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DOI 10.15517/RI.V30I1.35839
Ingeniería 30 (1): 75-94, enero-junio, 2020. ISSN: 2215-2652. San José, Costa Rica
Geoid Heights in Costa Rica,
Case of Study: Baseline Along the Central Pacic Zone
Alturas Geoidales en Costa Rica, caso de estudio:
Línea base a lo largo de la Zona Pacíco Central
Alonso Vega Fernández
Surveying Engineering School, University of Costa Rica, San José, Costa Rica
alonso.vega_f@ucr.ac.cr
Oscar H. Lücke Castro
Central American School of Geology, University of Costa Rica, San José, Costa Rica
oscar.luckecastro@ucr.ac.cr
Jaime Garbanzo León
Surveying Engineering School, University of Costa Rica, San José, Costa Rica
jaimegarbanzo@gmail.com
Recibido: 8 de enero 2019 Aceptado: 23 de setiembre 2019
_________________________________________________________
Abstract
A precise orthometric height (H) and orthometric height difference (ΔH) determination is required in
many elds like Construction, Geodesy and Geophysics. H is often obtained from an ellipsoidal height
(h) and geoid height (N) of a geopotential model (GM) because this computation does not have the spirit
leveling restrictions on long distances. However, the H accuracy depends on the GM local area adaptation,
and current global geopotential models (GGMs) have not been yet evaluated for Costa Rica. Therefore, this
paper aims to determine which GGM maintains a better t with a GPS/levelling baseline that contains the
gravity full spectrum. A 74 km baseline was measured using GPS, spirit leveling and gravity measurements
to validate the N computed from EGM2008, EIGEN-6C4, GECO, EGM96, GGM05C and GOCO05C. First,
an absolute N assessment was made, where geoid height from the GGMs (N
GGM
) were directly compared to
the geometric geoid heights (N
geo
) obtained from GPS and spirit levelling. A bias t (N
bias
) was computed
from this comparison to the GGMs with respect to the local vertical reference surface (W
0
). This N
bias
value
differs from model to model but the best t is given by GECO. By subtracting the N
bias
, a relative geoid height
(ΔN) assessment was designed to compare the differences between GGM relative geoid height (ΔN
GGM
)
and geometric relative geoid height (ΔN
geo
) on segments along the baseline. The ΔN comparison shows that
EGM2008, EIGEN-6C4 and GECO better represent the Costa Rican Central Pacic Coastal Zone and over
long distances, ΔH can be computed with a decimeter to centimeter precision.
Keywords:
Geodesy, Geoid, Geodetic Levelling, Ortometric Hight, Hight Systems.
VEGA, LÜCKE Y GARBANZO: Geoid Heights in Costa Rica, Case of Study...
76
Resumen
La determinación de alturas ortométricas (H) y diferencias de altura ortométrica (ΔH) precisas es requerida
en campos como construcción, geodesia y geofísica. La H puede obtenerse midiendo altura elipsoidal (h) y
calculando altura geoidal (N) de un modelo geoidal (GM), evitando limitaciones de la nivelación geodésica en
largas distancias. Sin embargo, la precisión de H dependerá de la adaptación del GM al sitio, y la adaptación
de los modelos de geopotencial globales (GGMs) actualmente se desconoce para Costa Rica. Por tanto, el
presente artículo busca determinar cuál GGM mantiene mayor congruencia con una línea base de GPS /
Nivelación que contenga el espectro de gravedad completo. Para esto, se midió una línea base de 74 km
utilizando GPS, nivelación geodésica y mediciones gravimétricas, para validar el N calculado con EGM2008,
EIGEN-6C4, GECO, EGM96, GGM05C y GOCO05C. Primero, se evaluó el valor absoluto de N, donde
N de los GGMs (N
GGM
) se comparó directamente con N geométrica (N
geo
) obtenida con GPS y nivelación
geodésica. Se obtuvo un sesgo (N
bias
) de los GGMs respecto de la supercie de referencia vertical local (W
0
).
Este valor diere de modelo a modelo, pero GECO se ajusta más a la zona. Al sustraer el Nbias, se aplicó una
evaluación de la altura geoidal relativa (ΔN
GGM
) respecto de la altura geoidal geométrica relativa (ΔN
geo
) en
segmentos a lo largo de la línea base. La comparación de ΔN mostró que EGM2008, EIGEN-6C4 y GECO
representan de mejor forma la zona costera del Pacíco Central de Costa Rica, y que en largas distancias es
posible obtener ΔH con precisiones decimétricas a centimétricas.
