Acta Geod. Geoph. Hung., Vol. 47(1), pp. 1–11 (2012)
DOI: 10.1556/AGeod.47.2012.1.8
DEVELOPMENT OF A SITE-SPECIFIC ZHD MODEL
USING RADIOSONDE DATA
D Singh, J K Ghosh, D Kashyap
Department of Civil Engineering, Indian Institute of Technology Roorkee, 247667, India,
e-mail: [email protected]
[Manuscript received June 8, 2011; accepted January 4, 2012]
Estimation of precipitable water vapor (PWV) in the atmosphere using ground
based GPS (Global Positioning System) data requires an appropriate model for computation of zenith hydrostatic delay (ZHD). Presented herein is a site-specific ZHD
model (SSM) for a station at New Delhi, India. The model has been developed by
regressing one-year atmospheric vertical profile data collected through radiosonde.
The model based on surface atmospheric pressure at the station, has been validated
invoking data of three more years. The ZHD values estimated through the model
disagree at the 0.3 mm level with ZHD values obtained from raytracing of radiosonde
data. Further, Saastamoinen ZHD model provides an error about 0.23 mm rms while
about 0.19 mm by the developed model (SSM). Thus, developed SSM can be used
for precise estimation of PWV.
Keywords: GPS; PWV; site specific; zenith hydrostatic delay
1.
Introduction
Water vapor plays an important role in several atmospheric and geophysical processes such as transfer of energy, formation of clouds, weather system etc. Therefore,
an accurate estimation of water vapor with high spatial and temporal resolution is
required for operational weather forecasting and climate research. Various instruments like radiosonde, ground based remote sensors, radiometers are long been used
for estimation of water vapor. These operational meteorological measurements do
not have adequate resolutions. Thus, restricts accuracy in short-range precipitation
forecast. However, ground based GPS receiver provides continuous and near real
time estimates of precipitable water vapor (PWV), ensuring all weather coverage
and cost effectiveness. This requires model based computations of zenith hydrostatic delay (ZHD). However, the ZHD model is a major source of error. Estimation of GPS PWV with a sub-millimeter accuracy thus requires proper modeling of
ZHD (Bosser et al. 2007). At present, several global and regional ZHD models are
available. Traditionally, ZHD is estimated by invoking Saastamoinen (1972) global
ZHD model. But, most of the existing ZHD models including Saastamoinen have
been derived using the available radiosonde data from European and North American continent (Satirapod and Chalermwattanachai 2005). However, ZHD model
c
1217-8977/$ 20.00 2012
Akadémiai Kiadó, Budapest
D SINGH et al.
2
Table I. ZHD models and their input parameters
Models and year
Saastamoinen (1972)
Hopfield (1969)
Black (1978)
Baby (Baby et al. 1988)
Davis (Davis et al. 1985)
MOPS Hydrostatic delay model (MOPS 1998)
Improved mean gravity model (Bosser et al. 2007)
Unified model (Raju et al. 2007a)
Site Specific model (SSM) (Raju et al. 2007a)
Nature
Parameters
global
global
global
global
global
global
global
regional
site specific
ps , Φ, h
Ts , ps , β
ps , Ts
ps , Φ, h
ps, Φ, h
Φ, day of year
Φ, hsfc , t
ps
ps
ps – atmospheric surface pressure (mbar)
Φ – latitude
h – ellipsoidal height
Ts – surface temperature (K)
β – temperature lapse rate (◦ C/km)
hsfc – surface height (orthometric height (m))
t – time (month)
parameters are region specific due to strong spatial heterogeneity and temporal
variability of the atmospheric constituents. Therefore, a model developed for one
region may not be applicable to another region. In this paper, a site-specific ZHD
model has been proposed based on radiosonde data of New Delhi.
2.
Literature review
Any error in ZHD affects computation of zenith wet delay (ZWD) and hence GPS
based PWV. Thus, estimation of GPS based PWV with sub millimeter accuracy
requires sub millimeter accuracy in ZHD models. Prominent ZHD models developed
so far are depicted in Table I along with their input requirements.
