A high-resolution gravimetric quasigeoid model for Vietnam

Abstract

A high-resolution gravimetric quasigeoid model for Vietnam and its surrounding areas was determined based on new gravity data. A set of 29,121 land gravity measurements was used in combination with fill-in data where no gravity data existed. Global Gravity field Models plus Residual Terrain Model effects and gravity field derived from altimetry satellites were used to provide the fill-in information over land and marine areas. A mixed model up to degree/order 719 was used for the removal of the long and medium wavelengths and the calculation of the quasigeoid restore effects. The residual height anomalies have been determined employing the Stokes integral using the Fast Fourier Transform approach and deterministic kernel modification proposed by Wong–Gore, as well as by means of Least-Squares Collocation. The accuracy of the resulting quasigeoid models was evaluated by comparing with height anomalies derived from 812 co-located GNSS/levelling points. Results are very similar; both local quasigeoid models have a standard deviation of 9.7 cm and 50 cm in mean bias when compared to the GNSS/levelling points. This new local quasigeoid model for Vietnam represents a significant improvement over the global models EIGEN-6C4 and EGM2008, which have standard deviations of 19.2 and 29.1 cm, respectively, for this region.

Full text

Vuetal. Earth, Planets and Space (2019) 71:65
https://doi.org/10.1186/s40623-019-1045-3
FULL PAPER
A high-resolution gravimetric quasigeoid
model forVietnam
Dinh Toan Vu1*, Sean Bruinsma1,2 and Sylvain Bonvalot1
Abstract
A high-resolution gravimetric quasigeoid model for Vietnam and its surrounding areas was determined based on new
gravity data. A set of 29,121 land gravity measurements was used in combination with fill-in data where no gravity
data existed. Global Gravity field Models plus Residual Terrain Model effects and gravity field derived from altimetry
satellites were used to provide the fill-in information over land and marine areas. A mixed model up to degree/order
719 was used for the removal of the long and medium wavelengths and the calculation of the quasigeoid restore
effects. The residual height anomalies have been determined employing the Stokes integral using the Fast Fourier
Transform approach and deterministic kernel modification proposed by Wong–Gore, as well as by means of Least-
Squares Collocation. The accuracy of the resulting quasigeoid models was evaluated by comparing with height
anomalies derived from 812 co-located GNSS/levelling points. Results are very similar; both local quasigeoid models
have a standard deviation of 9.7 cm and 50 cm in mean bias when compared to the GNSS/levelling points. This new
local quasigeoid model for Vietnam represents a significant improvement over the global models EIGEN-6C4 and
EGM2008, which have standard deviations of 19.2 and 29.1 cm, respectively, for this region.
Keywords: Regional quasigeoid, Gravity anomalies, Least-Squares Collocation, Stokes FFT, GNSS/levelling
© The Author(s) 2019. This article is distributed under the terms of the Creative Commons Attribution 4.0 International License
(http://creat iveco mmons .org/licen ses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium,
provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license,
and indicate if changes were made.
Introduction
e quasigeoid is defined as a surface that is the clos-
est to mean sea level (Hofmann-Wellenhof and Moritz
2006). It serves as a reference surface for the vertical sys-
tem (Torge and Müller 2012). e normal height (H*),
the geometric distance between the quasigeoid and the
Earth’s surface, has traditionally been determined by
spirit levelling and adding the normal correction in order
to transform the levelled height into the normal height,
but it has taken much time and cost. anks to GNSS
technology, accurate ellipsoidal heights (h) are now easily
accessible and the normal height can also be determined
by subtracting the height anomaly (ζ) from the geodetic
height as follows:
For local or regional applications, the efficiency of this
approach is valid only if the height anomalies are known
(1)
H∗=h−ζ
with an accuracy of few centimetres. In Vietnam, Global
Gravity field Models (GGM) have been used in GNSS
applications since the late 1990s: EGM96 (Lemoine
etal. 1998) at first and currently EGM2008 (Pavlis etal.
2012). However, EGM2008 is inadequate for GNSS level-
ling over Vietnam. Its accuracy is insufficient to comply
with fourth-order levelling specifications (a misclosure
of
25√k
mm over a distance of kkm). A local gravimet-
ric quasigeoid of Vietnam was so far never calculated.
So, there is a strong need for a high-accuracy and high-
resolution gravimetric quasigeoid model of Vietnam and
its vicinity for the purpose of modernizing the height
system using GNSS instead of spirit levelling, as well as
for other applications such as geology, geophysics, and
oceanography. Several neighbouring countries make con-
tinuous efforts to determine and improve their geoid or
quasigeoid model successfully. For comparison, Table1
shows the accuracy of several local geoid or quasigeoid
models, and notably of neighbouring countries in Asia.
e resulting standard deviations (STD) obtained for
the most recent geoids models range from a few cm
up to 30cm, depending on the quality of the available
Open Access
*Correspondence: dinhtoan.vu@get.omp.eu
1 Géosciences Environnement Toulouse (GET), Obser vatoire Midi-
Pyrénées, Toulouse, France
Full list of author information is available at the end of the article

