GTm-III: a new global empirical model for mapping zenith wet delays onto precipitable water vapour

Abstract

The weighted mean temperature (T_m) is a critical parameter in research of GPS meteorology. Recently, we have established two global T_m models, GTm-I and GTm-Π, using ground-based radiosonde data. These two models, which directly use the location and the day of year to calculate T_m, consider the annual cycle only. In this study, we further take into account the variation characteristics of T_m in semi-annual and diurnal periodicity and estimate the initial phase of each cycle. A more accurate global empirical T_m model called GTm-III is constructed using high-precision Global Geodetic Observing System (GGOS) Atmosphere T_m grid data, and then validated using a new set of T_m grid data, radiosonde data and Constellation Observation System for Meteorology, Ionosphere, and Climate (COSMIC) radio occultation data. Results indicate that compared to the existing global T_m models, GTm-III provides T_m estimates of high accuracy on a global scale and the predicted values are close to the true values in every moment of every day.

Spatial analysis, Satellite geodesy, Ionosphere/atmosphere interactions

1 INTRODUCTION

Understanding the spatiotemporal characteristics of water vapour is of great scientific and practical significance to the study of global climate change and weather forecast. Conventional methods for measuring water vapour mainly include radiosonde and water vapour radiometer, which are not able to meet the increasing demands of meteorological development due to heavy workload, high equipment costs and low spatiotemporal resolution. Askne & Nordius (1987) first deduced the relationship between atmospheric wet delay and precipitable water vapour (PWV), and proposed a method to detect the atmosphere using ground-based global positioning system (GPS). Bevis et al. (1992) implemented the measurement of atmospheric water vapour using ground-based GPS by estimating the weighed mean temperature of the atmosphere using surface temperature (T_s), and first proposed the concept of GPS meteorology. In ground-based GPS meteorology, zenith atmospheric PWV is estimated from signal delays measured by terrestrial GPS stations (Rocken et al.1993; Bevis et al.1994; Duan et al.1996; Chen 1998; Ding 2009). Compared to conventional methods, using ground-based GPS to monitor precipitable water is advantageous in providing perceptible water with high-precision and high spatiotemporal resolution (Yu 2011). Presently, GPS precise point positioning (PPP) can estimate zenith wet delay (ZWD) and other unknown parameters using precise satellite ephemeris and clock corrections (Xu et al.2011). PWV refers to the height of the column of liquid water that would result if it were possible to condense all the water vapour in the overlying column of the atmosphere (Yao et al.2012). The relationship between ZWD and PWV can be expressed as follows:

\begin{equation} {\rm PWV} = \Pi \times {\rm ZWD,} \end{equation}

(1)

\begin{equation} \Pi = \frac{{10^6 }}{{\rho _{\rm w} R_{\rm v} \left[ {\left( {k_3 /T_{\rm m} } \right) + k_2^{'} } \right]}}, \end{equation}

(2)

where Π is the conversion factor of water vapour; ρ_w is the density of liquid water; R_v is the gas constant for water vapour, k₂′ and k₃ are the atmospheric refractivity constants (Bevis et al.1992), and T_m is the weighed mean temperature of the atmosphere, the only variable in calculating Π. The precision of PWV is decided by ZWD and Π. Because International GNSS Service (IGS) is able to provide new zenith total delay (ZTD) product with typical formal errors of 1.5–5 mm (Byun & Bar-Sever 2009), and zenith hydrostatic delay (ZHD) can be modelled with an accuracy of a few millimetres or better given surface pressure measurements (Saastamoinen 1972), the ZWD can achieve a high accuracy. Assuming the ZWD error is 5 mm, the error of PWV caused by ZWD error is only about 0.7 mm, hence consideration should be given towards improving the accuracy of estimating T_m under this circumstance. According to the law of error propagation, we obtain the following approximate formula by differential on Π:

\begin{equation} \frac{{\sigma _\Pi }}{\Pi } \approx \frac{{\sigma _{T_{\rm m} } }}{{T_{\rm m} }}, \end{equation}

(3)

where |$\sigma _\Pi$| is the error of Π, and |$\sigma _{T_{\rm m} }$| is the error of T_m. From formula (3), it can be learnt that precision of T_m importantly affects quality of PWV calculated from ZWD. For instance, when ZWD is 400 mm, if |$\sigma _{T_{\rm m} }$| is 5 K, it would cause error of 1.1 mm to PWV.