Most of the ZHD models are global in nature. However, Hopfield and
Saastamoinen models are most widely used and provide reasonably accurate ZHD.
These models treat air as an ideal gas, which is not too critical. However, the assumptions of a unique temperature lapse rate and height-independent gravity are
not very realistic. Saastamoinen (1972) estimated the accuracy of the hydrostatic
components with 2–3 mm RMS. Mendes and Langley (1998) found that ZHD could
be estimated with sub millimeter accuracy from the Saastamoinen model if accurate
measurement of surface pressure is available. The limitation of the Saastamoinen
ZHD model is that it requires ellipsoidal height. Further, the temporal variability
of surface pressure and temperature have not been considered in the Saastamoinen
ZHD model (Bosser et al. 2007). Since, the variation of mean gravity is nearly sinusoidal, accuracy of Saastamoinen gravity consideration is 0.001 m/s2 which leads
to an error of 0.2–0.4 mm (Bosser et al. 2007).
Other sources of errors in ZHD models are error in refractive constant and
dry gas constant (Bosser et al. 2007). Elgered et al. (1991) reported that the
Acta Geod. Geoph. Hung. 47, 2012
SITE-SPECIFIC ZHD MODEL
3
(rms) error in the refractivity constant contributes to 2.4 mm error in ZHD computation. Davis et al. (1985) improved the Saastamoinen model by incorporating
Thayer (1974) refractivity constant. Mendes (1999) concluded that ZHD can be
predicted from surface measurement of pressure having an error below 0.5 mm and
concluded that among Hopfield, Saastamoinen, Baby et al. (1988), Davis et al.
(1985), models performance of the Saastamoinen model is far better than the other
hydrostatic models. He further observed that the accuracy of the Saastamoinen
ZHD model is at sub-millimeter level whereas the other models agreed at the millimeter level. In another study, Janes et al. (1991) have reported an accuracy at
the 2–3 mm level for ZHD predictions from empirical models (Saastamoinen and
Hopfield) when compared to ray tracing through standard atmosphere and Radio
soundings. The precision of the ZHD model also depends on the quality of meteorological sensor. Hauser (1989) has estimated that gravity waves could induce 1.7
hpa (mbar) at ground i.e. 4 mm error in ZHD. Turbulence in the boundary layer
is responsible for a smaller (∼0.1 mbar) deviation (Bock and Doerflinger 2001). It
has has shown that a ZHD error of 1 mm leads to a PWV estimation accuracy of
0.15 mm, and 0.40 mbar pressure error causes ∼1 mm ZHD error (Bai and Feng
2003). Janes et al. (1991) has mentioned that sensitivity of ZHD to surface pressure
is 2 mm/hpa. Moreover, the sensitivity of zenith delay to temperature and relative humidity is about 5–20 mm/◦ C and 1–3 mm/% respectively. Therefore, a mm
level accuracy in ZHD thus achievable with precise meteorological sensor (Bock and
Doerflinger 2001).
A study conducted over the Indian subcontinent using the upper air data for
three years (1995–1997) has shown that a unified ZHD model (regional model) is
inferior to the site-specific models (SSMs). The mean of absolute difference (MAD)
obtained from a unified ZHD model is about ∼0.96 cm, 0.90 cm from Hopfield
model and 1.57 cm from Saastamoinen model. The site-specific ZHD models based
on atmospheric surface pressure have shown a mean absolute difference from 0.17 cm
to 1.80 cm over the Indian subcontinent. This performance is comparable with that
of the Hopfield model (Raju et al. 2007a). The mean absolute difference of the
site-specific ZHD model developed by Raju et al. (2007a) for New Delhi is 0.27 cm
with a standard deviation of 0.2 cm when compared with ZHD from radiosonde
data. The Radiosonde data comprised of vertical profiles of p, T up to 25 km and
above (Raju et al. 2007b). Thus, site-specific ZHD model provides better precision
than unified ZHD model over the Indian subcontinent. However, radiosondes can
generally ascend up to about 20 km and atmosphere above this height may provide
a significant contribution to ZHD (Vedel et al. 2001). Such delay arising above
the known atmospheric profile is termed as upper tropospheric correction (Mekik
1997). Thus, the objective of this research work is to develop a site-specific ZHD
model incorporating the upper air correction and treating the gravity as a function
of height.