Palabras clave:
Geodesia, Geoide, Nivelación, Nivelación Geodésica, Altura Ortométrica, Sistemas de Altura.
Ingeniería 30 (1): 75-94, enero-junio, 2020. ISSN: 2215-2652. San José, Costa Rica DOI 10.15517/RI.V30I1.35839
77
1. INTRODUCTION
A precise determination of an orthometric height (H) is required in many elds like Construc-
tion, Geodesy and Geophysics. Orthometric heights are often obtained using the GNSS positioning
technique and a geopotential model to replace conventional leveling techniques because there is no
restriction in distances, and spirit leveling can be time-consuming. For these reasons, the behavior
of these models, in terms of accuracy, is always a concern. Geopotential models (GMs) are used
to correct ellipsoidal heights (h) value, which allows to obtain an H because it can supply a geoid
height (N). This N value is a difference between h and H that occurs because of the heterogeneities
in distribution of Earth’s masses (Vaníček, Kingdon, & Santos, 2012).
There are many Global Geopotential Models (GGMs) such as the EGM2008, EIGEN-6C4,
GECO (Förste et al., 2014; Gilardoni, Reguzzoni, & Sampietro, 2016; Pavlis, Holmes, Kenyon,
& Factor, 2012), which provide high global accuracy. However, these accuracies are derived from
error estimates of the least squares adjustment results called “internal error” and the challenge lies
on the determination of the “external error” (Gruber, 2004). The external error determines how
similar is a GM to the real geoid. Thus, a GGM sometimes does not provide a good representation
for local areas, especially if the GGM does not have terrestrial or aerial gravity of the local area.
Consequently, there are adaptations of the models for local areas which provides a more precise
representation (Sánchez, 2003; Sobrino, Mourón, & Fernández, 2009). However, this situation is
not the case for some developing countries in Central America and among them Costa Rica.
Currently, there is no local GM adapted to Costa Rica, and the performance of GGMs has not
yet been evaluated in detail for the country. Köther et al. (2012) provide a regional evaluation of
EGM2008 in terms of gravity anomalies compared to surface gravity data. Moreover, there has
been also an attempt to measure the quality of the OSU-91A by comparing a N computed from
GPS measurements and known H Benchmarks (BMs) of the National Geographic Institute (Diaz,
1997). However, it was found that these N were not comparable, and a quality measure could not
be done. There were studies carried out on local geoid determination (Cordero, 2010; Moya &
Dörries, 2016) but the areas taken into account for these studies were less than 50 km
2
. Thus, the
t of different geoid models in most of the country remains unknown.
GNSS measurements and spirit leveling are frequently used for quality assessment by com-
puting a geometric N (N
geo
). This GNSS measurements are available with their respective (H) in
areas such as United States, Canada, Japan, Brazil and the European Countries. In addition, GGMs
accuracy determination studies are abundant.
For instance, Gruber (2009) tested EGM08, EGM96 and other GGMs in Canada, Japan, and
Europe; Szűcs (2012) compared GOCE and GRACE derived models to GNSS/H corrected with
EGM08 to match the spectral contents; Guimarães et al. (2012) assessed various models for the
State of Sao Pablo, Brazil. All these studies either tried to match the spectral contents or estimate
an omission error for each model.
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78
Regarding, Kotsakis and Katsambalos (2010), they used the full spectral contents of the GNSS/H
to evaluate EGM08, EGM96 and other EIGEN derived models and, they found a bias after a
least-squares determination. This bias was corrected to compare the different studied models.
Although to match the spectral contents of the model is a more rigorous approach, the full spectrum
of gravity effects is contained in measurements in practical applications. Thus, scientist, engineers,
geographers and other common GGM users could chose a determined GGM model knowing what
uncertainties are in their measurements.
Consequently, this paper aims to determine which geoid maintains a better t with a GPS/
leveling baseline that contains the full spectrum of gravity contents for engineers and researchers
having better criteria for choosing between GGMs for their application. Also, GPS/leveling baseline
was independently measured to avoid problems presented by other studies (Cordero, 2010; Diaz,
1997; Moya & Dörries, 2016), which relied on the erroneous local leveling network or studied a
small area to be compared to GGMs.
2. METHODS
2.1 Data collection
The required data for this evaluation is: latitude, longitude, ellipsoidal height, spirit leveling
and gravity measurements. These data are used as listed below.