Previous research of T_m has made substantial achievements worldwide. Using 2-yr radiosonde data from 13 stations in the United States, Bevis et al. (1992) proposed a regression equation for calculating T_m from T_s, which is suitable for mid-latitudes. However, the application of Bevis T_m–T_s relationship has been limited because few GPS stations are equipped with instruments to measure T_s. Ross & Rosenfeld (1997) analysed 23-yr radiosonde data from 53 stations in the world and found that the T_m–T_s relationship changes with station location and season. Li et al. (1999) proposed a regression formula suitable for estimation of T_m in eastern China, and Wang et al. (2011) also established regionally applicable linear models. Yu and Liu (2009) found that the accuracy of T_m calculated using Bevis T_m–T_s relationship has a strong correlation with the altitude. Yao et al. (2012) established a globally applicable empirical T_m model GTm-I, using radiosonde data from 135 stations in 2005–2009, which considered seasonal and geographical variations and was independent of surface temperature. In GTm-I, T_s is expressed by the global pressure and temperature (GPT) model (Boehm et al.2007), and seasonal variations in T_m are considered. To solve the anomalies of GTm-I in some oceanic areas, Yao et al. (2013) simulated T_m in sea areas where observation is unavailable through a combination of GPT model and Bevis T_m–T_s relationship; and the simulated data, together with radiosonde data, were used to recalculate the model coefficients of GTm-I; finally, an improved model GTm-Π was established.

For higher global T_m accuracy, this study built and validated a new global T_m model GTm-III. Compared with the existing T_m models, GTm-III is superior by taking into account the semi-annual and diurnal variation in T_m, in addition to the annual cycle considered in GTm-I and GTm-Π. The proposed GTm-III estimates the initial phase of each cycle and uses high-precision Global Geodetic Observing System (GGOS) Atmosphere T_m (http://ggosatm.hg.tuwien.ac.at/) grid data as data source.

2 METHODS FOR COMPUTING T_m

2.1 Computing T_m based on atmospheric profile information

The T_m can be calculated using an integral formula with vapour pressure and temperature profile information along zenith direction over the stations, which can be expressed as:

\begin{equation} T_{\rm m} = \frac{{\int {(e/T){\rm d}z} }}{{\int {(e/T^2 ){\rm d}z} }}, \end{equation}

(4)

where e is the vapour pressure (hPa), and T is the absolute temperature (K). In practice, T_m is calculated using the following discrete integral formula:

\begin{equation} T_{\rm m} = \frac{{\int {(e/T){\rm d}z} }}{{\int {(e/T^2 ){\rm d}z} }} = \frac{{\sum {\frac{{e_i }}{{T_i }}} \Delta h_i }}{{\sum {\frac{{e_i }}{{T_i^2 }}} \Delta h_i }}, \end{equation}

(5)

where e_i and T_i are, respectively, the average vapour pressure and average temperature of the atmosphere at the ith layer, and Δh_i is the atmosphere thickness at the ith layer (m).

Both radiosonde and Constellation Observation System for Meteorology, Ionosphere, and Climate (COSMIC) data can provide atmospheric profile information. Presently, there are approximately 1000 radiosonde stations worldwide, at which radiosondes are usually launched twice daily at UTC 00:00 and 12:00 and provide atmospheric profile products (e.g. pressure, temperature, dew point temperature and relative humidity). COSMIC is a space-based observing network consisting of six low-earth-orbit satellites, which provides approximately 2000∼2500 global occultation events every day. European Centre for Medium-Range Weather Forecasts (ECMWF) began operationally assimilating GPSRO measurements from the COSMIC constellation of six satellites on 2006 December 12 (Anthes et al.2008). The WetPrf products from COSMIC Data Analysis and Archive Center provide information of air temperature, barometric pressure, vapour pressure and refractive index for computing PWV with 0.1 km vertical interval from mean sea level to 40 km above sea level. COSMIC data provide vapour pressure profiles, whereas radiosonde data provide the dew point temperature T_d and pressure P, which can be used to calculate the vapour pressure e (hPa) as follows (Bolton 1980):

