Esta obra está bajo una Licencia de Creative Commons. Atribución - No Comercial - Compartir Igual

DOI 10.15517/ri.v30i1.35839

Ingeniería 30 (1): 1-20, 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, Costa Rica

alonso.vega_f@ucr.ac.cr

Oscar H. Lücke Castro

Central American School of Geology, University of Costa Rica, Costa Rica

oscar.luckecastro@ucr.ac.cr

Jaime Garbanzo León

Surveying Engineering School, University of Costa Rica, Costa Rica

jaimegarbanzo@gmail.com

Recibido: 8 de enero 2019 Aceptado: 24 de octubre 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.

Esta obra está bajo una Licencia de Creative Commons. Atribución - No Comercial - Compartir Igual

DOI 10.15517/ri.v30i1.35839

Ingeniería 30 (1): 1-20, enero-junio, 2020. ISSN: 2215-2652. San José, Costa Rica

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.

3

1. INTRODUCTION

A precise determination of an orthometric height (H) is required in many elds like Construction,

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.

VEGA, LÜCKE Y GARBANZO: Geoid Heights in Costa Rica, Case of Study...

4

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)

5

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 programed 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).

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

VEGA, LÜCKE Y GARBANZO: Geoid Heights in Costa Rica, Case of Study...

6

limitations. 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.

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

Ingeniería 30 (1): 1-20, enero-junio, 2020. ISSN: 2215-2652. San José, Costa Rica DOI 10.15517/ri.v30i1.35839

7

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

hypothesis of these test, states that the variances are statistically similar and follow the relation

shown by Equation 6.

VEGA, LÜCKE Y GARBANZO: Geoid Heights in Costa Rica, Case of Study...

8

(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-equality 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): 1-20, enero-junio, 2020. ISSN: 2215-2652. San José, Costa Rica DOI 10.15517/ri.v30i1.35839

9

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”

VEGA, LÜCKE Y GARBANZO: Geoid Heights in Costa Rica, Case of Study...

10

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.

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