were complete Bouguer anomalies. One other way (not tested in this paper) of attempting an agreement with EGM96 in gravity gaps would be to remove EGM96 generated gravity anomalies from point Faye gravity anomalies before gridding. In this approach, gridding procedures which attempt to replicate a zero value in gravity data gaps (such as collocation, but not splines) might enforce the agreement with EGM96. Still, no gridding method will replace the need for real data on Cuba.
Figure 1 displays a color image of the CARIB97 geoid height model. The geoid heights range from a low (magenta) of -71.0 m over the Puerto Rico trench to a high (red) of +17.1 m in Costa Rica.
4. The GPS/Tidal Benchmark Data Set
NGS performed a GPS survey of numerous airports throughout the Caribbean as part of a contract with the FAA. At 31 surveyed points (averaging 2 or 3 per island), a tidal height H' (height above local mean sea level) was available in addition to the observed GPS ellipsoidal height (h). Each tidal height on a benchmark has ostensibly been obtained by leveling from a nearby tide gauge. The location of these 31 GPS/tidal benchmarks is found in Figure 2.
A tidal height (H') is not a true orthometric height. A tidal height is a height relative to local mean sea level. Local mean sea level does not correspond to the level of the geoid because of permanent ocean circulation patterns, and other oceanographic signals. This difference between local mean sea level and the geoid level is known as local permanent ocean dynamic topography (PODT, or "V"). We may thus consider V as a bias in H' relative to a true orthometric height, H:
|H = H' + V||(3)|
The value of V changes geographically, on long and short wavelength scales. It is below two meters in magnitude, and generally will not average (spatially) to zero over small geographic regions. As such, each tidal benchmark that refers to a different tide gauge than another benchmark will have a different "bias" (V) in its H' values. Therefore we do not have true orthometric heights, nor even consistently biased orthometric heights (as in NAVD 88 -- see (Smith and Milbert, 1998)) at these 31 points. Each tidal height will have its own bias, unless it shares a tide gauge with another benchmark.
EGM96/Bouguer geoid undulations were interpolated to each of these 31 points and residuals of the following type were formed:
|e = N - (h - H') = N - (h - H + V)||(4)|
|e = (NTRUE + eN) -[ (hTRUE + eh) - (HTRUE + eH) + V||(5)|
|e = eN - eh + eH - V||(6)|
where h is the ellipsoidal (GPS) height, H' is the tidal height, N is the EGM96/Bouguer geoid undulation, and V is the local PODT. The subscripts "TRUE" indicate errorless values, and the eN, eh and eH are the total (random, systematic and blunder) errors in the three heights. Even if eN, eh and eH were purely random, we can not expect the residuals "e" to be zero individually, nor that they will average to zero, because they contain V. The residuals "e" had an average of -98 cm with an RMS of 77 cm about that average. This represents primarily a combination of geoid error (eN), local leveling errors (eH) and local permanent ocean dynamic topography (V). To a lesser extent, GPS height error (eh) is also a component, but the internal error estimate for the GPS heights is 1.8 cm (Soler et al, 1997), which is nearly negligible next to the other error sources. Because geoid error is a significant component in the residuals, it was difficult to identify areas where we had problems in the tidal heights. Reducing the geoid error would leave only V and eH as primary error sources, and would allow a much more accurate determination of problem areas.
After computing CARIB97, residuals were formed as in equation 4, but using the CARIB97 geoid model. The average residual, e, changed to -51 cm with 62 cm RMS about the average. If, momentarily, we were to assume that the only systematic component of that -51 cm was V, then we arrive at an estimate for V of +51 cm (or +98 cm when using EGM96/Bouguer geoid heights). The +51 cm value has fair agreement with the estimate of V determined from the degree 20 spherical harmonic model of V as computed during the creation of EGM96 (Lemoine et al, 1997). In that model, a value of V in the Caribbean is approximately +25 cm. Of course V is not the only systematic quantity in the residuals, so our estimate will be biased by local leveling errors, and to a lesser extent, geoid and GPS errors. In addition, the +25 cm value in the EGM96 V model is extremely long wavelength in structure, and does not contain any information about local currents.