\begin{equation} \left\{ {\begin{array}{c} {e = 6.112 \times \exp \left( {\frac{{17.67T_{\rm d} }}{{T_{\rm d} + 243.5}}} \right) \times {\rm corr,}} \\ {{\rm corr} = (1.001 + ((P - 100)/900) \times 0.0034).} \end{array}} \right. \end{equation}

(6)

2.2 Computing T_m based on linear regression formula

In view of the strong correlation between T_m and T_s, many scholars propose that T_m can be expressed as a linear function of T_s at the observation station. Bevis et al. (1992) analysed 2-yr sounding data from 13 stations in the United States and first came up with a linear regression formula suitable for T_m estimation at mid-latitudes, as shown below:

\begin{equation} T_{\rm m} = 70.2 + 0.72T_{\rm s} . \end{equation}

(7)

According to Bevis T_m–T_s relationship, T_m is directly calculated from T_s. So far, this formula represents the most widely used T_m model. However, few GPS stations are equipped with instruments for measuring T_s in reality. Thus, it is not practical to calculate T_m based on Bevis T_m–T_s relationship. Additionally, Bevis T_m–T_s relationship is constructed based on radiosonde observation in North America, thus it is possible inapplicable on a global scale.

2.3 Existing global T_m models: GTm-I and GTm-II

Based on GPT model (Boehm et al.2007) and Bevis T_m–T_s relationship, Yao et al. (2012, 2013) established two global T_m models using radiosonde data, that is, GTm-I/GTm-II. These two models directly calculate global T_m with inputs of the location and the day of year (DOY). The expression is:

\begin{equation} T_{\rm m} = \alpha _1 + \alpha _2 h + \alpha _3 \cos \left( {\frac{{doy - 28}}{{365.25}}2\pi } \right), \end{equation}

(8)

in which

\begin{eqnarray} &&{\alpha _1 = \sum\limits_{i = 1}^{55} {[atm\_\, mean(i) \cdot a} P(i) + btm\_\,mean(i) \cdot bP(i)],} \nonumber\\ &&{\alpha _3 = \sum\limits_{i = 1}^{55} {[atm\_\, amp(i) \cdot aP(i) + btm\_\,amp(i) \cdot bP(i)],} } \end{eqnarray}

(9)

where doy is DOY; |$atm\_\,mean(i)$|⁠, |$btm\_\,mean(i)$|⁠, |$atm\_\,amp(i)$| and α₂ are model coefficients; and aP(i) and bP(i)are longitude- and latitude-related functions, respectively. However, the T_m calculated by GTm-I/GTm-II is almost identical at different times within a day, inconsistent with the actual situation.

3 CONSTRUCTION OF A NEW GLOBAL T_m MODEL: GTm-III

3.1 Expression of GTm-III

By studying characters of time-series of T_m, GTm-III considers the semi-annual and diurnal variation in T_m and takes the initial phase of the cycles as parameters. The expression is shown below:

\begin{eqnarray} T_{\rm m} &=& \alpha _1 + \alpha _2 h + \alpha _3 \cos \left( {\frac{{doy - C_1 }}{{365.25}}2\pi } \right)\nonumber\\ && +\; \alpha _4 \cos \left( {\frac{{doy - C_2 }}{{365.25}}4\pi } \right) + \alpha _5 \cos \left( {\frac{{hod - C_3 }}{{24}}2\pi } \right), \end{eqnarray}