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Vuetal. Earth, Planets and Space (2019) 71:65
gravity data for the quasigeoid determination. Note that a
hybrid model is constructed using a gravimetric geoid or
quasigeoid and GNSS/levelling data.
Presently, all conditions for accurate high-resolution
quasigeoid determination using the Remove–Compute–
Restore (RCR) technique (Sansò and Sideris 2013) of
Vietnam are met thanks to:
• A new generation of Global Gravity Field Mod-
els (GGM) based on Gravity field and steady-state
Ocean Circulation Explorer (GOCE; Drinkwater
etal. 2003) data was developed;
• High-resolution Digital Terrain Models (DTM);
• New gravity measurements covering the entire coun-
try, even if not homogeneously, as well as high-res-
olution altimeter-inferred gravity anomaly data over
sea.
e objective of this study is to compute a gravimet-
ric quasigeoid model of Vietnam with an accuracy that
allows GNSS levelling to comply with fourth- or maybe
third-order levelling specifications. High-quality GNSS/
levelling data were used to assess the accuracy of the
developed quasigeoid models.
is paper describes the development and validation
of quasigeoid solutions for Vietnam. Because gravity data
are not available for Vietnam’s neighbouring countries,
this paper will only focus on modelling the quasigeoid of
Vietnam. “Quasigeoid determination methodology” sec-
tion briefly presents the two methods used to compute
the quasigeoid, namely the Stokes’ integral—Fast Fourier
Transform (FFT), and Least-Squares Collocation (LSC).
“Data and pre-processing” section describes the data and
the procedure for pre-processing the data. “Quasigeoid
model estimation and validation” section presents and
discusses results of the quasigeoid computations. Finally,
“Conclusions” section gives the conclusions on the devel-
opment and accuracy of the quasigeoid of Vietnam.
Quasigeoid determination methodology
e RCR technique is a well-known method for gravi-
metric quasigeoid determination. It is realized by sum-
mation of three terms according to the formula:
where ζGGM is computed using a global geopotential
model, ζRTM expresses the Residual Terrain Model (RTM)
effect, and ζres is computed from residual gravity anoma-
lies employing the Stokes’ integral or LSC. e residual
gravity anomalies used to determine ζres are computed as
follows:
where
gFA
is the free-air gravity anomaly,
gGGM
is the
gravity anomaly computed with a GGM, and
gRTM
is
the RTM effect on the gravity anomaly.
In this study, we used the GRAVSOFT software (Fors-
berg and Tscherning 2008) developed at DTU (Technical
University of Denmark) to perform the quasigeoid com-
putations. e gravity anomaly and height anomaly were
computed using the GRAVSOFT GEOCOL program.
e RTM effects on the gravity anomaly and the height
anomaly were computed using the GRAVSOFT TC pro-
gram, with a radius of 20km for the detailed DTM and
200km for the coarse grid. us, we need three models
to calculate the RTM effects with TC program; these are
the detailed (SRTM3arc), coarse, and reference DTMs in
(2)
ζ=ζGGM +ζRTM +ζres
(3)
gres =gFA −gGGM −gRTM
Table 1 Statistics ofselected local geoid orquasigeoid models
No Country, region Name Years STD (cm) Geoid type References
1 Australia AUSGeoid98 1998 36.4 Gravimetric quasigeoid Featherstone et al. (2001)
AUSGeoid09 2009 22.2 Gravimetric quasigeoid Featherstone et al. (2011)
2 Argentina GAR 2007 29.0 Gravimetric geoid Piñón et al. (2018)
GEOIDEAR 2017 27.0 Gravimetric geoid Piñón et al. (2018)
3 Japan GSIGEO2000 2002 4.0 Hybrid geoid Kuroishi et al. (2002)
GSIGEO2011 2014 1.8 Hybrid geoid Miyahara et al. (2014)
4 South Korea KGEOID98 1998 42.2 Gravimetric geoid Yun (2002)
KNGeoid13 2013 5.4 Hybrid geoid Lee et al. (2017)
KNGeoid14 2014 5.2 Hybrid geoid Lee et al. (2017)
5 Thailand THAI12G 2012 15.1 Gravimetric geoid Dumrongchai et al. (2012)
6 Philippines PGM2014 2014 30.0 Gravimetric geoid Forsberg et al. (2014b)
PGM2016 2016 2.2 Hybrid geoid Gatchalian et al. (2016)
7Peninsular (Malaysia) VMGEOID04 2018 5.0 Hybrid geoid Ismail et al. (2018)
Sabah and Sarawak (Malaysia) EMGEOID05 2018 10.0 Hybrid geoid Ismail et al. (2018)

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Vuetal. Earth, Planets and Space (2019) 71:65
which the reference height grid was estimated by low-
pass filtering the detailed DTM in order to represent the
topographic signal above the maximum degree of the
GGM used (Forsberg 1984). e coarse and reference
DTM models were created as follows:
• e coarse grid is computed by simple averaging (e.g.
3′ × 3′ grid) of the detailed DTM model using the
SELECT program.
• e coarse grid (3′ × 3′) is then filtered with a mov-
ing average operator to the required resolution using
the TCGRID program; in the remove–restore proce-
dure, the required resolution is 27.8km (equivalent
to spherical harmonic d/o 720).
All computations have been performed with the
reference ellipsoid WGS84, of which the con-
stants are: a = 6,378,137.00 m, f = 1/298.257223563,
GM0 = 3.986004418 × 1014 m3s−2, and in the Tide Free
(TF) system. When a GGM is referred to the Zero Tide
(ZT) system or the Mean Tide (MT) system, the C2,0
coefficient is converted to the TF system using the for-
mula reported in Rapp (1989). In Vietnam, where the
height system refers to the MT system, the conversion
from MT system to the TF system is done according to
Ekman (1989).
Data andpre‑processing
e quasigeoid model was developed according to the
diagram shown in Fig. 1, which presents the different
steps, inputs, and modules for RCR operations. e input
data are presented in the following sections.
Global Geopotential Models (GGM)
Gravity data are reduced for the long and medium wave-
lengths, using GGMs, and the terrain effect, using DTMs,
in order for the residual gravity anomalies to be smooth
before gridding or prediction. e GGM has to best rep-
resent the gravity anomalies and height anomalies in the
selected area. GGMs, enhanced with RTM effects, are
also used to generate fill-in data where gravity measure-
ments are not available. e GGMs are available on the
International Center for Global Earth Models (ICGEM)
Web site.
e high-resolution EGM2008 model, developed up
to degree/order (d/o) 2190, has well-known errors, due
to datum inconsistencies and variability of the measure-
ment density and accuracy (Gilardoni etal. 2013), in the
low–medium frequency band because it is a pre-GOCE
model. Moreover, as it includes a large variety of sur-
face measurements, uncertainties also arise from datum
inconsistency in the compiled gravity database and from
variability of the measurement density and accuracy. In
Gilardoni etal. (2013), a GOCE model was successfully
used to improve the geoid model accuracy by combining
spherical harmonic coefficients of the EGM2008 model
with a GOCE gravity model. Following previous studies
that successfully used mixed GOCE and EGM2008 mod-
els for the removal of the long and medium wavelengths
to compute geoids of Malaysia (Jamil etal. 2017), Nepal
(Forsberg et al. 2014a), and the Philippines (Forsberg
etal. 2014b), we constructed a similarly combined model
to remove the long to medium wavelength components
of the gravity field up to d/o 719. We used the fifth release
of GOCE global potential model obtained from the direct
approach, named EGM-DIR-R5 (Bruinsma etal. 2014).
e combination was done in the following way:
• Degrees 2-260: EGM-DIR-R5.
• Degrees 270-2190: EGM2008.
• e coefficients of the mixed model from degrees
260 to 270 are computed by weighted mean of the
two models with the weights being determined as
the inverse of degree variances. ese coefficients are
computed as follows (Gilardoni etal. 2013):
where
TE
mn
and

