A globally applicable, season-specific
model for estimating the weighted mean
temperature of the atmosphere
YiBin Yao, Shuang Zhu & ShunQiang
Yue
Journal of Geodesy
Continuation of Bulletin Géodésique
and manuscripta geodaetica
ISSN 0949-7714
Volume 86
Number 12
J Geod (2012) 86:1125-1135
DOI 10.1007/s00190-012-0568-1
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J Geod (2012) 86:1125–1135
DOI 10.1007/s00190-012-0568-1
ORIGINAL ARTICLE
A globally applicable, season-specific model for estimating
the weighted mean temperature of the atmosphere
YiBin Yao · Shuang Zhu · ShunQiang Yue
Received: 18 October 2011 / Accepted: 17 April 2012 / Published online: 28 April 2012
© Springer-Verlag 2012
Abstract In GPS meteorology, the weighted mean temperature is usually obtained by using a linear function of the
surface temperature Ts . However, not every GPS station can
measure the surface temperature. The current study explores
the characteristics of surface temperature and weighted mean
temperature based on the global pressure and temperature
model (GPT) and the Bevis Tm –Ts relationship (Tm = a +
bTs ). A new global weighted mean temperature (GWMT)
model has been built which directly uses three-dimensional
coordinates and day of the year to calculate the weighted
mean temperature. The data of year 2005–2009 from 135
radiosonde stations provided by the Integrated Global Radiosonde Archive were used to calculate the model coefficients,
which have been validated through examples. The result
shows that the GWMT model is generally better than the
existing liner models in most areas according to the statistic
indexes (namely, mean absolute error and root mean square).
Then we calculated precipitable water vapor, and the result
shows that GWMT model can also yield high precision PWV.
List of abbreviations
GPS
Global positioning system
GPT
Global pressure and temperature
GWMT
Global weighted mean temperature
GNSS
Global navigation satellite system
ECMWF
European Centre for Medium-Range
ECMWF
Weather Forecasts
NCEP/NCAR National Centers for Environmental
Prediction/National Center for
Atmospheric Research
IGRA
Integrated Global Radiosonde Archive
IGS
International GNSS service
MAE
Mean absolute error
PWV
Precipitable water vapor
RMS
Root mean square
ZWD
Zenith wet delay
Keywords GPS meteorology · Weighted mean
temperature · GPT model · GWMT model
Because water vapor is very unevenly distributed in space and
in time, clarifying and quantifying the crucial roles that water
vapor plays in the weather and climate systems, particularly
in a predictive sense, requires meteorologists to measure the
space–time distribution of water vapor as well as possible.
This challenging task is achieved in a coordinated international effort using radiosondes, space-based sensing systems
including microwave radiometers, and, more recently, the
Global Positioning System (GPS). The technique known as
ground-based GPS meteorology, often shortened to ‘GPS
met’, uses networks of continuously operating GPS stations
to infer the vertically integrated or ‘total column’ quantity
of water vapor overlying each station in the GPS network
(Bevis et al. 1992, 1994; Duan et al. 1996). This integral measure, known as precipitable water or (rather redundantly) as
Electronic supplementary material The online version of this
article (doi:10.1007/s00190-012-0568-1) contains supplementary
material, which is available to authorized users.
Y. Yao (B) · S. Zhu · S. Yue
School of Geodesy and Geomatics, Wuhan University,
Wuhan 430079, China
e-mail: ybyao@whu.edu.cn
S. Zhu
e-mail: shuangzhu@whu.edu.cn
S. Yue
e-mail: shqyue@whu.edu.cn
1 Introduction
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Y. Yao et al.
precipitable water vapor (PWV), is expressed in length units
and refers to the height of the column of liquid water that
would result if it were possible to condense all the water vapor
in the overlying column of atmosphere. While GPS meteorology cannot, like a radiosonde profile, describe the vertical
distribution of water vapor, GPS meteorology is attractive
because it is an ‘all-weather’ system, it operates continuously in time, it does not suffer from significant instrumental
drifts, and GPS networks are now ubiquitous over most land
masses, and are continuing to expand their spatial coverage.