(10)

in which

\begin{eqnarray} \alpha _1 &= & \sum\limits_{i - 1}^{55} {[atm\_\,mean(i)\,aP(i) + btm\_\,mean(i)\,bP(i)],}\nonumber\\ \alpha _3 &= & \sum\limits_{i - 1}^{55} {[atm\_\,amp1(i)\,aP(i) + btm\_\,amp1(i)\,bP(i)],}\nonumber\\ \alpha _4 &= & \sum\limits_{i - 1}^{55} {[atm\_\,amp2(i)\,aP(i) + btm\_\,amp2(i)\,bP(i)],}\nonumber\\ \alpha _5 &= & \sum\limits_{i - 1}^{55} {[atm\_\,amp3(i)\,aP(i) + btm\_\,amp3(i)\,bP(i)],} \end{eqnarray}

(11)

where α₁ is the average value; α₂ is elevation correction; α₃, α₄ and α₅ are the amplitudes of annual, semi-annual and diurnal periodicity, respectively; C₁, C₂ and C₃ are the initial phases of annual, semi-annual and diurnal periodicity, respectively; doy is DOY; and hod is UTC time. In formula (11), aP(i) and bP(i) are, respectively, the longitude- and latitude-related functions, and the others are model coefficients. Our previous analyses attested that globally average accuracy of T_m would be improved about 0.4 K after adding semi-annual and diurnal periodicity terms and estimating their initial phases as parameters.

3.2 Data source for constructing GTm-III

GGOS Atmosphere can provide T_m grid data with a time resolution of 6 hr (at UTC 00:00, 06:00, 12:00 and 18:00) and a spatial resolution of 2.5° × 2°; the T_m grid data are obtained based on relevant information provided by ECMWF. Here we first test the accuracy of the above-mentioned products. A total of 341 radiosonde stations uniformly and globally distributed are selected and their T_m at UTC 00:00 and 12:00 of every day in 2010 are taken as the reference values. The T_m at the same station and time is calculated using the bilinear interpolation method with GGOS Atmosphere grid data, and then compared with the reference value. The statistical results of annual average accuracy at all radiosonde stations are shown in Table 1. The histograms of mean bias error (MBE), mean absolute error (MAE) and root mean square (RMS) of T_m gridded data are presented in Fig. 1.

Figure 1.

Histograms of MBE, MAE and RMS of the ‘GGOS Atmosphere’ T_m grid data in comparison with respect to radiosonde data in 2010.

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Table 1.

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Statistical results of the ‘GGOS Atmosphere’ T_m grid data tested by radiosonde data in 2010 /K.

Variable	Average	Max.	Min.
MBE	−0.45	2.90	−7.82
MAE	1.46	7.82	0.54
RMS	1.93	7.99	0.71

Table 1.

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Statistical results of the ‘GGOS Atmosphere’ T_m grid data tested by radiosonde data in 2010 /K.

Variable	Average	Max.	Min.
MBE	−0.45	2.90	−7.82
MAE	1.46	7.82	0.54
RMS	1.93	7.99	0.71

Table 1 shows that the MBE averages −0.45 K and ranges from −7.82 to 2.9 K; the MAE averages 1.46 K and ranges from 0.54 to 7.82 K; the RMS averages 1.93 K and ranges from 0.71 to 7.99 K. Fig. 1 shows that MBEs between −3 K and 3 K (T_m with such error causes about 0.7 mm error in PWV when ZWD is 400 mm) account for 94 per cent of the total data in consistence with that MAEs below 3 K account for 94 per cent, and RMS below 4 K account for 95 per cent of the total data. Through the above-mentioned analysis and former study experience, T_m grid data provided by GGOS Atmosphere are highly precise and reliable, thus can be used as data source for constructing GTm-III.