1. GNSS latitude (φ) and longitude (λ) are used to obtain the geoid heights for each GGM
(N
GGM
).
2. Orthometric heights (H) are obtained from gravity measurements, spirit leveling, and a
known point referred to the mean see level.
3. A geometric N (N
geo
) can be computed from an ellipsoidal height (h) and a H.
These GNSS measurements were obtained on 6 BMs set along 74 km of the Central Pacic of
Costa Rica; two GNSS receivers (Dual-frequency Topcom GR-3) collected the data simultaneously
using the relative static method. This technique allows to have a correlation between the two recei-
vers to calculate relative position with the highest precision in data post-processing. Even, it is use
to determine geometric height difference referred to an ellipsoid (Instituto Geográco Agustín
Codazzi, 1997).
The eld survey was planned in order to measure a GNSS baseline during at least 1 hour of
continuous observation on two consecutive BMs. One of the receivers remains in observation
while the other receiver is transferred to the next BM to link the consecutive GNSS baselines. In all
cases, a 15º elevation cut-off angle, PDOP values of less than 7, and 15 seconds sampling rate were
used in data collection. The data post-processing took into account the IGS precise orbits (epoch:
2016.270) with respect to the WGS84 ellipsoid (National Imagery and Mapping Agency, 1997)
79
Moreover, several GGMs (see Table 1) were chosen based on the following selection criteria:
the popularity, the geoid model maximum degree (
max
), maximum spatial resolution (ψ
max
) and the
satellite missions used for its calculation. Most are recent GGMs that have been developed with
combined satellites missions, satellite altimetry data, and some GGMs have terrestrial gravity data.
The International Centre for Global Earth Models (ICGEM) provided these GGMs, which can be
downloaded from the online computation platform, where N is determined from the height anomaly
plus spherical shell approximation of the topography (Barthelmes, 2013). The calculation service
uses a web-interface (http://icgem.gfz-potsdam.de/) to calculate gravity eld functionals from the
spherical harmonic models on grids (for this study, grids are referring to WGS84, and unmodied
model tide system). In order to obtain a N for each GGM in the exact location a script was progra-
med to apply an Inverse Distance Weighted Interpolation (IDW) on the grid.
Table 1. GGMs used on N evaluation.
GGM
max
ψ
max
(km) Data Reference
EGM2008 2190 9.132 S (Grace), G, A (Pavlis et al., 2012)
EIGEN-6C4 2190 9.132 S (Goce, Grace, Lageos), G, A (Förste et al., 2014)
GECO 2190 9.132 S (Goce), EGM2008 (Gilardoni et al., 2016)
EGM96 360 55.556 S, G, A (Lemoine et al., 1998)
GGM05C 360 55.556 S (Grace, Goce), G, A (Ries et al., 2016)
GOCO05C 720 27.778 S, G, A
(Fecher, Pail, & Gruber,
2016)
S = Satellite Tracking Data, G = Terrestrial Gravity Data, A = Altimetry Data.
The EGM2008 is a high resolution GGM that has been well studied worldwide, showing good
results for many regions (Dawod, Mohamed, & Ismail, 2009; Gruber, 2009; Kotsakis & Katsam-
balos, 2010; Łyszkowicz, 2009). EIGEN-6C4 and GECO use the EGM2008 terrestrial gravimetric
data as calculation input thus there is a correlation between these GGMs.
Furthermore, both EIGEN-6C4 and GECO also include satellite data that was not available
for EGM2008, so the comparison of the N results with these GGM is necessary. The GECO geoid
model combines the GOCE satellite-only global model and EGM2008 to improve the model accu-
racy in the low to medium wavelengths in some areas (Gilardoni et al., 2016). EIGEN-6C4 uses
LAGEOS, GRACE and GOCE satellite data and incorporates an EGM2008 geoid height grid for
the continents (Förste et al., 2014).
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80
In the case of the spirit leveling, it was carried out following the NOAA standards to match the
maximum precision (NOAA, 1981, 1995), except for the staff material because of resource limi-
tations. NOAA requires IDS (Invar Doubled Scale) but a bar-code metal staff was used. This staff
add some uncertainties associated with thermal expansion. Moreover, the automatic level used in
this study was a Leica Sprinter 250M. The accuracy of the instruments used in spirit leveling was
veried with ISO 17123-2:2001 test (International Organization for Standarization, 2001).