5. Investigation of localized height problems
CARIB97 was computed as a tool to aid in the identification of datum biases and height errors in the Caribbean. However, the presence of V in the ocean, combined with a lack of highly accurate models of V means that we are unable to clearly identify height discrepancies between islands. However, we may draw some conclusions about the heights on individual islands. Two relevant examples are given below in this section.
At this point it is necessary to comment on the reliability of the tidal heights in the Caribbean. Some of the countries surveyed had existing height control in the form of a single benchmark with a given (historical) elevation. Scarce resources have prevented many countries from pursuing more rigorous geodetic control. In some cases a tide gauge existed, but no supporting documentation could show it being tied to a benchmark. In other cases there were multiple benchmarks on various parts of an island, with no tie between them. And, in most cases, the documentation (if it existed) about the original source of an elevation was vague at best (e.g., elevations on one island are documented as "trigonometric heights and/or leveling", without further elaboration). Given all this, it is also impossible to determine what permanent tide system any of the tidal heights are referenced to. The point here is that it will be difficult to draw strong conclusions about the sources of internal inconsistences without knowing more about whether and how each benchmark is related to the others.
5.1 Grand Bahama Island
Figure 3 shows the location and centered residuals of three GPS/tidal benchmarks on the southwest side of Grand Bahama Island. The difference in these residuals ranges over 54 cm in approximately 5 km. This is a 108 ppm tilt. In identifying the cause of this tilt, we can immediately rule out GPS error (with an internal consistency of 1.8 cm RMS for this survey). Although V changes geographically, even the most energetic currents (such as the Gulf Stream) only have 10-20 ppm tilts perpendicular to the flow. Because these points lie along the shoreline (not shown in Figure 3), there would need to be an extremely energetic current flowing directly into the island. This is clearly not the case. Geoid error might be a small contributor; gravity coverage is sparse on Grand Bahama Island, but the altimetry data is dense. We do not anticipate geoid errors in excess of 10 ppm in such a situation. This leaves only errors in the local tidal heights. Whether these errors occurred from leveling or from errors at the tide gauges is unclear. What is clear is that there is an internal disagreement that we do not feel comes from GPS, geoid or oceanographic phenomena. Running a level line between the tidal benchmarks should help identify why this tilt exists, and how to correct it.
Figure 4 shows the location and centered residuals of three GPS/tidal benchmarks near the central portion of Curaçao. Here we see 18 cm of tilt across the island (about 10 km) yielding an 18 ppm tilt. However, we also see 16.2 cm of tilt across half the island (about 5 km), for 32 ppm tilt. Gravity coverage on Curaçao is fair (about 2-3 km nominal spacing) and therefore geoid tilts should be around 5-8 ppm. We can not, on this island, so easily dismiss V as the cause for these tilts because two of the benchmarks are on opposite sides of the island. It may be possible for prevailing atmospheric and oceanographic conditions to significantly differ between east and west sides of the island, and thus lead to different V values. However, that does not explain the 32 ppm tilt seen between the west and central benchmarks. If they refer to the same tide gauge, then we can assume that errors in the reported orthometric heights (in addition to moderate geoid errors) are causing the tilt. If they refer to differing tide gauges, different definitions of mean sea level at the two gauges might be the cause. Again, a level line is a reasonable method for independently checking the internal consistency of these points.
6. Comparison of CARIB97 with EGM96
Aside from comparing geoid undulations at GPS/tidal benchmarks, it is useful to compare the EGM96/Bouguer gridded geoid undulations with CARIB97. Figure 5 displays the differences of CARIB97 minus EGM96/Bouguer geoid undulations. These differences have an average of -16 cm, with an RMS of 42 cm about that mean. Maximum and minimum differences are -2.7 meters (magenta) in Colombia and +2.5 meters (red) in Haiti. Of singular importance in this comparison are not localized extremes, but broad disagreements (over 50 km in spatial resolution). The most obvious example of this is found in northern Colombia, near Santa Marta peak (11 North latitude, 286 East longitude). This mountain rises nearly 5800 meters from the surrounding lowlands, and spreads out over an area 150 km in diameter. In the combined NIMA/NGS gravity data set, there were only two points on this mountain, both at elevations below 900 meters (see Figure 6). NIMA did not have access to additional gravity data on Santa Marta for EGM96 (David Hughes [NIMA], personal communication), so the geoid differences probably result from differences between NIMA and NGS methods of interpolating across gravity data gaps. Future cooperative efforts between NGS and NIMA are scheduled, and solving these issues are important to both agencies.