GPS signal delays are typically characterized in terms of
the zenith delay (i.e., the delay accumulated along a vertical
path through the atmosphere) and two gradient parameters
that quantify the azimuthal variability of delay. GPS networks actually sense the total propagation delay imposed by
the neutral atmosphere, often referred to as the neutral delay.
The neutral delay is composed of the hydrostatic delay, which
is proportional to surface pressure, and the wet delay which is
very nearly proportional to the total quantity of water vapor
along the raypath (Saastamoinen 1972; Hogg et al. 1981;
Davis et al. 1985). Thus the zenith neutral delay (ZND) is the
sum of the zenith hydrostatic delay (ZHD) and the zenith wet
delay (ZWD). The central procedure of GPS meteorology is:
(i) infer the ZND time series at each GPS station as part of
a geodetic analysis, (ii) estimate the ZHD from a measure
or a prediction of surface pressure at each GPS station, (iii)
deduce the ZWD via the identity ZWD = ZND − ZHD, and
(iv) transform the PWV estimate into an estimate of PWV
using the relation
PWV = · ZWD
(1)
When the surface pressure is predicted using a weather model,
rather than locally measured using a barometer, then the
resulting error in the ZHD estimate tends to dominate the
error in the PWV estimate (particularly in regions with significant topography). If surface pressure is measured using
a well-calibrated barometer, and if modern mapping functions (Boehm et al. 2007) are used in the geodetic analysis of
delay, then the largest source of error in the PWV estimate
often derives from the value assigned to the factor , in Eq.
(1). Reducing this particular error source is the purpose of
our study.
The conversion factor in Eq. (1) can be expressed as
=
ρw Rv
106
(k3 /Tm ) + k2
(2)
where ρw is the density of water, Rv is the specific gas constant, k2 and k3 are atmospheric refractivity constants (Davis
et al. 1985; Bevis et al. 1994), and Tm is the weighted mean
temperature of the atmosphere (Davis et al. 1985) which varies in space and in time. The uncertainty in is dominated
by the uncertainty in Tm (Bevis et al. 1994). Therefore, our
goal in this paper is to provide a new and convenient means
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of estimating Tm so as to reduce its error level, and thereby
minimize the error associated with the prediction of .
While Tm can be computed using radiosonde profiles, or
using numerical weather models (Bevis et al. 1994), most
published studies to date estimate Tm utilizing the suggestion of Bevis et al. (1992) that Tm is linearly related to surface
temperature, Ts , i.e.,
Tm = a + bTs
(3)
The main advantage of this approach is that it is relatively
simple to implement. Bevis et al. (1992) noted that to get the
best results, the constants a and b should be ‘tuned’ to specific areas and seasons, but went on to offer some ‘average’
values for a and b, suitable for use in mid latitudes, based
on an analysis of 8,718 radiosonde profiles. This led to the
equation Tm = 70.2 + 0.72Ts , in which 70.2, Tm and Ts are
specified in Kelvin. In a later conference presentation, Bevis
et al. (1995) revised this equation to Tm = 85.63 + 0.668Ts
(85.63 is specified in Kelvin), based on an analysis of about
250,000 radiosonde profiles with a nearly global distribution.
These two sets of equations look quite different, but in fact,
they make fairly similar predictions for the range of surface
temperatures (say −20 to 35 ◦ C) commonly encountered at
GPS stations. The second version of the equation has been
most widely adopted.
Ross and Rosenfeld (1997) emphasized the advantages of
tuning the constants a and b to site and season. Wang et al.
(2005) compared and analyzed global estimates of Tm from
three different data sets from 1997 to 2002: ECMWF 40year reanalysis (ERA-40), the NCEP/NCAR, and the IGRA,
and analyzed the limitations of the Bevis Tm –Ts relationship.
Later, many practitioners of GPS meteorology have tuned
the constants a and b, so as to optimize equation (3) for their
regions, and often by season as well. In China, these studies
include Li et al. (1999); Liu et al. (2000), Gu et al. (2006),
Li et al. (2006), Wang et al. (2007, 2011), Lv et al. (2008).