It was necessary to use gravity measurements in order to apply orthometric corrections to the
levelling height differences along the baseline. These corrections can be found in the work of Kao
and Ning (2008). The gravity survey was carried out with a relative Burris gravity meter (B-106).
The survey started on a known gravity value of a station at University of Costa Rica (UCR) to set
up a secondary gravity station near to the baseline middle, on the EST BM (see Figure 1). The gra-
vity values were translated to the other stations.
The initial and nal measurements of the gravity campaign were made on the gravity station
at UCR, and three others were performed on the secondary station (EST). Both procedures were
made to control the instrumental drift and the effects caused by atmospheric pressure changes
(Seigle, 1995).
Finally, observed gravity values were corrected to get the gravity value on the ground surface.
The corrections related with variation in latitude and elevation were applied on the eld. Other as
earth tides correction and instrument drift correction were applied after the measurement works.
For each BM, the N
geo
was calculated applying Equation 1 (Hofmann-Wellenhof & Moritz,
2006).
(1)
Figure 1. The BM distribution in the Central Pacic Coastal Zone. Coordinates are shown in WGS84.
Ingeniería 30 (1): 75-94, enero-junio, 2020. ISSN: 2215-2652. San José, Costa Rica DOI 10.15517/RI.V30I1.35839
81
The h values were obtained as described above, and H of the BMs with the orthometric height
differences (ΔH) measurements. The orthometric heights were referred to the National Vertical
Network reference frame (W
0
unknown). The vertical geodetic datum of Costa Rica was referred
to a mean sea level calculated in the Puntarenas’ port, using tide-gauge measurements during the
period between 1942 - 1969 (Camacho, 1997).
These values provide the dataset for the GGMs evaluation. The GGMs omission error was not
estimated, applying an evaluation with unmodied GGMs. In addition, the BMs distribution on the
Central Pacic Coastal Zone are shown in Figure 1.
2.2 Assessment of the GGMs
For the N
GGM
assessment, there are two methods to determine the GGM performance in terms of
accuracy. In the rst one, an absolute N assessment was carried out where N
GGM
values were directly
compared with the N
geo
values. The subtraction of these two values results in an approximation of
the bias from GGMs respect to the local vertical reference (N
bias
) as shown in Equation 2. For the
second method, a relative geoid height (ΔN) assessment was designed to compare the differences
between relative values of segments along the baseline (S). The ΔNs were computed subtracting
values from the same dataset taking one point as reference (see Equation 3) and depending on the
separation of the benchmarks in order to assess the GGMs behavior with different baseline lengths.
(2)
(3)
In the case of ΔN
geo
, this calculation uses the height differences between two BMs based on
height systems relationship of Equation 1, using orthometric and ellipsoidal height difference to
obtain Equation 4 (Hirt et al., 2011; Torge & Müller, 2012).
(4)
Then, the relative geoid height (ΔN) assessment uses the residuals between ΔN
GGM
and ΔN
geo
(ΔN
GGM-geo
) as shown in Equation 5.
(5)
2.3 Statistical testing
When comparing two different datasets, it is highly relevant to know whether these datasets
have the same precision. Thus, an F-test of equality of the variances must be performed. The null
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82
hypothesis of these test, states that the variances are statistically similar and follow the relation
shown by Equation 6.
(6)
where S
1
y S
2
are the sample variances with S
1
> S
2
.
This hypothesis can be rejected when F, computed from Eq 6, is larger than the critical value
(F
α
), computed from the Fisher distribution (Ghilani, 2011).
First, this F-test of equality of the variances was used in order to test if the differences in the
geoid heights (ΔN
GGM-geo
) are comparable between geoids, which provides some criteria to decide
which model performs better.
Second, if the null hypothesis (H
0
) is accepted, a two sampled t test is performed to know if the
ΔN
GGM-geo
means are signicantly different. This test is computed from Equations 7 and 8.
(7)
(8)
where n
1
and n
2
are the sample sizes; S
1
and S
2
are the standard deviations; and are the
means for sample 1 and sample 2 respectively.
H
0
of this test states that the differences of the mean population are 0 ((μ
1
-μ
2
)= 0); The non-equa-
lity of the means is chosen as H
a
(Ayyub & McCuen, 2011).
2.4 RMS quality assessment
An alternative quality assessment can be performed by computing the root mean squared (RMS),
where the residuals are the ΔN
GGM-geo
values. The RMS is computed using Equation 9.