Acta Geod. Geoph. Hung. 47, 2012
D SINGH et al.
4
3.
Background theory of zenith hydrostatic delay
As an electromagnetic wave propagates in the atmosphere, it gets continuously
refracted due to the varying index of refraction between different layers of the air
starting from the top of the atmosphere up to the ground. Since, the troposphere is
non-dispersive to GPS signals, its refractive index (n) is the ratio of the speed of light
in vacuum to the phase velocity of the signal. As n is just slightly larger than one, a
more convenient quantity (namely refractivity N ) is defined as N = 106 (n−1 ). The
delay experienced by a signal in traveling through the neutral (non-ionized) part of
the atmosphere is generally referred to as tropospheric delay. This is about 80% of
the total delay. The tropospheric path delay, attributed mainly due to retardation,
is described by the following equation (Bock and Doerflinger 2001)
−6
N (s)ds ,
(1)
∆L = [n(s) − 1]ds = 10
where, ds is the incremental path length of the signal. Refractivity of air is usually
described by empirical equations related to thermo dynamical state variables of the
air viz. temperature, pressure, and water vapor pressure. Smith and Weintraub
(1953) suggested the following relationship
e p
,
(2)
+ 3.73 · 105
N = 77.6
T
T2
where p is the total atmospheric pressure (in mbar), T is the atmospheric temperature (in degree Kelvin), and e is the partial pressure of water vapor (in mbar). This
expression is assumed to hold up to about 0.5% under normal atmospheric conditions. In most contexts, the first term in Eq. (2) is considerably larger than the
second. Thayer (1974) provides a more accurate formula for refractivity as follows
p e e
d
−1
N = K1
Zw
Zd−1 + K2
+ K3
,
(3)
T
T
T2
where Ki are constants that are empirically determined in laboratory, pd is the
partial pressure due to dry gases (mbar), e is the partial water vapor pressure
−1
are the compressibility
(mbar), T is the temperature of air (K), Zd−1 and Zw
factor for dry air and water vapor respectively. The compressibility factors, which
are corrections for non-ideal gas behavior, have nearly constant values that differ
from unity by a few parts per thousand (Mendes et al. 2000). The uncertainties
in the constants of Eq. (3) limit the accuracy with which the refractivity can be
computed, to about 0.02% (Davis et al. 1985). Equation (3) can alternatively be
written as follows
e e
−1
Zw
+ K3
,
(4)
N = K1 Rd + K2
T
T2
where
K2 = K2 − K1
Acta Geod. Geoph. Hung. 47, 2012
Rd
Rw
.
(5)
SITE-SPECIFIC ZHD MODEL
5
Rd is the specific gas constant for dry air, Rw is the specific gas constant for water
vapor, and is the density of moist air. Rueger (2002) has also given a very accurate
formula for computing the refractivity. Several authors have determined refractivity
constants empirically and theoretically. However, the estimates are not always in
good agreement for the water vapor content (Davis et al. 1985). The first term
in Eq. (4) is no longer a pure dry component as the total mass density contains
the contribution of water vapor. Hence, this term is referred to as hydrostatic
component of refractivity. Integration of the hydrostatic component of refractivity
along zenith direction constitutes the zenith hydrostatic delay, ZHD, which can be
expressed as (Bosser et al. 2007) follows
ZHD = 10
−6
ra
K1 Rd
dz ,
(6)
rs
where rs is the geocentric radius of the receiver antenna (m), ra is the geocentric
radius of the top of the neutral atmosphere (m), and z is the integration variable.