In this work, we take a new approach to estimating Tm
based on the GPT model (Boehm et al. 2007) and the Bevis
Tm –Ts relationship.
2 Computing the weighted mean temperature, Tm ,
from radiosonde profiles
The weighted mean temperature Tm was defined by Davis
et al. (1985) as the ratio of two vertical integrals though the
atmosphere:
(e/T ) dz
Tm = (4)
e/T 2 dz
where e is the vapor pressure, T is absolute temperature,
and both integrals extend from the surface to the top of the
atmosphere. Since there is very little water vapor above the
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Model for estimating the weighted mean temperature
troposphere, Tm can be thought of as the (unconventionally
defined) mean temperature of water vapor in the troposphere,
overlying any given point on the ground.
If Tm is computed from radiosonde profiles, or using the
output fields of a numerical weather model, the integrals in
Eq. (4) will be discretized into sums of contributions from
successive atmospheric layers, thus:
n ei 1 Ti h i
(5)
Tm =
n ei
h
i
1 T2
i
where
ei
Ti
=
ei
Ti
ei ei−1 + 2
Ti2 Ti−1
ei
i−1
, h i is
,
=
2
2
2
T
e
+ Ti−1
i
the thickness of the atmosphere at the ith layer and ei and
Ti are the water vapor pressure and temperature at the top of
the atmosphere at the ith layer, respectively. ei−1 , and Ti−1
are the water vapor pressure and temperature at the bottom
of the atmosphere at the ith layer, respectively, and n is the
number of layers. More layers result in higher accuracy. This
method applies only if the station sounding data are available;
however, the practicality is not strong.
Bevis et al. (1992, 1995) use this approach to compute Tm
for a large suite of radiosonde profiles, and correlate Tm with
surface temperature (i.e, the first temperature reported by
the radiosonde). Having computed large numbers of (Ts , Tm )
pairs, they performed a linear regression based on Eq. (1),
minimizing the difference between observed and predicted
Tm .
3 GPT model that calculates Ts
The Bevis Tm –Ts relationship shows that Tm can be expressed
as a linear function of Ts . However, in practical application, the real-time global surface temperature at any location
cannot be obtained; this problem has directly obstructed the
development of GPS meteorology. The GPT model, which
was established by Boehm et al. (2007), solves this problem. The model is based on degree and order nine spherical
harmonics.
In consideration of the relationship between surface temperature and location and season of station, the GPT model
can provide pressure and temperature at any station on the
Earth’s surface using the station’s three-dimensional coordinates and day of the year as input parameters.
1127
temperature and measured weighted mean temperature, is
greatly affected by station location and time. Therefore, the
linear model is not considered in the establishment of the
GWMT model.
To further validate the above conclusion, in the present
study, 377 radiosonde stations in the world were randomly
selected (the data can be obtained from the IGRA website
for free). The bad radiosonde data with less than 15 layers
or observation heights lower than 12 km have been excluded.
Tm was calculated by the integral method using Eq. 4. It
was assumed true to evaluate the accuracy of Tmb , calculated
using the station surface temperature Ts and the Bevis Tm –
Ts relationship. The results of four stations from different
regions are shown in Fig. 1.
The information about the four stations is shown in Table 1.
Figure 1 shows that there is a significant time-varying feature in the residuals of Tmb (the Bevis Tm –Ts relationship).
It indicates that the simple linear model developed by Bevis
does not respond well to the annual feature of Tm ; it even
weakens to some extent.
To separately analyze the relationship between elevation
and the residuals of Tmb , The residuals of 136 stations in the
mid-latitudes (N35◦ –N45◦ ) were analyzed. The mean residuals of Tmb at different stations are shown in Fig. 2.
Figure 2 shows that, in the same latitude zone, the mean
residual grows with elevation. This confirms that the Bevis
Tm –Ts relationship cannot fully reflect the relationship
between station elevation and the weighted mean temperature.