(9)
The RMS values can be computed for each baseline length. In addition, all residuals for every
baseline length can be computed in order to obtain a global RMS value, which shows the GGM
(ΔN
GGM
) t relative to the standard of comparison (ΔN
GGM-geo
).
Ingeniería 30 (1): 75-94, enero-junio, 2020. ISSN: 2215-2652. San José, Costa Rica DOI 10.15517/RI.V30I1.35839
83
3. RESULTS AND ANALISIS
3.1 Absolute Assessment
There are two alternatives to determine the GGMs characteristics (see the Assessment of the
GGMs section). The rst alternative implies the existence of a bias t while the second one implies
the removal of this bias. First, we assess the absolute bias of the evaluated models. Table 2 presents
the different heights obtained to each BM along the baseline that were necessary to N
geo
calculation.
H values are referred to Costa Rica’s ofcial vertical network.
Table 2. The H, h and N
geo
values obtained to each BM; the results are given in meters.
BM h σ H σ N
geo
σ
1Q 16.516 ±0.011 4.8832 ±0.000 11.6328 ±0.011
861B 18.011 ±0.000 6.2869 ±0.004 11.7241 ±0.004
PS 24.291 ±0.008 13.0252 ±0.006 11.2658 ±0.010
EST 18.641 ±0.014 8.2167 ±0.008 10.4243 ±0.016
HERM 16.475 ±0.019 6.0694 ±0.011 10.4056 ±0.022
HERR 21.089 ±0.022 10.5023 ±0.011 10.5867 ±0.026
Table 3. GNSS position of each BM in the baseline.
BM LAT LON
1Q 09°25’29.31309” -84°10’12.36605”
861B 09°29’27.03686” -84°12’33.26513”
PS 09°32’03.28487” -84°17’03.33745”
EST 09°31’56.93949” -84°27’11.30845”
HERM 09°34’29.53570” -84°36’11.46800”
HERR 09°39’17.77866” -84°38’54.46786”
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84
Table 4. N
GGM
values obtained whit each GGM; the results are given in meters.
BM EGM96 EGM2008 EIGEN-6C4 GECO GGM05C GOCO05C
1Q 11.5056 11.7567 11.6115 11.6056 12.3335 11.7497
861B 11.6954 11.8427 11.6807 11.6675 12.5576 12.0240
PS 11.4139 11.4343 11.2550 11.2502 12.1767 11.7127
EST 10.3639 10.5824 10.4121 10.4183 10.8111 10.6738
HERM 10.0183 10.5157 10.3615 10.3889 10.2546 10.3479
HERR 10.3644 10.8089 10.6561 10.7087 10.5956 10.5072
Table 5. Differences between N
geo
and N
GGM
(N
bias
), and mean bias approximation;
the results are given in meters.
BM EGM96 EGM2008 EIGEN-6C4 GECO GGM05C GOCO05C
1Q -0.127 0.124 -0.021 -0.027 0.701 0.117
861B -0.029 0.119 -0.043 -0.057 0.834 0.300
PS 0.148 0.169 -0.011 -0.016 0.911 0.447
EST -0.060 0.158 -0.012 -0.006 0.387 0.250
HERM -0.387 0.110 -0.044 -0.017 -0.151 -0.058
HERR -0.222 0.222 0.069 0.122 0.009 -0.079
Bias t -0.113 0.15 -0.01 0 0.449 0.106
STD 0.182 0.042 0.042 0.062 0.443 0.173
Moreover, the positions for this study where obtained Fixing the 861B BM point in a relative
static post-process (see Table 3). The 861B position was computed using the NRCAN-PPP service
(https://webapp.geod.nrcan.gc.ca/geod/tools-outils/ppp.php). A centimetric accuracy of these posi-
tions is enough, since the geoid is a very smooth surface and the maximum resolution of a GGM is
about 9 km. This information was needed to calculate the N
GGM
shown in Table 4.
Ingeniería 30 (1): 75-94, enero-junio, 2020. ISSN: 2215-2652. San José, Costa Rica DOI 10.15517/RI.V30I1.35839
85
The mean value of the subtraction of N
GGMs
and N
geo
results in the least-squares constant bias
value for each GGM as shown in Table 5. These values show that this vertical offset varies from
model to model. However, EIGEN-6C4 and GECO t the best to this leveling base line.
Likewise, this result shows the inuence of the degree and order of a model despite the smooth
variation of the pacic leveling baseline. The small bias t of EIGEN-6C4 and GECO is an evi-
dence of the improvement made with the inclusion of GOCE data. Figure 2 shows a box-whisker
chart, where the distribution of the values can be better seen. Despite these results, the determina-
tion of a bias t for the whole country remains as an open question because a better distribution
and a greater number of samples are needed.