3.3 Calculate of GTm-III coefficients

When solving the coefficients of GTm-III, a first step is to linearize the non-linear formula (10) as follows:

\begin{eqnarray} T_{\rm m} &=& \alpha _1 + \alpha _2 h + \alpha _3 \cos \left( {\frac{{doy}}{{365.25}}2\pi } \right)\cos \left( {\frac{{C_1 }}{{365.25}}2\pi } \right) \nonumber\\ &&+\; \alpha _3 \sin \left( {\frac{{doy}}{{365.25}}2\pi } \right)\sin \left( {\frac{{C_1 }}{{365.25}}2\pi } \right) \nonumber\\ && +\; \alpha _4 \cos \left( {\frac{{doy}}{{365.25}}4\pi } \right)\cos \left( {\frac{{C_2 }}{{365.25}}4\pi } \right) \nonumber\\ &&+\; \alpha _4 \sin \left( {\frac{{doy}}{{365.25}}4\pi } \right)\sin \left( {\frac{{C_2 }}{{365.25}}4\pi } \right) \nonumber\\ && +\; \alpha _5 \cos \left( {\frac{{hod}}{{24}}2\pi } \right)\cos \left( {\frac{{C_3 }}{{24}}2\pi } \right) \nonumber\\ &&+\; \alpha _5 \sin \left( {\frac{{hod}}{{24}}2\pi } \right)\sin \left( {\frac{{C_3 }}{{24}}2\pi } \right). \end{eqnarray}

(12)

Let

\begin{eqnarray} \alpha _3^{\prime} &=& \alpha _3 \cos \left( {\frac{{C_1 }}{{365.25}}2\pi } \right), \nonumber\\ \alpha _4^{\prime} &=& \alpha _3 \sin \left( {\frac{{C_1 }}{{365.25}}2\pi } \right), \nonumber\\ \alpha _5^{\prime} &=& \alpha _4 \cos \left( {\frac{{C_2 }}{{365.25}}4\pi } \right), \nonumber\\ \alpha _6^{\prime} &=& \alpha _4 \sin \left( {\frac{{C_2 }}{{365.25}}4\pi } \right), \nonumber\\ \alpha _7^{\prime} &=& \alpha _5 \cos \left( {\frac{{C_3 }}{{24}}2\pi } \right), \nonumber\\ \alpha _8^{\prime} &=& \alpha _5 \sin \left( {\frac{{C_3 }}{{24}}2\pi } \right). \end{eqnarray}

(13)

Then, we get:

\begin{eqnarray} T_m &=& \alpha _1 + \alpha _2 h + \alpha _3 '\cos \left( {\frac{{doy}}{{365.25}}2\pi } \right)\nonumber\\ &&+\; \alpha _4 '\sin \left( {\frac{{doy}}{{365.25}}2\pi } \right) + \alpha _5 '\cos \left( {\frac{{doy}}{{365.25}}4\pi } \right) \nonumber\\ &&+\; \alpha _6 '\sin \left( {\frac{{doy}}{{365.25}}4\pi } \right) + \alpha _7 '\cos \left( {\frac{{hod}}{{24}}2\pi } \right) \nonumber\\ &&+\; \alpha _8 '\sin \left( {\frac{{hod}}{{24}}2\pi } \right). \end{eqnarray}

(14)

The model coefficients are calculated using the least-square method. Here we use the 2005–2011 T_m grids data provided by GGOS Atmosphere to fit the model. The globally absolute values of α₃, α₄ and α₅ can be calculated after getting the coefficients of eq. (14). The global distribution of α₁, absolute values of α₃, α₄ and α₅ are shown in Fig. 2 (constant coefficient α₂ ≈ −0.0051).

Figure 2.

Global distribution of coefficients in a new global model for T_m, GTm-III.

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The distributions of α₁ in GTm-III and GTm-II are similar. Fig. 2(a) shows that α₁ near the equator is relatively larger, and it decreases with the increase of latitude, because annual average T_m decreases with the increase of latitude, and so decreases coefficient α₁. Amplitudes of annual periodicity α₃ also show similarity between GTm-III and GTm-II. In Fig. 2(b), absolute values of α₃ in the Northern Hemisphere seems larger than those in the southern, and the largest value locates in Siberian area, where temperature varies significantly between different years. Because of linear correlation between T_m and temperature, one can infer that T_m in Siberian area changes largely in different years, which explains the largest value of α₃ here. In Fig. 2(c), amplitudes of semi-annual periodicity α₄ are larger in the polar areas, which caused by phenomena of polar day and night. For amplitudes of diurnal periodicity α₅, Fig. 2(d) shows that it is larger in most continental areas, especially in Africa, while smaller in oceanic areas. We believe amplitudes of diurnal periodicity α₅ closely relates to temperature difference between day and night.