To separately analyze the relationship between latitude
and residuals of Tmb , excluding the effect of station elevation, the current study analyzed the mean residuals of Tmb
from 290 stations whose elevations are less than 500 m in
the S10◦ –N45◦ area. The result is shown in Fig. 3.
Figure 3 shows that, as the latitude becomes higher, the
overall residuals of the Tmb become larger. It proves that the
Bevis Tm –Ts relationship cannot fully reflect the relationship
between station latitude and the weighted mean temperature.
In summary, based on the analysis above, we notice the
correlation between the weighted mean temperature and station location and time variation. A simple linear model cannot
truly reflect that. Therefore, considering a refinement of the
linear relationship between Tm and Ts , the GWMT model has
been established.
According to the Bevis Tm –Ts relationship, the relationship between the station surface temperature Ts and the
weighted mean temperature Tm is
4 GWMT model and its determination
Most studies on weighted mean temperature had focused on
establishing a local model by using regional sounding station
data, but not to provide a globally applicable model. Moreover, the linear model, which is based on the station surface
Tm = b1 + b2 Ts
(6)
According to the GPT model, the equation for calculating
the surface temperature is
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10
(a)
15
94120
(b)
41112
10
DeltaTm(K)
DeltaTm(K)
5
0
-5
5
0
-5
-10
-10
-15
01
02
03
04
05
06
07
08
09
10
11
03
04
05
06
Year
(c)
20
10
0
-10
01
20
47401
DeltaTm(K)
DeltaTm(K)
30
02
03
04
05
06
07
08
09
10
11
Year
07
08
09
10
(d)
10
0
-10
01
11
17351
02
03
04
05
06
07
08
09
10
11
Year
Year
Fig. 1 Residuals at four stations. Delta Tm is the residual of Tmb and Tm , Ts used in the Bevis Tm –Ts relationship is surface temperature
Table 1 Information of the four
stations used in Fig. 1
Number
Name
Latitude (◦ )
Longitude (◦ )
94120
DARWIN
−12.43
130.87
29
Height (m)
41112
ABHA
18.23
42.65
2,093
47401
WAKKANAI
45.42
141.68
11
17351
ADANA
37.05
35.35
28
10
16
12
Mean of DeltaTm/K
m
Mean of Delta T (K)
14
10
8
6
4
2
0
-2
0
500
1000
1500
2000
2500
0
3000
-5
-10
Height(m)
Fig. 2 Relationship between the residuals of Tmb and station elevation.
Mean of Delta Tm is the mean residual of Tmb at one station from 2000
to 2010, Ts used in the Bevis Tm –Ts relationship is surface temperature
doy − 28
2π − 0.0065h
Ts = atm + ata · cos
365.25
55
i=1
123
0
10
20
30
40
latitude
Fig. 3 Relationship between the residuals of Tmb and latitude. Mean
of Delta Tm is the mean residual of Tmb at the stations from 2000 to
2010, Ts used in the Bevis Tm –Ts relationship is surface temperature
(7)
ata =
where
atm =
5
55
[at− amp(i) · a P(i) + bt− amp(i) · b P(i)] (8)
i=1
[at− mean(i) · a P(i) + bt− mean(i) · b P(i)]
doy is the day of the year, a P(i) and b P(i) are spherical harmonics obtained by latitude and longitude, and at− mean(i),
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Model for estimating the weighted mean temperature
1129
Fig. 4 135 radiosonde stations
used for the determination of
GWMT
bt− mean(i), at− amp(i), bt− amp(i) are vectors with 55 coefficients.
Substituting Eq. 7 into Eq. 6 gives:
doy − 28
2π
Tm = b1 + b2 atm + b2 ata · cos
365.25
−0.0065b2 h.
(9)
All elements of the first column of the vector a P(i) in the
GPT model are 1; therefore, b1 + b2 atm will eventually be
absorbed into α1
α1 =
55
[atm − mean(i) · a P(i) + btm − mean(i) · b P(i)].
i=1
(10)
In addition, we make α2 = 0.0065b2 and α3 = b2 ata; thus,
Eq. 9 can be simplified to
doy − 28
2π
(11)
T m = α1 + α2 h + α3 cos
365.25
where
α1 =
55
because some grids have no station, or some stations do not
have enough data, as shown in Fig. 4.