σ2
Tmn
E
are the coefficients and
degree variances, respectively, derived from
EGM2008.
TD
mn
and

σ2
Tmn
D
are the coefficients and
degree variances, respectively, derived from
EGM-DIR-R5.
All recent GGMs, such as GOCO05s (Mayer-Guerr
2015), EGM-TIM-R5 (Brockmann et al. 2014), and
EGM-SPW-R5 (Gatti et al. 2016), were tested in this
study, in steps of 10 degrees, to find out the best GGM
and its optimum maximum degree in combination with
EGM2008. Figure 2 indicates the STD of the differ-
ences between the GOCE GGMs in combination with
EGM2008 and the GNSS/levelling data. e EGM-DIR-
R5 at d/o 260 plus EGM2008 is the best model with
the smallest STD of 0.16m. Moreover, the EIGEN-6C4
model (Förste etal. 2014), computed from the combina-
tion of LAser GEOdynamics Satellite (LAGEOS), Grav-
ity Recovery And Climate Experiment (GRACE), GOCE,
and a reconstruction of EGM2008 beyond d/o 235, was
also tested. e combination model described above best
reproduces the gravity data. In particular, the Experi-
mental Gravity Field Model XGM2016 (Pail etal. 2018),
computed with improved terrestrial data especially over
continental areas such as South America, Africa, parts
(4)
T
mn =


TE
mn

σ2
T
mn 
E+
TD
mn

σ2
T
mn 
D






1

σ2
T
mn 
E+
1

σ2
T
mn 
D



−1

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Vuetal. Earth, Planets and Space (2019) 71:65
EGM-DIR-R5 (2-260)
GGM: Transition from 260 to 270
EGM2008 (270-2190)
DTM: SRTM3arc (land)
SRTM15plus (sea)
∆gGGM
|2
2159 (GEOCOL) ∆gRTM|2160
216000 (TC)
Measurement ∆gfill−inDTU15
COMBINATION ∆gFA
C
Subtract ∆gGGM|2
719 (GEOCOL)
Subtract ∆gRTM |720
216000 (TC)
Residual anomalies ∆gres
Grid ∆gres
grid (GEOGRID)
Residual ∆Nres(SPFOUR)
Parameters (EMPCOV, COVFIT)
Residual ∆Nres(GEOCOL)
Add∆NGGM|2
719(GEOCOL
(
Add∆NRTM|720
216000 (TC)
GeoidN
Validation GNSS/levelling
Fig. 1 Diagram of sequential steps (top to bottom) in the calculation of the quasigeoid

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Vuetal. Earth, Planets and Space (2019) 71:65
of Asia, and Antarctica, up to the same d/o 719, was also
used to compute quasigeoid for this region, but the result
was slightly worse than when using the mixed model.
Digital Terrain Models (DTM)
e DTM provides information on the short wavelengths
of the gravity field. RTM was selected to calculate the
terrain effects; using the RTM the smoothing effect on
gravity data can reach 50% if their elevations are accurate
(Forsberg 1984). Over land areas, the 90 m resolution
SRTM3arc_v4.1 (Farr etal. 2007) was used as the detailed
DTM. e 15” resolution Digital Bathymetry Model
(DBM) SRTM15arc_plus (Becker etal. 2009) was used
over sea, and after re-gridding to 3’’ it was then merged
with SRTM3arc_v4.1 using the full-resolution coastline
in Generic Mapping Tools (GMT) (Wessel and Smith
1998). Several DTMs, such as Earth2012 (Hirt 2013) and
DTM2006 (Pavlis etal. 2012), were also used for comput-
ing residual gravity anomalies on land, but the best model
is SRTM3arc_v4.1 (the STD of residual gravity anomalies
using this model is the smallest). To avoid the need to
distinguish between different density values (mass den-
sity of water ρw = 1030kg m−3 and mass density of rock
ρr = 2670kgm−3), we used the Rock-Equivalent Topogra-
phy (RET) approach (Balmino etal. 1973, 2012).
Gravity measurements andll‑in data
For the purpose of determining the quasigeoid, we col-
lected terrestrial gravity data obtained from the Institute
of Geophysics (IGP)—Vietnam Academy of Science and
Technology (VAST), the Vietnam Institute of Geodesy
and Cartography (VIGAC), and the Bureau Gravimé-
trique International (BGI; Bonvalot 2016). e total
number of gravity points is 31,102. A set of 19,267 land
gravity points was collected from IGP, but these surveys
have been done between 1961 and 1984 for the purpose
of geological survey, exploration geophysics, and mineral
prospecting when positioning was of poor quality, espe-
cially for heights, which were determined using barom-
eters (green points in Fig.3). As errors in the elevation
will propagate into the computed gravity anomaly, gross
error detection methods were first applied to clean up
the IGP data (see below). Fortunately, most of the coun-
tries have been re-surveyed from 2003 to 2011, through
the project “Measurement and Improvement of Viet-
nam National Gravity Data”, carried out in collaboration
with the Moscow Institute of Geodesy, Cartography and
Aerial Images, Moscow State University of Geodesy and
Cartography (MIIGAiK), Russia. is new dataset com-
prises 10,940 points, including an absolute gravity net-
work of 11 base reference stations with an accuracy of
better than ± 5 μGal and their tie points, 29 first-order
gravity stations with an accuracy of better than ± 15 μGal
and their tie points, and 92 third-order gravity points and
more than 10,000 detailed points measured from 2005
to 2009 (red points in Fig.3). e base reference stations
and first-order gravity network were determined from
absolute measurements using GBL instrument (Final
Report of VIGAC 2012). For the VIGAC gravity surveys,
GNSS has been used to determine coordinate and height,
Fig. 2 Standard deviation of the differences between the GOCE GGMs in combination with EGM2008 and the GNSS/levelling data