Authors state that a bias is an indicator of error in the geopotential model spherical harmonic
coefcients (Kotsakis & Katsambalos, 2010; Łyszkowicz, 2009). Figure 3 shows a pot of the N
values, which describe better the behavior among the models and the local orthometric height datum
with unknown geopotential reference value (W
0
). This approximation of the bias t gives a better
understanding of the reference surface used in the old Costa Rican height system.
Figure 2. Bias for the GGMs in meters.
Figure 3. A no signicant vertical offset is shown between local
reference geoid (W
0
unknown) and evaluated GGMs.
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86
3.2 Relative Assessment
The other method for the evaluation of GGMs is the removal of the bias by computing the
differences in N (ΔN). Here, the differences of the ΔNs subtraction (Equation 5) were analyzed for
all possible baseline lengths, grouping these residuals into different separation length categories
(<20 km, 21 to 40 km, 41 to 65 km).
Table 6. ΔN
GGM-geo
values with <20 km baselines length, mean (m) and standard deviation (S) of each
GGM; the results are given in meters.
Baseline EGM96 EGM2008 EIGEN-6C4 GECO GGM05C GOCO05C
1Q - 861B 0.098 -0.005 -0.022 -0.029 0.133 0.183
861B - PS 0.177 0.050 0.033 0.041 0.077 0.147
1Q - PS 0.275 0.045 0.011 0.012 0.210 0.330
PS - EST -0.208 -0.010 -0.001 0.010 -0.524 -0.197
EST - HERM -0.327 -0.048 -0.032 -0.011 -0.538 -0.307
HERM - HERR 0.165 0.112 0.114 0.139 0.160 -0.022
m 0.030 0.024 0.017 0.027 -0.080 0.022
S 0.240 0.057 0.053 0.060 0.352 0.243
Table 7. Fisher test on the variance (S
2
) for ΔN
GGM-geo
values of <20 km baseline length.
EGM2008 EIGEN-6C4 GECO GGM05C GOCO05C
EGM96 17.99 20.69 16.18 2.14 1.02
EGM2008 1.15 1.11 38.56 18.36
EIGEN-6C4 1.28 44.36 21.12
GECO 34.69 16.52
GGM05C 2.10
Critical value (F
95
) = 5.05; Degree freedom (n-1) = 5
Ingeniería 30 (1): 75-94, enero-junio, 2020. ISSN: 2215-2652. San José, Costa Rica DOI 10.15517/RI.V30I1.35839
87
In some cases, these separation lengths were less than the maximum spatial resolution (ψ
max
).
For instance, EGM96 and GGM05C have a ψ
max
greater than 55 km, which is longer than most
baselines. However, these baselines still provide useful data for studying the geoid surface beha-
vior. On the other hand, the overall precision estimated by ψ
max
is not absolute for the GGM’s entire
surface, since the data accuracy depends largely on the degree of the adaptation of the evaluated
area (Vanícek, Santos, Tenzer, & Hernández-Navarro, 2003). Despite this, the maximum degree
and order for each model is evaluated since knowledge of the error of the GGMs is important for
precise applications.
Table 6 shows the differences (Equation 5) of the values calculated for the baselines that are
less than 20 km. The mean (m) of ΔN
GGM-geo
is expected to be 0 because the ΔN are free from the
biases and the substation (ΔN
GGM-geo
) should reect a normal distribution. The results shown in Table
6 vary from a few mm (e.g. EGM2008/EIGEN-6C4), to more than half a meter (e.g. GGM05C).
The lowest mean value is obtained for EIGEN-6C4, where m = 1.72 cm and S = 5.28 cm.
However, EGM2008 and GECO also have similar values. Thus, a sher test is performed to assure
the comparison of values with the same precision (see Table 7). The test on variances (S
2
) separate
the GGMs into two groups which are statistically similar.
The rst group contains EGM2008, EIGEN-6C4 and GECO, the second one contains the
EGM96, GGM05C, GOGO05C. The rst group contains the lowest standard deviation therefore
we infer that this group should have a better t with the values obtained for the baseline measured
geometrically with gravity corrections.
Table 8. ΔN
GGM-geo
values of 21 to 40 km baselines length, mean (m) and standard deviation (S) of each
GGM; the results are given in meters.