By using more than 370,000 radiosonde profiles of 5 years
(January 2005 to December 2009) at 135 stations, we got the
atmospheric weighted mean temperature Tm through Eq. 4,
and considered it as a true value. Combined with Eq. 11 and
Eq. 12, the model coefficients can be estimated in a least
squares adjustment. We use the global distribution of α1 , α2 ,
and α3 to represent the results; however, α2 is a constant
coefficient.
α2 = −0.0041
Figure 5 shows that there are obvious problems in the
southern Pacific for α1 , as well as in the southern Pacific and
east coast of the African continent for α3 , which may lead to
lower accuracies in those areas. Combined with Fig. 4, we
inferred that it may due to the low observation density in these
areas. However, considering that GPS data in those areas is
hard to get and GPS meteorology is not applied there, it will
not affect the application of GWMT. In addition, we will use
other data which are more widely distributed to calculate the
model coefficients in future studies, to get better results.
[atm − mean(i) · a P(i) + btm − mean(i) · b P(i)]
i=1
α3 =
55
[atm − amp(i) · a P(i) + btm − amp(i) · b P(i)].
i=1
(12)
atm_mean(i), btm − mean(i), atm_amp(i), btm − amp(i),
and α2 are the model coefficients determined in a least squares
adjustment.
The key to obtain the coefficients of the model is to select
uniformly distributed radiosonde stations and high-quality
radiosonde data. We divided the globe into 10◦ × 20◦ global
grids evenly, and then selected one station from each grid to
build the model. Finally, only 135 stations have been selected
5 Validation of GWMT
To evaluate the accuracy of the GWMT model, we compared
the model with the widely used Bevis Tm –Ts relationship.
MAE and RMS were considered as the standard for accuracy evaluation, i.e,
N
1 Tm i − Tm i N
i=1
N
1 T − Tm 2
RMS =
mi
i
N
MAE =
(13)
(14)
i=1
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Global plots of α 3
Global plots of α 1
80
80
360
60
15
60
10
340
40
320
20
0
300
-20
280
-40
260
-60
Latitude/degree
Latitude/degree
40
5
20
0
0
-5
-20
-10
-40
-15
-60
-20
240
-80
-180 -150 -120
-80
-90
-60
-30
0
30
60
90
120
150
180
220
Longitude/degree
-25
-180 -150 -120
-90
-60
-30
0
30
60
90
120
150
180
Longitude/degree
Fig. 5 Global distribution of α1 and α3 in Eq. 11
where Tm i is the value calculated by the model, Tm i is the true
value derived from the integration of the radiosonde data, and
N is the number of sounding times. MAE reflects the mean
bias between the GWMT model and the true value, whereas
RMS reflects the applicability, reliability, and stability of the
GWMT model.
To test the internal and external accuracies of the GWMT
model, the data of two set of radiosonde stations are used to
evaluate the statistical precision of the GWMT model and
the Bevis Tm –Ts relationship, and the results were analyzed
and compared. The stations which used for establishing the
model were applied to evaluate the internal accuracy of the
model, and those did not were used to evaluate the external accuracy. There are two sources of surface temperature
for the Bevis Tm –Ts relationship for these stations: one is
calculated by the GPT model, and the other one is directly
extracted from the radiosonde data.
5.1 Testing of stations used for establishing the model
(internal accuracy testing)
The current study equally divided the globe into 18 belts
with respect to latitude (10◦ per latitude belt) and selected
one station from each belt. 18 stations were chosen from the
135 stations used for establishing the model. We calculated
Tmg (GWMT model) and Tmb (Bevis Tm –Ts relationship) for
2010, then calculated true values Tm using Eq. 4 and then
calculated their MAE and RMS, respectively. The results are
shown in Fig. 6.