The empirical ZHD models (i.e. Saastamoinen, Hopfield etc.) have evolved from
this equation under varying assumptions.
4.
Study area and source of radiosonde data
Radiosonde measures temperature, pressure, and humidity at various ascending
points. The radiosondes are expensive, and the cost of these devices restricts the
number of launches to twice daily (0000 and 1200 UTC) at a limited number of
stations (Bevis et al. 1992). Balloons are used to lift radiosonde devices into air
at the required ground station. These balloons usually can ascend a height up to
20 km. The radiosonde sends down the measured data, which gets stored in files.
The instruments measure temperature and relative humidity with accuracies of
∼0.2◦ C and ∼3.5%, respectively, with diminishing performance in cold, dry regions
(Elliot and Gaffen 1991). The pressure obtained from standard radiosonde has an
accuracy of 0.5 hPa (Guharay et al. 2010). These data contains vertical profiles
of Temperature, Pressure and relative humidity but poor in spatial and temporal
resolution. From these profiles, the altitude profiles of refractivity N can be obtained
(Eq. (4)) and further, from profiles of N , the tropospheric delay gets estimated.
In order to develop a site-specific ZHD model, a radiosonde launching station
located at New Delhi at 28.58 N, 77.20 E having elevation 216 m above MSL is
selected. Four years of radiosonde data for New Delhi station has been downloaded
from Wyoming university website (www.weather.uwyo.edu/upperair). Two files of
radiosonde data of each day (0000 UTC and 1200 UTC) are available. The records
that do not contain atmospheric pressure has been rejected at pre-processing stage
along with vertical profiles which are not available up to or beyond 20 km. Further,
there appears some horizontal drifting of radiosondes and termination of observation
after certain altitudes.
Acta Geod. Geoph. Hung. 47, 2012
D SINGH et al.
6
5.
Estimation of zenith hydrostatic delay from radiosonde profiles
Under the assumption that dry air is in hydrostatic equilibrium and the equation
of state, Eq. (6) can alternatively be written (Mekik 1997, Vedel et al. 2001) as:
ZHD = 10
−6
ps
K1 Rd
0
dp
,
g
(7)
where, ps is the surface pressure in mbar. As pressure and gravity (g) change
with height, both have been considered as variable in Eq. (7). Since the pressure
decreases with the ascent of balloons, ps is the upper limit of the integral. The
lower limit of zero pressure has been assumed to occur above 100 km from the earth
surface. The hydrostatic part of the integrals however need to be accounted for
after the last ascending point of the radiosonde profile, since the atmosphere will
still add to the integral in Eq. (7) (Mekik 1997). This can be obtained from Eq. (7)
denoted as upper tropospheric correction and is given as follows
pt
,
(8)
∆ZHD = 10−6 K1 Rd
gr
where, pt and gt are the pressure and gravity at the top of the radiosonde vertical
profile. Equation (8) can be obtained from Eq. (7) by substituting the lower pressure
as zero and the top pressure as pt . Equations (7) and (8) can be combined to yield
the total zenith hydrostatic delay as follows
ZHDRS = 10−6 K1 Rd
t
(pi − pi+1 )
i=1
gi
+ 10−6 K1 Rd
pt
,
gt
(9)
where i and i + 1 indicate the sequence of layers of radiosonde data profiles from
the surface towards vertical direction and t is the top vertical profile of the data.