Baseline EGM96 EGM2008 EIGEN-6C4 GECO GGM05C GOCO05C
861B - EST -0.032 0.039 0.031 0.051 -0.447 -0.050
1Q - EST 0.067 0.034 0.009 0.021 -0.314 0.133
PS - HERM -0.535 -0.058 -0.033 -0.001 -1.062 -0.505
EST - HERR -0.162 0.064 0.082 0.128 -0.378 -0.329
m -0.166 0.020 0.022 0.050 -0.550 -0.188
S 0.264 0.054 0.048 0.056 0.346 0.284
These results for the second group were expected because EGM96 was generated using older
data of orbit tracking, ocean topography and surface gravity (Lemoine et al., 1998). On the other
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hand, GGM05C and GOCO05C use ll-in gravity anomalies on the study area (not actual terres-
trial gravity data). This lack of terrestrial gravity data implies that these GGMs are able to resolve
long-wavelengths based solely on satellite data (Gruber, 2004) which results in high standard devia-
tions for the baseline of < 20 km (see Table 4).
For the 21 to 40 km baseline, the models EGM2008, EIGEN-6C4 and GECO still show when
comparing them to the other three GGMs (see Table 8 and Table 9). The test on the variances (Table
9) still shows a similar trend to Table 4 but the critical value has increased because a degree of
freedom is lost.
Table 9. Fisher test on the variance (S
2
) for ΔN
GGM-geo
of 21 to 40 km baseline length.
EGM2008 EIGEN-6C4 GECO GGM05C GOCO05C
EGM96 24.30 30.32 21.88 1.72 1.16
EGM2008 1.25 1.11 41.74 28.27
EIGEN-6C4 1.39 52.08 35.27
GECO 37.58 24.45
GGM05C 1.48
Critical value (F
95
) = 9.28; Degree freedom (n-1) = 3
Table 10. ΔN
GGM-geo
of 41 to 65 km baseline length, mean (m) and standard deviation (S) of each GGM;
the results are given in meters.
EGM96 EGM2008 EIGEN-6C4 GECO GGM05C GOCO05C
861B - HERM -0.359 -0.009 -0.001 0.040 -0.984 -0.358
1Q - HERM -0.260 -0.014 -0.023 0.011 -0.852 -0.175
PS - HERR -0.370 0.054 0.080 0.138 -0.902 -0.526
1Q - HERR -0.095 0.098 0.091 0.149 -0.692 -0.196
m -0.271 0.032 0.037 0.085 -0.858 -0.314
S 0.127 0.054 0.057 0.069 0.123 0.163
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Table 11. Fisher test on the variance (S
2
) for ΔN
GGM-geo
of 41 to 65 km baseline length.
EGM2008 EIGEN-6C4 GECO GGM05C GOCO05C
EGM96 5.63 4.96 3.38 1.07 1.65
EGM2008 1.14 1.67 5.26 9.28
EIGEN-6C4 1.47 4.63 8.17
GECO 3.15 5.56
GGM05C 1.76
Critical value (F
95
) = 9.28; Degree freedom (n-1) = 3
Table 12. Two-sampled t test to determine if two means are statistically similar
for the baseline of 41 to 65 km.
EGM2008 EIGEN-6C4 GECO GGM05C GOCO05C
EGM96 -4.39 -4.41 -4.90 6.63 0.41
EGM2008 -0.11 -1.19 13.26 4.02
EIGEN-6C4 -1.06 13.18 4.05
GECO 13.34 4.49
GGM05C -5.32
T value interval = -2.45, 2.45; D. freedom (v) = 6
In terms of Table 11, it shows that there are not statistical differences in the variances for baseli-
nes from 41 to 65 km. However, the differences in mean values increased for GGM05C, GOCO05C
and EGM96 (see Table 10). By applying a two-sampled t test, results show that the computed t
values are between the condence interval when comparing EGM2008, EIGEN-6C4 and GECO
(see Table 12). Thus, the null hypothesis that states that these means are statistically equal for these
GGMs is accepted.
Moreover, EGM96 shows no statistical difference when compared to GOCO05 for these base-
line lengths. The two-sampled t test was also applied to the baselines of <20 km and 21 to 40 km
but it doesn’t give any important information because the standard deviations of the dataset were
high, and the mean is close to 0 (see Table 6 and Table 8). However, it can be stated that there is
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90
not a signicant difference on the mean values for EGM2008, EIGEN-6C4 and GECO in all sta-
tistical tests performed.