Figure 6a, b shows that the precision of Tmg from the
GWMT model is uniform, MAE<5 K, and RMS<6 K, much
better than the precision of Tmb from the Bevis Tm –Ts relationship (Ts from the GPT model). The difference is more
significant in high latitudes. Figure 6c, d shows that the accuracy of Tmg and Tmb (Ts are true values) are generally at the
same level, while the GWMT model shows better stability
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by effectually restraining the greater error. Statistical data are
shown in Table 2.
Table 2 shows that the results of Tmg from the GWMT
model are much better than Tmb using GPT Ts and is slightly
better than Tmb using the true Ts . This shows that the approach
of the GWMT model is reasonable and feasible in theory and
in practice. And the model is independent of site surface temperature. Furthermore, when there is no surface observation
temperature available, it has a striking advantage compared
with the Bevis Tm –Ts relationship or with the other local
models.
5.2 Testing of stations not used for establishing the model
(external accuracy testing)
The testing of the stations which used for establishing the
model can only prove the rationality and effectiveness of the
GWMT model in some degree, but it cannot fully demonstrate the reliability and adaptability of the GWMT model.
Therefore, we examined the stations that have not used for
establishing the model. Based on Sect. 5.1, 59 stations were
selected (five stations from each belt) to test the model.
Not every belt, however, could provide five stations because
of bad data quality and station distribution. The results are
shown in Fig. 7.
Figure 7a, b shows that the accuracy of Tmg from the
GWMT model is generally better than Tmb from the Bevis
Tm –Ts relationship (Ts from GPT model) in the northern
hemisphere; however, the accuracy of the GWMT model
at several southern hemisphere stations is slightly worse.
Figure 7c, d shows the similar trends. The specific statistical
data are shown in Table 3.
Table 3 clearly shows that the accuracy of Tmg from the
GWMT model is still better than that of Tmb using GPT Ts
at the stations that have not used for establishing the model.
The accuracy is almost at the same level compared with Tmb
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(a)
(b)
20
Tmg.MAE
Tmb.MAE
15
10
5
0
-80
-60
-40
-20
0
20
40
60
RMS of GWMT(*) and Bevis(O) /K
MAE of GWMT(*) and Bevis(O) /K
20
16
14
12
10
8
6
4
2
80
Tmg.RMS
Tmb.RMS
18
-80
-60
-40
-20
latitude
(c)
40
60
80
8
Tmg.MAE
Tmb.MAE
7
6
5
4
3
2
-80
-60
-40
-20
0
20
40
60
RMS of GWMT(*) and Bevis(O) /K
MAE of GWMT(*) and Bevis(O) /K
20
(d)
8
1
0
latitude
6
5
4
3
2
80
Tmg.RMS
Tmb.RMS
7
-80
-60
-40
-20
latitude
0
20
40
60
80
latitude
Fig. 6 Results of the 18 stations used for modeling (K). Ts of the Bevis Tm –Ts relationship in a and b are calculated from the GPT model. Ts of
the Bevis Tm –Ts relationship in c and d are true values
Table 2 Statistical data of the
18 stations used for modeling
(K)
MAE (K)
Mean
RMS (K)
Max
Min
Mean
Max
Min
4.0
5.4
2.5
GWMT
3.1
4.3
1.5
Bevis Tm (GPT)
5.5
18.1
1.6
6.5
18.4
2.9
Bevis Tm (True Ts )
3.5
7.4
1.4
4.4
7.7
2.0
using the true Ts . Thus, the GWMT model can provide precise weighted mean temperature in the global scope.
5.3 Comparison between GWMT and Bevis Tm –Ts
relationship
Sections 5.1 and 5.2 have shown that the GWMT model has
higher accuracy compared with the other models. However,
the comparison is relative to the real Tm calculated by sounding data. Therefore, the GWMT model was then compared
with the Bevis Tm –Ts relationship by calculating the mean
values and standard deviations of the differences. We selected
all the 77 stations mentioned earlier. The result is shown in
Fig. 8.
Figure 8 shows, by latitude band, the mean values and
standard deviations of the differences of Tm calculated by the
GWMT and the Bevis Tm –Ts relationship. GWMT model is
closer to the Bevis Tm –Ts relationship when using true Ts .