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Vuetal. Earth, Planets and Space (2019) 71:65
so this is less prone to errors in positioning. e IGP data
are less accurate than the VIGAC data. However, the
combination of the IGP and VIGAC data enhances the
coverage considerably, especially in the South of Vietnam.
Finally, our land gravity data set was also complemented
by a set of gravity data provided by BGI for Vietnam and
surrounding areas (including 895 points in Vietnam, 229
points in Cambodia, and 351 points in ailand).
We have used two procedures to detect localized gross
errors in the data. e first uses SRTM3arc data to verify
the gravity observation elevations in the IGP data. e
results are listed in Table 2. e differences at GNSS/
levelling points show a STD of 6.86 m and an average
bias of − 3.13m, which are in line with results of Den-
ker (2005) in Germany. e VIGAC data show a STD of
23.46m and average bias of − 4.23m, while SRTM3arc
has a reported vertical accuracy of better than 16m (Farr
etal. 2007). is proves that SRTM provides very good
height information in the computation of the RTM effect.
e differences of the gravity observation elevations in
the IGP data with SRTM3arc are 73.50m in STD and
− 14.95m in mean bias. is indicates that the elevations
in the IGP data, which were determined from barometric
levelling, have gross errors. A horizontal error in gravity
data will also result in a discrepancy between the gravity
observation elevations in the IGP data and those derived
from DTM, especially in areas of steep elevation changes;
consequently, we cannot use elevation derived from the
DTM and we need a different procedure to detect gross
errors in IGP data.
e second procedure involves comparisons with the
EGM-DIR-R5 model. To reduce the effect of the omis-
sion error in the GGM, we have used EGM-DIR-R5 aug-
mented with high-resolution RTM effects beyond its
selected resolution (degree 260). e RTM effects were
computed using the GRAVSOFT TC program. e refer-
ence height grid was estimated in order to represent the
topographic signal above degree 260 of EGM-DIR-R5.
e 3” resolution of DTM is equivalent to spherical har-
monic d/o 216000, so the reduction in gravity data has
been evaluated in the following way:
Fig. 3 Distribution of land gravity data used in this study: red dots
are the VIGAC points, absolute points are indicated by triangles, green
dots are the IGP points, and blue dots are obtained from the BGI
database
Table 2 Statistics ofthedierences betweentheobservation elevations andtheSRTM3arc [Unit: (m)]
Data Mean STD Min Max
GNSS/levelling (812 points) − 3.13 6.86 − 50.13 32.94
VIGAC data (10,940 points) − 4.23 23.46 − 403.20 434.95
IGP data (all: 19,267 points) − 14.95 73.50 − 789.00 715.46
Rejected points of IGP data (1960 points) − 57.97 171.68 − 782.32 715.46
Accepted points of IGP data (17,307 points) − 10.08 49.42 − 789.00 491.80

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Vuetal. Earth, Planets and Space (2019) 71:65
e results are given in Table 3. e differences in
VIGAC data show a STD of 9.1 mGal and average bias of
− 0.6 mGal, whereas IGP data show a STD of 19.1 mGal
and average bias of 4.5 mGal. is again indicates that
the IGP gravity dataset contains gross errors. Aiming for
IGP data with the same precision as the VIGAC gravity
data (i.e. 9.1 mGal in STD), and assuming a normal dis-
tribution, we have eliminated those IGP data for which
differences are greater than three STD of the VIGAC
data (i.e. 27.3 mGal). ere are 1960 points greater than
this threshold, which we rejected. Table2 indicates that
the differences in elevation of these points (compared
to SRTM) have a STD of 171.68 m and average bias
of − 57.97 m, while the 17,307 accepted points have a
STD of 49.42m. ere is still a big difference in the IGP
cleaned data due to horizontal errors as indicated above.
To confirm that these gross errors in IGP gravity
anomaly are due to elevation errors, we have done tests
using different subsets of the gravity points according to
an elevation threshold (100m). e results of these tests
are also given in Table3. e higher elevation points in
the IGP data increase by 12 mGal in average bias and 9
(5)

g
res =
g
FA −
g
DIR5|260
2−
g
RTM|216000
261
mGal in STD while VIGAC results change by < 2 mGal.
For the lower altitude points, the accuracy of VIGAC and
IGP data is 8.4 and 9.5 mGal, respectively. is proves
that there is a small effect of elevation in VIGAC data,
whereas it is large in IGP data. After editing the IGP
dataset, they are at the same level as the VIGAC data
(about 9 mGal in STD when comparing with EGM-DIR-
R5 together with RTM effect).
As with the IGP data, we found 21 gravity points of
VIGAC data with large differences that were excluded
from the computation. us, a total of 1981 points
was eliminated. After cleaning, the difference between
the observed gravity anomalies with EGM-DIR-R5
(d/o = 260) together with RTM effects is − 0.3 mGal on
average and 9.0 mGal STD, whereas with only EGM-DIR-
R5 (d/o = 260) it is − 12.0 mGal and 21.7 mGal, respec-
tively. is result clearly shows that the terrain effect is
the most important parameter to consider in order to
enhance the consistency of available terrestrial gravimet-
ric data and GGMs and to produce a unified database.
e RTM data succeed in filling the spectral gap between
land gravity measurements and GGMs. Figure 4a also
indicates the presence of a height-correlated bias in our
Table 3 Statistics ofthedierences betweentheobserved gravity anomalies andtheGGM EGM‑DIR‑R5 [Unit: (mGal)]
Data Mean STD Max Min
VIGAC-(EGM-DIR-R5 + RTM) (10,940 points) − 0.7 9.1 158.4 − 62.7
IGP-(EGM-DIR-R5 + RTM) (19,267 points) 4.5 19.1 153.4 − 54.8
VIGAC-(EGM-DIR-R5 + RTM) (H < 100 m) (6980 points) − 0.1 8.4 158.4 − 35.6
IGP-(EGM-DIR-R5 + RTM) (H < 100 m) (13,497 points) − 0.6 9.5 121.8 − 54.8
VIGAC-(EGM-DIR-R5 + RTM) (H > 100 m) (3960 points) − 1.6 10.1 105.6 − 62.7
IGP-(EGM-DIR-R5 + RTM) (H > 100 m) (5770 points) 16.4 28.4 153.4 − 39.0
IGP cleaned-(EGM-DIR-R5 + RTM) (17,307 points) 0 8.9 27.3 − 27.1
All data (VIGAC cleaned, IGP cleaned, BGI)—EGM-DIR-R5 (29,121
points)
− 12.0 21.7 125.6 − 154.5
All data (VIGAC cleaned, IGP cleaned, BGI)—(EGM-DIR-R5 + RTM)
(29,121 points)
− 0.3 9.0 158.4 − 62.7
Fig. 4 Differences between measurements with: a EGM-DIR-R5 and b EGM-DIR-R5 + RTM