Pressures pi and pi+1 denote the atmospheric pressure at geo-potential height hi
and hi+1 . Here g1 denotes the absolute gravity at the site (ground). The zenith
hydrostatic delay estimated by radiosonde data is represented by ZHDrs . In the
present work Eq. (9) is used to estimate total zenith hydrostatic delay invoking
four years radiosonde data from New Delhi. The refractivity constant K1 has been
considered to be 77.604 K/mbar, the dry gas constant Rd as 287.05 J/kg.K, the
dry molecular weight Md as 28.9644 kg/kmol and the universal gas constant R as
8314.34 J/Kmol.K (Katsougiannopoulos et al. 2006). Another important parameter used in Eq. (9) is the gravity. But due to non-availability of absolute gravity
at the radiosonde launching station at New Delhi, the available data of absolute
gravity measured at NPL New Delhi, in 1971 by WRGRN (World Relative Gravity
Reference Network, USA, Air Force) has been used. The baseline distance between
the two stations is about 7 km and the elevation difference is 12 m. The spatial variation of absolute gravity is one mGal per mile (1.6 km) at mid latitude
(Griffiths and King 1965). Thus, the absolute gravity of the location (28◦ 34◦ 9’N,
77◦ 7E, 228 m) has been found to be 979 137.8 mGal (1 mGal = 0.00001 m/s2 )
invoking a gravity gradient of 0.3086 mGal/m (Li and Gotze 2001).
Acta Geod. Geoph. Hung. 47, 2012
SITE-SPECIFIC ZHD MODEL
7
Fig. 1. A scatter plot showing the variation of estimated zenith hydrostatic delay (2007)
Fig. 2. A scatter plot showing the variation of surface pressure annually (2007)
6.
6.1
Results
Results of estimated ZHD at New Delhi
Approximately 700 samples have been considered for the estimation of ZHD.
Applying the Eq. (9) and using vertical atmospheric pressure profiles from the
radiosonde data of the year 2007, ZHD has been estimated for the year at 0000
UTC and 1200 UTC. The annual mean of ZHD has been found to be 2.224 m
having a standard deviation of 0.0156 m. It has been observed that the minimum
value of estimated ZHD is during June–July and the maximum is during December–
January. The estimated ZHD varies from 2.184 m to 2.255 m corresponding to a
surface pressure of 958 mbar to 989 mbar. Therefore, the range of the estimated
ZHD is 0.071 m. Figure 1 shows the annual variation of the estimated ZHD for
station New Delhi for the year 2007 along with mean and standard deviation. This
difference in the variation of the estimated ZHD is mainly caused by large annual
variability in atmospheric dry constituents at the station New Delhi. The annual
mean, minimum and maximum of surface pressure (ps ) at the New Delhi station for
the year 2007 are 975 mbar, 958 mbar and 989 mbar respectively and its variation
is as shown in Fig. 2.
Acta Geod. Geoph. Hung. 47, 2012
D SINGH et al.
8
Fig.
3.
A scatter plot of estimated ZHD versus surface pressure along with regression line
(R2 = 0.9998)
Figures 1 and 2 reveal that the trends of both ZHD and surface pressure have
the same pattern and thus, can be inferred that variation in ZHD is related to the
variation in the surface atmospheric pressure and thus, can be considered as the
guiding parameter towards development of the proposed model.
6.2
Development of site specific ZHD model (SSM) for New Delhi
In order to develop a site-specific ZHD model, ZHD is estimated through numerical integration of respective atmospheric vertical profiles obtained from radiosonde
data using Eq. (9). A regression line represented by Eq. (10) gets best fitted to the
scatter plot of the estimated ZHD against surface atmospheric pressure (Fig. 3)
ZHDRS (m) = 0.0022677 · SurfacePressure (mbar) + 0.0121318 .
(10)
The model (Eq. (10)) provides the integrated hydrostatic delay all the way up
to surface. It has been found that the model provides highly correlated (0.9998)
value with that obtained from radiosonde observation with rms error as 0.19 mm
and it is accurate up to 0.13 mm.
6.3
Validation of site-specific ZHD model
In order to validate the developed model, site specific ZHD values have been
computed using the model and that from radiosonde data sets of the years 2006,
2008, 2009. Considering the ZHD values estimated from radiosonde data (as true
value), some significant statistical parameters have been computed as shown in
Table II. It can be found that rms error is always less than or equal to about
0.3 mm and mean absolute difference is less than 0.2 mm.