4 VALIDATION OF GTm-III

A different set of external data including T_m grid data provided by GGOS Atmosphere, radiosonde data and COSMIC occultation data, are used to evaluate the validity and applicability of the GTm-III, to test its accuracy and stability in global scale, and to examine its (dis)advantages compared with the existing models.

T_m from COSMIC should be examined before this validation. As described, 341 radiosonde stations were selected, further all COSMIC occultation events occurred at co-locations with the radiosonde stations in 2010 were selected. The co-location with radiosonde station means that location difference with nearby radiosonde station is less than 1° in longitude and latitude, less than 0.2 km in height at bottom of profiles and less than 1 hr in temporal resolution. Under this qualification, accuracy of T_m from COSMIC was examined by using 1036 selected co-located occultation events and by regarding T_m from radiosonde as the standard. The result shows that MBE between them is 0.7 K, MAE is 1.7 K and RMS is 2.3 K, which indicates that T_m from COSMIC also has high accuracy. Additionally, Yao et al. (2013) show that the GGOS Atmosphere grid data have an MBE of −0.06 K and an RMS of 1.94 K compared with COSMIC data.

4.1 T_m grid data from GGOS Atmosphere

As mentioned above, GTm-III is constructed using 2005–2011 T_m grid data. Here we validate the model using 2012 T_m grid data. With a spatial resolution of 2.5° × 2°, there are a total of 13 195 T_m gridpoints in the world. The T_m data of all grids directly provided by GGOS Atmosphere at four times (UTC 00:00, 06:00, 12:00 and 18:00) every day are taken as the reference values. The T_m of each gridpoint at the same time is calculated with GTm-III, GTm-II and Bevis T_m–T_s relationship (T_s calculated with GPT model for data unavailable in oceanic areas is referred to as Bevis&GPT). The obtained T_m estimates are compared with the reference values. The MAE and RMS of different models in 2012 are shown in Table 2 and Fig. 3.

Figure 3.

Global distribution of MBE, MAE and RMS of different models.

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Figure 3

(Continued.)

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Table 2.

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Statistical results of different models tested by the T_m grid data in 2012.

Model	MBE (in K)			MAE (in K)			RMS (in K)
	Mean	Max.	Min.	Mean	Max.	Min.	Mean	Max.	Min.
GTm-III	0.11	4.7	−3.8	2.5	5.5	0.8	3.2	6.5	1.0
GTm-II	−0.01	9.4	−8.9	3.4	9.4	0.9	4.1	10.4	1.1
Bevis&GPT	1.2	14.0	−5.8	4.0	14.0	0.8	4.8	14.9	1.1

Model	MBE (in K)			MAE (in K)			RMS (in K)
	Mean	Max.	Min.	Mean	Max.	Min.	Mean	Max.	Min.
GTm-III	0.11	4.7	−3.8	2.5	5.5	0.8	3.2	6.5	1.0
GTm-II	−0.01	9.4	−8.9	3.4	9.4	0.9	4.1	10.4	1.1
Bevis&GPT	1.2	14.0	−5.8	4.0	14.0	0.8	4.8	14.9	1.1

Table 2.

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Statistical results of different models tested by the T_m grid data in 2012.