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Vuetal. Earth, Planets and Space (2019) 71:65
data, but this bias in residual anomaly is significantly
reduced by taking RTM effects into account (Fig.4b).
e mixed DIR/EGM model up to d/o 2159 together
with topographic effects was used to fill-in data on land.
e fill-in data have been evaluated following a spectral
enhancement approach as:
e use of the mixed DIR/EGM model instead of
EGM-DIR-R5 only in combination with RTM effect is
because of EGM2008, which performs better than RTM
effect within the spectral window 260-2159 in Vietnam.
is issue will be further clarified in the next section.
e DTU15 gravity field model (Andersen and Knud-
sen 2016) was used for marine areas. Altimetric gravity
is of good quality over the open seas. However, coastal
zones remain problematic because most altimeters
cannot measure up to the coast (Hirt 2013). Airborne
(Forsberg and Olesen 2010) or shipborne (Featherstone
2010) gravimetry is used preferentially to close the gap
between gravity data on land and marine altimetric
gravity fields if it is available. ese observations are
(6)
�
g
fill−in =
�g
DIR/EGM|2159
2+
�g
RTM|216000
2160
not available for Vietnam’s coastal zones, so the accu-
racy of the quasigeoid in coastal zones is a difficult
problem. DTM and DBM provide information on the
short wavelengths of the gravity field in coastal zones
and can be used to augment and improve global grav-
ity fields (Hirt 2013). In this study, we used RTM effects
together with GGM in coastal zones instead of using
the altimetric gravity field.
Finally, these heterogeneous gravity data were merged
in a complex procedure described below:
(1) For the Stokes FFT program, a 5′ × 5′ grid is inter-
polated with the GRAVSOFT GEOGRID program
using the LSC method on the gravity measure-
ments. en, only grid nodes lying within 5′ radius
circles centred on each of the terrestrial gravity
points were kept;
(2) e grid nodes lying 50km and more beyond the
5′ radius circles were filled in with the mixed DIR/
EGM model together with RTM effect over land
(green points in Fig.5a) and with DTU15 over sea
(blue points in Fig. 5a). e full-resolution GMT
Fig. 5 Final gravity data compilation for the study area. a Geographic display of combination of the gravity data: red dots are grid from land gravity
points, orange dots are tapered transition points from fill-in data on land or on the sea to land gravity data, and green dots are fill-in points on land
and blue dots are DTU15 marine gravity points; b gravity anomalies ∆gFA; and c grid of residual gravity anomalies

Page 9 of 16
Vuetal. Earth, Planets and Space (2019) 71:65
coastline was used to determine marine and land
regions.
(3) e transition areas between observations and fill-
in models (land transitions starting at the 5′ radius
circles to 50km beyond, coastal transitions starting
at the coastline to 50km on sea) were filled using a
combination of data and models.
From measurement points, the differences between
observations and fill-in data were calculated in the
following way:
whereas the differences in the fill-in grids (beyond
50km from the 5′ radius circles) were set to zero.
e LSC method in GEOGRID program was then
used to interpolate these differences to the transi-
tion points (

g
grid
dif
). Gravity anomalies of transition
points were then constructed by adding

g
grid
dif
to
the fill-in data of transition points as follows:
(4) For collocation, the actual measurements were
used; however, the same fill-in and transition data
as in (2) and (3), respectively, were used.
e combination of the heterogeneous gravity data is
shown in Fig.5a. Statistics of the merged gravity anoma-
lies are given in Table4 and shown in Fig.5b. ese grav-
ity anomalies were reduced using the GGM and RTM
effects according to Eq.(3). Table4 and Fig.5c show the
grid of residual gravity anomalies with three types of
gridded data (measurement grid, fill-in grid on land, and
DTU on sea) that were merged. ese residual gravity
anomalies are generally small (< 30mGal in magnitude).
Large residual gravity anomalies occur in mountain-
ous regions (e.g. the northwest and central parts of the
study area) where the altitude is greater than 1000m. e
reason for the large residuals is that errors of the DTM
and terrestrial gravity in mountainous regions are larger
those in flat regions.
(7)
�
g
dif =
�g
FA −
�g
DIR/EGM|2159
2−
�g
RTM|216000
2160
(8)
�
g
grid
trans =
�g
DIR/EGM|
2159
2+
�g
RTM|
216000
2160 +
�g
grid
dif .
GNSS/levelling points
From 2009 to 2010, Vietnam Department of Survey-
ing and Cartography carried out GNSS observations on
the levelling points. e GNSS baselines were observed
using dual-frequency instruments in static mode with
a minimum measurement time of 6h per session. e
GNSS data were processed with Bernese software to
obtain ellipsoidal heights referred to the WGS84 ellip-
soid. e total number of GNSS/levelling stations used in
this study is 812, and these are shown in Fig.6. GNSS/
levelling data include horizontal coordinates (latitude,
longitude) and the computed height anomaly. Of the 812
Table 4 Statistics of the gravity anomalies and their
residuals [Unit: (mGal)]
Mean STD Max Min

g
C
FA
(measurements, DIR/
EGM + RTM, DTU15)
− 16.1 26.7 163.7 − 174.0

g
C
FA
− ∆gDIR/EGM − ∆gRTM (∆gres)− 0.9 10.2 117.2 − 127.2
∆gres (5′ × 5′ grid) − 0.8 7.8 − 59.2 49.1
Fig. 6 GNSS/levelling data: yellow dots are first and second order of
the national levelling networks, whereas purple dots are third order