The differences are close to zero in most latitude belts. On the
other hand, the differences with the Bevis Tm –Ts relationship
when using GPT Ts are larger. Between the latitudes N20◦
and N80◦ , the differences are more significant. It proves that
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(a)
(b)
25
Tmg.MAE
Tmb.MAE
20
15
10
5
0
-80
-60
-40
-20
0
20
40
60
RMS of GWMT(*) and Bevis(O) /K
MAE of GWMT(*) and Bevis(O) /K
25
Tmg.RMS
Tmb.RMS
20
15
10
5
0
80
-80
-60
-40
-20
latitude
(c)
40
60
80
15
Tmg.MAE
Tmb.MAE
12
10
8
6
4
2
-80
-60
-40
-20
0
20
40
60
RMS of GWMT(*) and Bevis(O) /K
MAE of GWMT(*) and Bevis(O) /K
20
(d)
14
0
0
latitude
Tmg.RMS
Tmb.RMS
10
5
0
80
-80
-60
-40
latitude
-20
0
20
40
60
80
latitude
Fig. 7 Results of 59 stations not used for establishing the model (K). Ts of the Bevis Tm –Ts relationship in a and b are calculated from the GPT
model. Ts of the Bevis Tm –Ts relationship in c and d are true values
Table 3 Statistical results of 59
stations not used for establishing
the model (K)
MAE (K)
Mean
Max
Min
Mean
Max
Min
GWMT
3.7
13.1
1.4
4.6
14.2
1.7
Bevis Tm (GPT)
4.8
20.1
1.3
5.7
21.0
1.7
Bevis Tm (True Ts )
3.3
10.5
1.0
4.1
12.0
1.5
the GWMT model is closer to the Bevis Tm –Ts relationship
when using the true Ts .
5.4 Comparison between GWMT and GGOS atmosphere
data
The results above indicate that the GWMT model has higher
accuracy in areas where the station density is greater. However, Fig. 5 shows that the accuracy in some areas with no
stations like the southern Pacific is lower. In order to evaluate the accuracy, GMWT is also compared with the grids
123
RMS (K)
of weighted mean temperature provided by project GGOS
Atmosphere, which provides Tm globally on 2◦ × 2.5◦ grids
every 6 hour (http://ggosatm.hg.tuwien.ac.at/). For example,
we used the epoch 0 UT on 27 January 2012.
Figure 9 shows that the differences between GWMT and
GGOS Atmosphere in most areas are small, mostly between
−10 and 10 K, which means that the accuracy of Tmg in most
areas is good. However, the accuracy in southern Pacific
is worse, similar to the abnormal areas of α1 and α3 in
Fig. 5. This confirmed our hypothesis in Sect. 4: it is because
the absence of the radiosonde station data. However, the
Author's personal copy
Model for estimating the weighted mean temperature
True Ts
1133
GPT Ts
will be coming from these areas. We will use other data sets
which are globally available to calculate the model coefficients in future studies, expecting to get more reliable results
globally.
50
40
30
dTm/K
20
5.5 Validation of PWV
10
0
-10
-20
-30
-40
-80
-60
-40
-20
0
20
40
60
80
Latitude/degree
Fig. 8 Differences between the GWMT model and the Bevis Tm –Ts
relationship. Ts used in the red line are calculated from the GPT model,
Ts used in the blue line are true values of Ts
90
80
Latitude/degree
70
50
60
30
40
10
20
-10
0
In GPS meteorology, our purpose is to improve the accuracy
of PWV. Therefore, we use the GWMT model to calculate
the PWV to verify its practicality. Wuhan station was not
used for establishing the model. Figure 10 shows the PWV
comparison at Wuhan, China, in 2010.
Figure 10 shows that the GPS/PWV (GWMT Tm ) and RS
PWV are very close in value. In addition, they have the same
changing trend. The same holds for GPS PWV (RS Tm ).The
differences of PWV (mean value and standard deviations)
are shown in Fig. 11.