Acta Geod. Geoph. Hung. 47, 2012
SITE-SPECIFIC ZHD MODEL
9
Table II. The statistical parameters of the developed site-specific ZHD model for New Delhi
Year
R2
RMS
m
Bias
m
MAD
m
2007
2006
2008
2009
0.9998
0.9996
0.9995
0.9996
0.00019
0.00028
0.00033
0.00027
0.000003
0.000097
0.000140
0.000100
0.00013
0.00016
0.00019
0.00016
2007
0.9997
Saastamoinen
0.00023
0.000110
0.00018
Fig. 4. A scatter plot comparison of ZHD using different models for 2007 at New Delhi
The ZHD values using the Saastamoinen model and that estimated from radiosonde profile data at Delhi for the year 2007 is given in Table II. It can be observed that the rms is 0.00023 m and MAD as 0.00018 m for Saastamoinen model
while for site specific ZHD model rms is 0.00019 m and MAD is 0.00013 m for
2007. Hence, the site-specific ZHD model provides a slightly better result than the
Saastamoinen ZHD model. Hence, the developed model can be used for a precise
estimation of the precipitable water vapor (PWV) for New Delhi.
6.4
Comparison with other ZHD models
A comparison of the ZHD values obtained from the developed site-specific model
(SSM developed), the SSM model developed by Raju et al. (2007a), the Saastamoinen ZHD model (Saastamoinen) and the ray-traced radiosonde data (Est ZHD
(RS)) have been done, as shown in Fig. 4. It can be observed that ZHD values
from Saastamoinen and from the site-specific model are close to each other and also
coincide with the ray-traced ZHD, while there is a large systematic difference of
about 19 mm to the SSM model by Raju et al. (2007a).
The large deviation might be due to a difference of the methods of estimation of
ZHD. In case of Raju’s SSM, ZHD is estimated by numerical integration of pressure
and temperature vertical profiles of radiosonde data available up to 25 km and
Acta Geod. Geoph. Hung. 47, 2012
D SINGH et al.
10
above. While, in this study, ZHD is estimated by numerical integration of pressure
and gravity profiles available up to 20 km and for atmosphere above 20 km, an
upper tropospheric correction has been considered.
7.
Discussions
In this study, an account of the ZHD of GPS signals has been explained briefly.
This delay plays a crucial role in GPS PWV estimation. The study focuses on the
development of a site-specific zenith ZHD model based on radiosonde profiles. The
model permits estimation of the delay due to dry air present in the atmosphere.
The hydrostatic delay introduced by troposphere depends on the geographic location
and atmospheric condition. Therefore, in this study a site-specific ZHD model has
been developed that is applicable to a particular place viz. the radiosonde launching
station at New Delhi. The model treats the ZHD as a function of surface pressure
and thus, permits continuous estimation of ZHD. The model has been validated
by comparing its values with the estimated ZHD obtained from radiosonde data
collected for the years 2006, 2008 and 2010 respectively at the New Delhi station.
The precision of the model is about 0.3 mm (RMS) and 0.2 mm in respect of
the mean absolute difference. The model invoking the surface pressure, performs
slightly better than the Saastamoinen ZHD model. However, the study reveals that
the ZHD estimate is quite sensitive to the surface pressure. An error of 1 hPa
in the surface pressure measurement introduces an error of about 2.3 mm error
in ZHD estimate. Thus, it is important to measure the pressure accurately for
ZHD estimation.
8. Conclusions
A site-specific ZHD model has been developed as a function of surface atmospheric pressure model. The model considers the geometrical height same as the
geo-potential height i.e., neglects their difference (Wallace and Hobs 1977). Absolute gravity of the station has been extrapolated from that of a station that is about
7 km away and has 12 m difference elevation. In addition, for different parameters
such as the vertical gravity gradient, refractivity constant, dry molecular weight,
universal gas constant standard values have been considered. In spite of all these
assumptions/approximations, the site-specific model performs slightly better than
Saastamoinen (1972) ZHD model.
Acknowledgements
Authors are thankful to the Department of Science and Technology for funding of the
project. Further, authors acknowledge with thanks the Wyoming University, for making
available Radiosonde data online.
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