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Vuetal. Earth, Planets and Space (2019) 71:65
GNSS/levelling points, 428 points are first and second
order and 384 points are third order of the national level-
ling networks. First-, second-, and third-order levelling in
Vietnam allows misclosure of
5√k
,
12√k
, and
25√k
mm
over a distance of kkm, respectively. Normal height is
currently used in the national height system of Vietnam.
Figure 6 shows that gravity measurements of VIGAC
have been made alongside the first- and second-order
levelling. e GNSS/levelling geometric height anoma-
lies were compared with those derived from the EGM-
DIR-R5 or the mixed DIR/EGM model together with
RTM effects. e RTM effect is used as augmentation of
GGMs beyond their selected resolution as:
e results are listed in Table5. ese results clearly
show significant improvement (2.5 cm in STD) when
using the mixed DIR/EGM instead of using EGM-DIR-
R5 only in combination with RTM effects. is demon-
strates that EGM2008 performs better than RTM effects
computed with TC within the spectral window d/o 260-
2159 in Vietnam, even if fill-in data were used.
Quasigeoid model estimation andvalidation
e residual height anomalies have been determined
using the regular grid of residual gravity anomalies
employing the Stokes’ integral in the 1D-FFT approach
(Haagmans et al. 1993) implemented in the GRAVS-
OFT SPFOUR program with the Wong–Gore modifi-
cation (WG) of the Stokes’ kernel function (Wong and
Gore 1969). WG removes low harmonics up to degree
N1, so the influence of the local data at long wavelengths
is eliminated and then linearly tapered to N2 (Forsberg
and Tscherning 2008). N1 and N2 are selected accord-
ing to data and the GGM used in the remove step, but
they should be less than or equal to degree nmax of the
GGM. To find out the optimum N1 and N2 degree, the
quasigeoid was computed by the Stokes FFT using WG
with N1 and N2 being tested from 100 to 260 (maximum
degree of the EGM-DIR-R5 model used in combination
with EGM2008) in steps of 10°. e computed quasigeoid
(9)
�ζ =
ζ
GNSS/levelling −
ζ
DIR5|260
2−
ζ
RTM|216000
261
(10)
�ζ =
ζ
GNSS
/
levelling −
ζ
DIR/EGM|2159
2−
ζ
RTM|216000
2160
models were then compared to GNSS/levelling data.
Finally, the best quasigeoid model was obtained when the
low harmonics were completely removed from Stokes’
function up to degree N1 = 220 and then linearly tapered
to N2 = 230.
Residual height anomalies were also calculated with the
LSC method, using the GRAVSOFT GEOCOL program.
Computation of the empirical and fitted covariance func-
tions of the gravity anomalies is required in LSC to esti-
mate the residual height anomalies. We used the error
degree variances of the mixed DIR/EGM model up to 719
and the fourth model of Tscherning and Rapp (1974) for
the degree variances of degree > 719. Degree 719 agrees
best with the empirical data for fitting the model covari-
ance function. e empirical covariance function of the
data has been computed using the GRAVSOFT EMP-
COV program and was fitted to the Tscherning and Rapp
model using the GRAVSOFT COVFIT program. We
have determined the following optimum parameters for
inputs of GEOCOL: the depth to the Bjerhammar sphere
R − RB = − 0.028 km and the variance of the gravity
anomalies at zero height VARG = 131.06 mGal2. Figure7
shows the plot of the empirical covariance (blue line) and
fitted covariance functions for the residual gravity anom-
alies Δgres (red line).
e residual height anomalies

�ζ
FFT
res 
computed with
the SPFOUR program vary from − 0.701 to 0.402m. e
height anomalies, obtained by restoration of ζGGM and
ζRTM, vary from − 35.097 to 16.684m (GEOID_FFT solu-
tion). e residual height anomalies were also computed
with the GEOCOL program:
�ζ LSC
res
varies from − 0.779
to 0.432m and ζLSC from − 34.969 to 16.688m (GEOID_
LSC solution).
Figure 8a, b shows the residual height anomalies

�ζ
FFT
res 
and GEOID_FFT, and the differences between
GEOID_FFT and GEOID_LSC are shown in Fig.8c. e
differences range from − 0.232 to 0.244m. e large dif-
ferences between GEOID_FFT and GEOID_LSC occur
in the regions where the residual gravity anomalies are
large (> 30 mGal). is issue will be discussed later.
For validation, the height anomalies were compared
with those inferred from 812 GNSS/levelling reference
points (ζGNSS/levelling). Figure9a–c shows the plots of the
differences of ζFFT, ζLSC, and ζEGM2008 with ζGNSS/levelling,
and the statistics are listed in Table6. e differences for
GEOID_FFT range from 0.136 to 0.816m with a STD of
0.097m and average bias of 0.506m and for GEOID_LSC
from 0.138 to 0.815m with a STD of 0.097m and aver-
age bias of 0.508m. e results show that both methods
reach the same precision, with a STD at the 9.7cm level.
e reason for the large average bias is datum incon-
sistencies and long wavelength quasigeoid errors. ese
quasigeoids refer to a global reference system while
Table 5 Statistics of the dierences between the GNSS/
levelling points andtheGGM [Unit: (m)]
GGMs d/o ofGGM Mean STD Max Min
DIR-R5 260 0.455 0.235 1.281 − 0.305
DIR-R5 + RTM 260 0.543 0.184 1.138 − 0.155
DIR/EGM + RTM 2159 0.515 0.157 1.018 − 0.103

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Vuetal. Earth, Planets and Space (2019) 71:65
Fig. 7 Empirical and fitted covariance functions for residual gravity anomalies Δgres
Fig. 8 Residual height anomalies

�ζ
FFT
res 
(a), GEOID_FFT (b), and difference between GEOID_FFT and GEOID_LSC (c)