Figure 11 shows that the differences between GPS/PWV
(GWMT Tm ) and RS PWV are small, and the mean value is
1.2 mm. It is the same for the differences between GPS/PWV
(RS Tm ) and RS PWV. This strongly proves that high-accuracy PWV can be calculated by GWMT, which means that
GWMT can play an important role in the application of global
remote sensing of the atmosphere.
-30
-50
-20
-70
-40
-90
-180 -150 -120 -90 -60 -30
0
30
60
90
120 150 180
Longitude/degree
Fig. 9 Comparison of global Tm between GGOS Atmosphere and the
GWMT model on 27 January 2012 at 0 UT
lower accuracy in those areas does hardly affect the application of the GWMT model, because very little GPS data
6 Conclusions
By studying the characteristics of weighted mean temperature, based on the GPT model and the Bevis Tm –Ts relationship, this paper proposes a new model called GWMT. This
model fully takes the effects of the station location (longitude, latitude, and elevation) and the season (day of the year)
when calculating the weighted mean temperature Tm into
account. The model coefficients have been solved based on
(a)
(b)
80
GPS PWV(GWMT Tm)
RS PWV
70
60
60
50
50
PWV/mm
PWV/mm
70
80
40
30
40
30
20
20
10
10
0
Jan10 Feb10Mar10 Apr10 May10 Jun10 Jul10 Aug10 Sep10 Oct10 Nov10Dec10
Date/day
Fig. 10 PWV comparison at Wuhan station. GPS PWV (GWMT Tm )
is calculated by using ZTD (provided by IGS) and the GWMT model,
RS PWV is calculated by using IGRA sounding data, GPS PWV (RS
GPS PWV(RS Tm)
RS PWV
0
Jan10 Feb10Mar10 Apr10 May10 Jun10 Jul10 Aug10 Sep10 Oct10 Nov10 Dec10
Date/day
Tm ) is calculated by using ZTD (provided by IGS) and IGRA Tm . a
The comparison between GPS PWV (GWMT Tm ) and RS PWV, b the
comparison between GPS PWV (RS Tm ) and RS PWV
123
Author's personal copy
1134
Y. Yao et al.
(GWMT Tm)
Mean=1.2
(b)
RMS=4.1
30
25
25
20
20
15
15
dPWV/mm
dPWV/m
(a)
30
10
5
Mean=1.3
RMS=4.0
10
5
0
0
-5
-5
-10
-10
-15
(RS Tm)
-15
Jan10 Feb10 Mar10 Apr10 May10 Jun10 Jul10 Aug10 Sep10 Oct10 Nov10Dec10
Date/day
Jan10 Feb10 Mar10 Apr10 May10 Jun10 Jul10 Aug10 Sep10 Oct10 Nov10 Dec10
Date/day
Fig. 11 Differences of PWV at Wuhan station. a The differences between GPS/PWV (GWMT Tm ) and RS PWV, b the differences between GPS
PWV (RS Tm ) and RS PWV
5 years of radiosonde data at 135 stations and evaluated the
internal and external accuracies of the model. The result demonstrates that the model can directly calculate the weighted
mean temperature Tm through station locations and day of
the year and has a higher accuracy than the Bevis Tm –Ts
relationship. Thus, we also get high-precision PWV result
by GWMT. With strong reliability and practicality, it does
not rely on ground-measured temperatures, and can be more
extensively applied than any other local model. The model is
the original result of calculating weighted mean temperature
and strongly pushes forward the development and application of numerical weather prediction. However, we find that
in some areas such as the southern Pacific, the accuracy of Tm
is lower than that in the continent, which is caused by low distribution density of radiosonde stations. We cannot avoid the
problem, however, it would not affect the worldwide application of the GWMT and subsequent research because very
little GPS data will be coming from these areas. We will use
other data sets which are more widely distributed to amend
the model coefficients in the future studies.
Acknowledgments The authors would like to thank IGRA for providing access to the web-based IGRA data and GGOS Atmosphere
for providing grids of weighted mean temperature. We also appreciate the helpful comments of Michael Bevis, Johannes Boehm, Roland
Klees and the anonymous reviewers.This research was supported by the
National Natural Science Foundation of China (41021061; 41174012).
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