Model	MBE (in K)			MAE (in K)			RMS (in K)
	Mean	Max.	Min.	Mean	Max.	Min.	Mean	Max.	Min.
GTm-III	0.11	4.7	−3.8	2.5	5.5	0.8	3.2	6.5	1.0
GTm-II	−0.01	9.4	−8.9	3.4	9.4	0.9	4.1	10.4	1.1
Bevis&GPT	1.2	14.0	−5.8	4.0	14.0	0.8	4.8	14.9	1.1

Model	MBE (in K)			MAE (in K)			RMS (in K)
	Mean	Max.	Min.	Mean	Max.	Min.	Mean	Max.	Min.
GTm-III	0.11	4.7	−3.8	2.5	5.5	0.8	3.2	6.5	1.0
GTm-II	−0.01	9.4	−8.9	3.4	9.4	0.9	4.1	10.4	1.1
Bevis&GPT	1.2	14.0	−5.8	4.0	14.0	0.8	4.8	14.9	1.1

Table 2 and Fig. 3 show that for GTm-III, MBE averages 0.11 K and ranges from −3.8 to 4.7 K; MAE averages 2.5 K and ranges from 0.8 to 5.5 K; RMS averages 3.2 K and ranges from 1.0 to 6.5 K. The average MAE and RMS of GTm-III are the smallest, while the average MBE of GTm-II is the smallest. Though the mean of MBE of GTm-II is only −0.01 K, Fig. 3 reflects great positive errors near the South Pole and negative errors in some oceans and on land, resulting in the small average MBE. Therefore, GTm-III improves accuracy globally comparing with GTm-II. The accuracy and stability of GTm-III are slightly higher at mid- and low latitudes than at high latitudes and better in oceanic areas than on land. For Bevis T_m–T_s relationship, large errors exist in parts of the Himalayas and Antarctic regions. Similar errors are observed in the results of GTm-II, which uses the Bevis T_m–T_s relationship as supplemental data source; additionally, large errors are observed in the results of GTm-II in some areas of Chile. Together, the above-mentioned results demonstrate that GTm-III has higher accuracy for estimating T_m with no large errors on a global scale.

4.2 Radiosonde data

A total of 461 radiosonde stations in the globe are selected; of which, more stations are distributed in the Northern Hemisphere than in the Southern Hemisphere (Fig. 4). As described in Section 2.1, the T_m calculated from radiosonde data at UTC 00:00 and 12:00 every day in 2012 are treated as the reference values. The T_m estimates are calculated using GTm-III, GTm-II and Bevis T_m–T_s relationship (T_s is the actual surface temperature), and are statistically analysed. MAE and RMS of different models are shown in Table 3.

Figure 4.

Global distribution of radiosonde stations.

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Table 3.

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Statistical results of different models tested by the radiosonde data in 2012.

Model	MBE (in K)			MAE (in K)			RMS (in K)
	Mean	Max.	Min.	Mean	Max.	Min.	Mean	Max.	Min.
GTm-III	0.7	8.6	−3.1	3.3	8.6	1.0	4.2	11.0	1.3
GTm-II	−0.7	6.3	−7.2	3.5	7.5	1.0	4.4	10.0	1.3
Bevis	0.2	9.0	−8.2	3.4	9.1	1.1	4.2	11.2	1.4

Model	MBE (in K)			MAE (in K)			RMS (in K)
	Mean	Max.	Min.	Mean	Max.	Min.	Mean	Max.	Min.
GTm-III	0.7	8.6	−3.1	3.3	8.6	1.0	4.2	11.0	1.3
GTm-II	−0.7	6.3	−7.2	3.5	7.5	1.0	4.4	10.0	1.3
Bevis	0.2	9.0	−8.2	3.4	9.1	1.1	4.2	11.2	1.4

Table 3.

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Statistical results of different models tested by the radiosonde data in 2012.

Model	MBE (in K)			MAE (in K)			RMS (in K)
	Mean	Max.	Min.	Mean	Max.	Min.	Mean	Max.	Min.
GTm-III	0.7	8.6	−3.1	3.3	8.6	1.0	4.2	11.0	1.3
GTm-II	−0.7	6.3	−7.2	3.5	7.5	1.0	4.4	10.0	1.3
Bevis	0.2	9.0	−8.2	3.4	9.1	1.1	4.2	11.2	1.4