Page 12 of 16
Vuetal. Earth, Planets and Space (2019) 71:65
heights that have been determined from levelling refer
to national mean sea level. It should also be noted that
here the degree-0 term (e.g. − 41cm for EGM2008 with
the reference ellipsoid WGS84; http://earth -info.nga.
mil/GandG /wgs84 /gravi tymod /egm20 08/egm08 _wg s84
.html) is not included in this average bias. e results
of the comparison indicated significant improvement
in the local quasigeoids over EGM2008 and EIGEN-
6C4 in Vietnam, which have STD of 29.1 and 19.2cm,
respectively. Figure 9 shows this improvement over
EGM2008, especially in the north and the mountainous
regions.
We have also compared these quasigeoids with
GNSS/levelling points split according to order: 428
points of first- and second-order levelling and 384
points of third-order levelling. e results of GEOID_
LSC show a STD of 8.7cm for the first- and second-
order points (where gravity was measured) and 10.8cm
for the third-order points (where gravity was not
measured), while GEOID_FFT has a STD of 9.1 cm
for the first- and second-order points and 10.4cm for
the third-order points (Table 7). ese results show
that the LSC method is a little more precise than the
1D FFT where gravity data are available. To further
clarify this issue, we have evaluated the quasigeoid in
two areas (two rectangles in Fig.3) where there is suf-
ficient terrestrial gravity as well as GNSS/levelling data
(136 GNSS/levelling points in a northern area defined
by 20° ≤ φ ≤ 22° in latitude and 105° ≤ λ ≤ 108° in longi-
tude and 120 GNSS/levelling points in a southern area
defined by 8° ≤ φ ≤ 11° in latitude and 104° ≤ λ ≤ 108° in
longitude). For the northern area, the STD of the LSC
and 1D FFT methods is 7.4cm and 8.2cm, respectively;
Fig. 9 Differences between the developed quasigeoid and GNSS/levelling. a GEOID_FFT, b GEOID_LSC, and c EGM2008
Table 6 Statistics of the quasigeoid and their validation
withGNSS/levelling data [Unit: m]
Mean STD Max Min
�ζ FFT
res − 0.005 0.084 0.402 − 0.701
ζFFT − 16.169 11.781 16.684 − 35.097
�ζ LSC
res
0 0.080 0.432 − 0.779
ζLSC − 16.164 11.778 16.688 − 34.969
ζGNSS/levelling − ζFFT 0.506 0.097 0.816 0.136
ζGNSS/levelling − ζLSC 0.508 0.097 0.815 0.138
ζGNSS/levelling − ζEIGEN-6C4 0.514 0.192 1.057 − 0.348
ζGNSS/levelling − ζEGM2008 0.428 0.291 1.272 − 0.516

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Vuetal. Earth, Planets and Space (2019) 71:65
for the southern area, the STDs are 9.2cm and 10.0cm,
respectively. e results of the comparison indicated
that the LSC method is more precise than the 1D FFT
method, 0.8cm of STD for each area. However, quality
of the available gravity data in Vietnam is not homoge-
neous (bias, precision between IGP, VIGAC, and fill-
in data) and not enough information on the accuracy
is available, which is challenging with the LSC method
while with the Stokes FFT method a good data density
is required. It is the reason why these two methods have
the same accuracy for the whole study area. e circles
in Figs. 8c, 9a, b show the area where the difference
between the two quasigeoid solutions is significant (and
where terrestrial gravity data is available). e higher
accuracy of LSC, which uses all observations, may be
due to the higher density of measurements in these
areas than for other areas; a denser residual gravity grid
could have been computed for SPFOUR. is hypoth-
esis was confirmed by computing and using denser
grids (2.5′ × 2.5′) for the two test areas with the Stokes
FFT method. e results are shown in the last 2 rows in
Table7, and they indicate that the Stokes FFT method
has the same accuracy as the LSC method over these
two areas when GEOID_FFT is computed with the
denser grids.
e STD of the differences between EGM2008 and
EIGEN-6C4 with 384 GNSS/levelling points of the
third-order levelling is 27.4cm and 18.6cm (Table7),
respectively, whereas the STD of GEOID_FFT and
GEOID_LSC is 10.8 and 10.4 cm, respectively. ese
numerical findings signify that the addition of the RTM
effects to DIR/EGM has significantly improved the
accuracy of the height anomalies in the area where no
data existed.
Moreover, a quasigeoid was computed using DTU15
data (ζFFT-DTU) instead of using the mixed DIR/EGM
model together with RTM effect within 50 km from
the coastline. e height anomalies derived from these
quasigeoids were compared with those derived from 69
GNSS/levelling points near the coast (ζGNSS/levelling_coast in
Table7). An improvement can be seen when using RTM
effects together with the mixed DIR/EGM model instead
of using DTU15 gravity within 50km from the coastline.
is suggests that RTM effects together with the DIR/
EGM model can be used to fill the gap between gravity
data on land and marine altimetric gravity if airborne or
shipborne gravity is not available in coastal zones.
To assess the applicability of the developed quasigeoid
model in GNSS levelling, a test was performed in a rela-
tive sense. e relative fits of GEOID_LSC to GNSS/
levelling data were determined over 21,161 baselines
(baseline length < 100 km). Figure 10a, b shows the dis-
crepancies between height anomaly differences derived
from EGM2008 (d/o 2190) and GEOID_LSC with those
derived from GNSS/levelling data. e tolerance of
fourth-order spirit levelling was used for verification.
With GEOID_LSC, 19,237 (90.91%) baselines comply,
while with EGM2008 only 10,158 (52.00%) baselines do.
With EGM2008, the accuracy of GNSS levelling cannot
reach fourth-order levelling requirements. With GEOID_
LSC, there are still 1924 baselines that lie outside the
tolerance of fourth-order levelling, probably due to
errors in GNSS/levelling data; error detection methods
must be applied to clean the GNSS/levelling data before
Table 7 Dierences betweenthequasigeoid andGNSS/levelling data according totheorder oflevelling network [Unit:
m]
Number ofpoints Mean STD Min Max
ζGNSS/levelling − ζFFT (first and second order) 428 0.508 0.091 0.217 0.781
ζGNSS/levelling − ζLSC (first and second order) 428 0.515 0.087 0.182 0.807
ζGNSS/levelling − ζFFT (third order) 384 0.503 0.104 0.136 0.816
ζGNSS/levelling − ζLSC (third order) 384 0.500 0.108 0.116 0.815
ζGNSS/levelling − ζEIGEN-6C4 (third order) 384 0.488 0.186 − 0.348 0.990
ζGNSS/levelling − ζEGM2008 (third order) 384 0.402 0.274 − 0.512 1.143
ζGNSS/levelling − ζFFT (northern area) 136 0.551 0.082 0.290 0.781
ζGNSS/levelling − ζLSC (northern area) 136 0.540 0.074 0.287 0.707
ζGNSS/levelling − ζFFT (southern area) 120 0.465 0.100 0.224 0.776
ζGNSS/levelling − ζLSC (southern area) 120 0.462 0.092 0.230 0.765
ζGNSS/levelling_coast − ζFFT 69 0.532 0.090 0.333 0.719
ζGNSS/levelling_coast − ζFFT-DTU 69 0.524 0.098 0.322 0.746
Quasigeoid computed with Stokes FFT with grid 2.5′ × 2.5′ over two test areas
ζGNSS/levelling − ζFFT (northern area) 136 0.575 0.073 0.319 0.762
ζGNSS/levelling − ζFFT (southern area) 120 0.446 0.096 0.234 0.829