Other examples of broad disagreement with EGM96 occur near Venezuela (with impacts reaching as far as Trinidad and Tobago), and in the vicinity of Cuba. Even though we attempt to force an agreement with EGM96 through zeroing the residual free air anomalies, discrepancies between CARIB97 and EGM96 are noticeable (Edwards, 1997). Future improvements to CARIB97 will center around fixing these known areas of disagreement.
Most other areas of disagreement have distinct "ringing" signals, where highs and lows are adjoined. This is indicative of the different resolutions of the two grids (50 km for EGM96/Bouguer and 3 km for CARIB97). The local tilts induced by this ringing are the sort of geoid error (mostly EGM96 omission) which we were trying to identify and remove by computing CARIB97.
A gravimetric, geocentric geoid model, CARIB97, was computed by means of a 1-D FFT convolution about the EGM96 spherical harmonic model of the geopotential. Comparison against 31 GPS/tidal benchmarks indicates that geoid omission error has been reduced from previously used EGM96/Bouguer geoid grids. This reduction of geoid error has allowed us to identify island-by-island height problems. Two such problems on Grand Bahama and Curaçao have been identified, but each of the islands has its own datum issues that must be identified individually.
Gravity data gaps around Cuba and Venezuela required that the geoid be artificially forced to agree with EGM96. Coverage varies island by island; sparsely covered areas may have relative geoid errors at the 5-8 ppm level across an island; more densely covered islands should expect relative geoid errors closer to 3-4 ppm. Methods of interpolation across large (100+ km) gravity data gaps are consistently unreliable. We see additional data acquisition in such areas as the only viable solution to removing geoid errors induced by data gaps.
The residuals between CARIB97 and GPS/tidal heights have an average of -51 cm, with a 62 cm RMS. This is a significant improvement over the residuals between the EGM96/Bouguer geoid and GPS/tidal heights, whose average was -98 cm, with an RMS of 77 cm. The -51 cm average residual leads to a +51 cm estimate for the average permanent ocean dynamic topography in the area. This estimate is biased by local leveling errors, and local mean sea level definitions. However, it is in fair agreement with the EGM96 PODT model (degree 20 spherical harmonic expansion), which shows long-wavelength PODT averaging +25 cm in this region. If the systematic errors and sea-level datum definition problems in the area were reduced, the RMS of 62 cm can be expected to reduce significantly.
Further progress in identifying height errors, datum biases and geoid errors will only be possible if we are able to merge accurate models of PODT with our geoid models. Without PODT models, we will always have an unknown bias (or biases) on each island, and be unable to identify whether its source is oceanographic, definitional or measurement error.
8. Recent experimental results
Some recent computational tests were performed after the initial release of CARIB97, which represent our latest attempts to improve the geoid in the Caribbean region. Helicopter borne gravity surveys were performed in 1985 over the Bahamas region and the east coast of Florida, and these data were part of the NIMA gravity holdings in this region. Processing problems which precluded the incorporation of the airborne data into CARIB97 were solved by Bob Moose ([NGS], personal communication), and over 60,000 points, ranging in altitude from 500 to 2000 meters were used in more recent tests. The Bahamas data were especially important, as there are few terrestrial gravity measurements on any of the Bahama islands, and the local depths are shallow enough (under 20 meters) to make altimetry unreliable.
The airborne gravity measurements were added to the terrestrial and altimetry holdings, and an experimental geoid, CARIB98, was produced in the same fashion as CARIB97. This experimental geoid was then compared to the GPS/Tidal benchmarks, as per CARIB97. The average difference changed from -51 cm to -40 cm, with the RMS about the average shrinking from 62 cm to 59 cm. Although these statistics seem to point to an improved geoid, they are inconclusive. Some islands show improved results with CARIB98, while others show degraded results. The overall effect is a Caribbean-wide tilt induced by the local data in the Bahamas. Because CARIB98 remains an experimental geoid, it is not being released publicly yet. Further studies may allow more conclusive answers about the use of the airborne gravity.
Burša, M, 1995: Report of Special Commission SC3, Fundamental constants, International Association of Geodesy, Paris.
Edwards, K., 1997: EGM96/CARIB97 investigations in Trinidad, Presented at the 1997 meeting of the IAG, Rio de Janeiro, Brazil.