Another alternative to test the GGMs is evaluating the RMS values and taking the residuals as
a subtraction of ΔN
geo
to ΔN
GGM
(ΔN
GGM-geo
). If all residuals for the different baseline lengths are
combined to compute the RMS, then a global evaluation for each GGM can be performed which
provides extra information and validation of the analysis. There is a clear separation of the two
GGMs groups; the rst group showed RMS values bellow 15 cm and, the second group showed
RMS values above this value (see Table 13). Moreover, GGM05C resulted in the worst RMS, which
is twice as much as the closest RMS value obtained for GOCO05C.
Figure 4 shows the RMS, where EGM2008, EIGEN-6C4 and GECO presented the best cross-
comparison results for most evaluation cases, with RMS magnitudes similar for each baseline
length. These results indicate that the baseline length variation does not signicantly change the
differences between ΔN
geo
and ΔN
GGM
for these GGMs.
Finally, this project is a consequence of a problem that researchers will have when evaluating
a GM. The leveling heights from the old altimetric system are unreliable because the accuracy is
unknown. Furthermore, this altimetric system is very old and has not been maintained correctly
in a country that has strong surface modication because tectonic forces. Therefore, Costa Rica
lacks an independent and reliable validation method, which will a high uncertainty when any new
GGMs is computed.
Table 13. Global RMS values for all baseline lengths of each GGM. No represents
the baseline number; the results are given in meters.
Baseline length
(km)
No EGM96 EGM2008 EIGEN-6C4 GECO GGM05C GOCO05C
<20 6 22.1 5.7 5.1 6.1 33.1 22.3
21 to 40 4 28.2 5.1 4.7 7.0 62.6 30.9
41 to 65 4 29.3 5.7 6.2 10.4 86.4 34.4
Global 14 26.2 5.8 5.7 8.2 58.1 29.2
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91
Figure 4. RMS comparison for GGMs for each different long baseline evaluation. EIGEN-6C4 and EGM2008 presented similar
RMS values in each case with and, there is an evident of the RMS values for GGMs that are derived solely from satellite data.
4. CONCLUSIONS
An approximation of the bias t (N
bias
) was computed for global geopotential models (GGMs),
which show a correlation with the local reference geopotential value (W
0
) of old local Costa Rican
reference datum. The bias t is not constant among the models; however, the best results are shown
by EIGEN-6C4 and GECO. Because of the subtraction of the geometric geoid separation (N
geo
) to
the GGMs geoid separation (NGGM) is close to 0 and the standard deviation (S) is high for some
models, a two-sampled t test will result in a null hypothesis. Consequently, a sher test on the
variance (S
2
) can give information about which geoids are comparable depending on the precision.
In baseline from 45 to 65km, each tested geoid obtains a Fisher value equal or less than the criti-
cal value. This means that all tested geoids were equally precise. Thus, a two-sampled test could
be applied. This statistical test shows that EGM2008, EIGEN-6C4 and GECO have no signicant
difference, when comparing the subtraction of the ΔN (N
GGM-geo
). This tendency is also shown in the
graphical representation of the RMS, which can also be good for geoid evaluation. Theoretically,
VEGA, LÜCKE Y GARBANZO: Geoid Heights in Costa Rica, Case of Study...
92
after the bias removal, these differences should be 0 if the geoids computed from GGMs are equal
to the geometrically computed baseline of the local geoid.
The EGM2008, EIGEN-6C4 and GECO geoid models better represent the Central Pacic
Coastal Zone. Also, each of these geoids are more suitable for use in local engineering or scientic
projects such as leveling work to obtain orthometric height difference (ΔH) over long distances
with GNSS/levelling technique which requires decimeter to centimeter precision. Finally, more
surveys should be done for to have better testing of geoid models for this country because of the
lack of reliable leveling data in Costa Rica. This issue will appear whenever a new geoid determi-
nation is available.
ACKNOWLEDGMENTS
The authors would like to thank Survey Engineering School and Central American School of
Geology, from University of Costa Rica, for the support during spirit levelling, satellite positioning
and gravity measurements.
SYMBOLS
GM Geopotential model
GGM Global Geopotential Model
GPS Global Positioning System
H Ellipsoidal height
H Orthometric height
M Mean
N Geoid height
RMS Root Mean Squared
S Standard deviation
S
2
Sample variance
Δh Ellipsoidal height difference
ΔH Orthometric height difference
ΔN Geoid height difference
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