Page 14 of 16
Vuetal. Earth, Planets and Space (2019) 71:65
fitting with a gravimetric quasigeoid. Moreover, a large
bias was also found between gravimetric quasigeoid and
GNSS/levelling data (50cm) in which the degree-0 term
also needs to be taken into account to determine the
true vertical datum offsets for Vietnam with respect to
a global equipotential surface. Such offsets must be cor-
rected for before using a quasigeoid in GNSS levelling.
ese issues will be solved in future research. Figure10c
shows the discrepancies between height anomaly differ-
ences derived from GEOID_LSC and those derived from
GNSS/levelling data in the northern test area (defined by
20.5° ≤ φ ≤ 21.5° in latitude and 106° ≤ λ ≤ 107.5° in lon-
gitude), but with the tolerance set by third-order spirit
levelling (
12√k
mm over a distance of kkm was used to
compare). ere are 201 (81.05%) baselines (over a total
of 248 baselines) that lie inside the tolerance of third-
order levelling. We also tested for the southern area
(defined by 11° ≤ φ ≤ 12° in latitude and 106° ≤ λ ≤ 108.5°
in longitude); there are 172 (78.89%) baselines (over a
total of 218 baselines) that lie inside this tolerance. is
suggests that with gravity data for the entire country of
similar quality and distribution as for these areas, the
resulting quasigeoid may allow GNSS levelling to comply
with third-order levelling specifications.
Conclusions
A new quasigeoid model has been generated for Viet-
nam and surrounding areas from the combination of
heterogeneous data including 29,121 terrestrial grav-
ity points, global gravity models, and high-resolution
topographic and bathymetric data. Two gravimetric
quasigeoid solutions, called GEOID_FFT and GEOID_
LSC, were computed with the Stokes’ integral using the
FFT-1D approach and deterministic kernel modifica-
tion as proposed by Wong–Gore and the LSC method,
respectively. ese quasigeoid models were validated
through a comparison with 812 GNSS/levelling points.
Our results show that both models lead to very similar
results reaching a STD at the 9.7cm level with a mean
bias of 50cm. e results of the comparison indicated
significant improvement in these models over the com-
monly used EGM2008 and EIGEN-6C4 for Vietnam in all
areas, covered or not by land gravity measurements. Such
regional models are thus likely to be used for GNSS lev-
elling applications, with accuracy satisfying fourth-order
levelling for the entire country and third order in areas
where there are sufficient data (north and south areas),
and should also contribute to the modernization of Viet-
nam’s height system. A significant improvement for areas
with poor data coverage proves that the recent GOCE/
GRACE GGM in combination with EGM2008 and RTM
effects may be used to improve quasigeoid determination
in the areas where gravity data are not available or insuf-
ficient, especially in mountainous regions and coastal
zones. e best agreement in Vietnam is observed for
EGM-DIR-R5 used up to d/o 260, EGM2008 used up
from d/o 270 to 2159 and RTM effects used equivalent to
d/o 216000.
Land gravity data are not available for large parts of the
mountainous region, and consequently, the gravimetric
quasigeoid solutions are significantly less accurate there.
Improvement in the proposed quasigeoid model will
require better data coverage over land and sea in Vietnam
and its vicinity. ese regions have to be covered with
preferentially airborne data and shipborne data.
Abbreviations
GNSS: Global Navigation Satellite System; FFT: Fast Fourier Transform; LSC:
Least-Squares Collocation; GGM: Global Gravity field Model; STD: Standard
deviation; RCR : Remove–Compute–Restore; GOCE: Gravity field and steady-
state Ocean Circulation Explorer; DTM: Digital Terrain Model; DBM: Digital
Bathymetry Model; GMT: Generic Mapping Tools; RET: Rock-Equivalent Topog-
raphy; DTU: Technical University of Denmark; IGP: Institute of Geophysics;
VAST: Vietnam Academy of Science and Technology ; BGI: Bureau Gravimé-
trique International; VIGAC : Vietnam Institute of Geodesy and Cartography;
RTM: Residual Terrain Model; ZT: Zero Tide; MT: Mean Tide; FT: Tide Free; ICGEM:
Center for Global Earth Model; d/o: Degree/Order; LAGEOS: LAser GEOdynam-
ics Satellite; GRACE: Gravity Recovery And Climate Experiment.
Acknowledgements
We are very grateful to the VIGAC, IGP-VAST, and BGI (http://bgi.obs-mip.fr/)
that supplied data for this study. This study was also supported by CNES. The
authors would like to thank the authors of GRAVSOFT (DTU) for providing their
software. The SRTM3arc_v4.1 is available via http://srtm.csi.cgiar .org/SELEC
TION/input Coord .asp. The SRTM15arc_plus model is available via https ://topex
.ucsd.edu/pub/srtm1 5_plus/. All used GGMs in this paper are available via
http://icgem .gfz-potsd am.de/tom_longt ime. The DTU15 gravity is available
via https ://ftp.space .dtu.dk/pub/. In this paper, we used the Generic Mapping
Fig. 10 Magnitude of relative differences between EGM2008 (d/o 2190) and GEOID_LSC with GNSS/levelling data over 21,161 baselines (blue),
fourth-order tolerance (orange), a EGM2008, b GEOID_FFT, and c magnitude of relative differences between GEOID_LSC and GNSS/first- and
second-order levelling data in north area over 248 baselines (blue), third-order tolerance (green)

Page 15 of 16
Vuetal. Earth, Planets and Space (2019) 71:65
Tools (GMT) for producing some of figures. We thank the anonymous review-
ers for their constructive comments and helpful suggestions.
Authors’ contributions
DTV performed all the data processing and drafted the manuscript. All authors
analysed and discussed the preliminary results. SB and SB provided critical
comments for this study and supported the observations. All authors read and
approved the final manuscript.
Funding
Dinh Toan VU receives funding of the Vietnamese Government’s 911 project
and University Paul Sabatier (UPS)-GET. This study is also supported by CNES.
Availability of data and materials
The data that support the findings of the present study are available from the
corresponding author upon request.
Competing interests
The authors declare that they have no competing interests.
Author details
1 Géosciences Environnement Toulouse (GET), Obser vatoire Midi-Pyrénées,
Toulouse, France. 2 Centre National d’Etudes Spatiales (CNES), Toulouse, France.
Received: 1 February 2019 Accepted: 25 May 2019
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