Forsberg, R., 1984: Local covariance functions and density distributions, Report no. 356, Department of Geodetic Science and Surveying, The Ohio State University, Columbus, Ohio.
Haagmans, R., E. de Min, and M. van Gelderen, 1993: Fast evaluation of convolution integrals on the sphere using 1-D FFT, and a comparison with existing methods for Stokes' integral, Manuscripta Geodaetica, 18(5), 227-241.
Heiskanen, W.A. and H. Moritz, 1967: Physical Geodesy. W.H. Freeman, San Francisco.
Kearsley, A.H.W., 1986: Data requirements for determining precise relative geoid heights from gravimetry, Journal of Geophysical Research, 91(B9), 9193-9201.
Knudsen, P. and O.B. Andersen, 1997: Improved recovery of the global marine gravity field from the GEOSAT and the ERS-1 geodetic mission altimetry, Gravity, Geoid and Marine Geodesy, International Symposium No. 117, Tokyo, Japan, September 30- October 5, 1996 (GraGeoMar 1996), Springer Verlag.
Lemoine, F.G. et al., 1997: The development of the NASA GSFC and NIMA joint geopotential model. Gravity, Geoid and Marine Geodesy,
International Symposium No. 117, Tokyo, Japan, September 30- October 5, 1996
(GraGeoMar 1996), Springer Verlag.
Martinec, Z., C. Matyska, E.W. Grafarend, and P. Vanicek, 1993: On Helmert's 2nd condensation method. Manus. Geod., 16(5), 417-421.
Morelli, C., 1971: The international gravity standardization net 1971, Special publication no. 4, International Association of Geodesy, Paris.
Moritz, H. 1988: Geodetic Reference System 1980, Bulletin Géodésique, 62(3), pp. 348-358.
National Imagery and Mapping Agency, 1996:
WGS 84 EGM96 15-Minute Geoid Height File, distributed at
Rapp, R.H., 1997: Use of potential coefficient models for geoid undulation determinations using a spherical harmonic representation of the height anomaly/geoid undulation difference, Journal of Geodesy, 71, 282-289.
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.
Small, H.J. 1992: A comparison of techniques for the gridding of satellite altimeter data, Thesis, The Ohio State University, Columbus, OH, 91 pp.
Smith, D.A., and D.G. Milbert, 1998:
The GEOID96 high resolution geoid height model for the United States,
Journal of Geodesy, Vol. 73, No. 1, p. 1-9, 1999)
Smith, W.H.F., and P. Wessel, 1990: Gridding with continuous curvature splines in tension. Geophysics, 55(3), 293-305. Soler, T., L. Hall and C. Reed, 1997: Establishment of a GPS high accuracy reference geodetic network in the Caribbean, submitted for publication to
Surveying and Land Information Systems. Tscherning, C.C. et al, 1992:GRAVSOFT -- a software system for physical geodesy, in Proceedings of the 1st continental workshop on the geoid in
Europe, Prague. United States Geological Survey, 1997: GTOPO30, Global 30 arc-second elevation data set, WWW distribution at
http://edcwww.cr.usgs.gov/landdaac/gtopo30/gtopo30.html Wichiencharoen, C., 1982: Fortran programs for computing geoid undulations from potential coefficients and gravity anomalies. Internal Report of
the Department of Geodetic Science and Surveying, The Ohio State University, Columbus, OH.
Smith, W.H.F., and P. Wessel, 1990: Gridding with continuous curvature splines in tension. Geophysics, 55(3), 293-305.
Soler, T., L. Hall and C. Reed, 1997: Establishment of a GPS high accuracy reference geodetic network in the Caribbean, submitted for publication to Surveying and Land Information Systems.
Tscherning, C.C. et al, 1992:GRAVSOFT -- a software system for physical geodesy, in Proceedings of the 1st continental workshop on the geoid in Europe, Prague.
United States Geological Survey, 1997: GTOPO30, Global 30 arc-second elevation data set, WWW distribution at http://edcwww.cr.usgs.gov/landdaac/gtopo30/gtopo30.html
Wichiencharoen, C., 1982: Fortran programs for computing geoid undulations from potential coefficients and gravity anomalies. Internal Report of the Department of Geodetic Science and Surveying, The Ohio State University, Columbus, OH.