Model	MBE (in K)			MAE (in K)			RMS (in K)
	Mean	Max.	Min.	Mean	Max.	Min.	Mean	Max.	Min.
GTm-III	0.7	8.6	−3.1	3.3	8.6	1.0	4.2	11.0	1.3
GTm-II	−0.7	6.3	−7.2	3.5	7.5	1.0	4.4	10.0	1.3
Bevis	0.2	9.0	−8.2	3.4	9.1	1.1	4.2	11.2	1.4

Table 3 shows that in terms of the accuracy and stability, GTm-III has the smallest means of MAE (3.3 K) and RMS (4.2 K) among the three models. Bevis T_m–T_s relationship has the smallest mean of MBE of 0.2 K. The mean of MBE is 0.7 K for GTm-III and is −0.7 K for GTm-II, meaning that they have equal absolute value but opposite signs. GTm-II and Bevis T_m–T_s relationship have relatively high accuracy as well, because the radiosonde stations involved in GTm-II modelling basically cover the inspection area; Bevis T_m–T_s relationship uses the North American radiosonde stations and surface temperature measurements. Radiosonde only provides observations at two times with minor changes every day, although GTm-III adds diurnal periodicity, its advantage relative to GTm-II is not obvious in terms of the overall accuracy. Analysis of the accuracy and stability of various models on different DOYs are described later. The daily testing results at all stations of each model were statistically analysed, the MBE, MAE and RMS of different models varying with DOY are shown in Fig. 5.

Figure 5.

Results of GTm-III, GTm-II and the Bevis T_m–T_s relationship tested by radiosonde data at different days of year.

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From Fig. 5(a), one can see that biases of Bevis T_m–T_s relationship are generally the smallest, and quantity of positive biases of GTm-III is larger than that of GTm-II. For each model, GTm-III has larger biases in winter, GTm-II has larger biases in spring and Bevis T_m–T_s relationship has larger biases in February, March and April. Figs 5(b) and (c) show that model accuracies are different in different seasons and highest in summer. It also shows that accuracies of T_m calculated by Bevis T_m–T_s relationship are least affected by seasonal differences, and those calculated by GTm-II and GTm-III are lowest in February and in March, which are also lower than that calculated by Bevis T_m–T_s relationship. Because radiosonde stations used in the experiment are mainly located at high latitudes of the Northern Hemisphere, where T_m changes little in summer but greater in winter, and GTm-II and GTm-III are filtered with multiyear T_m observations, one sees larger changes and weak model accuracy in winter. However, the accuracy and stability of each model varies at different time. In DOY 1∼100, accuracies (in terms of RMS) of GTm-III and GTm-II are equivalent, and Bevis T_m–T_s relationship are generally more accurate than them. However, in DOY 100∼300, GTm-III and GTm-II are more accurate than the Bevis T_m–T_s relationship. A statistical comparison shows that during the whole year period, GTm-III is better than GTm-II in 71 per cent cases, and GTm-III is better than the Bevis T_m–T_s relationship in 58 per cent cases.

Further, the accuracy of the models is tested in radiosonde stations at different latitudes by integrating data from the stations at similar latitudes (Fig. 6).

Figure 6.

Results of GTm-III, GTm-II and the Bevis T_m–T_s relationship tested by radiosonde data in different latitudes.

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From Fig. 6(a), one can see that GTm-III has relatively smaller biases in each latitude and small differences among biases in different latitudes, verifying general excellence of GTm-III than GTm-II and Bevis T_m–T_s relationship. Figs 6(b) and (c) show that regardless of modelling method, the accuracy and stability of all models are poorer at high latitudes than at low latitudes. For GTm-III, the accuracy and stability are highest in low-latitude regions and slightly better than GTm-II but inferior to Bevis T_m–T_s relationship at high latitudes of the Northern Hemisphere. These results reflect the advantage of Bevis T_m–T_s relationship as a regional model, for it has the highest accuracy and stability in the mid-latitudes of the Northern Hemisphere. Though the Bevis T_m–T_s relationship is better than GTm-III in mid-latitude of the Northern Hemisphere, their differences are small. Moreover, GTm-III is capable of calculating T_m without temperature observations, which also represents its practicality.

4.3 COSMIC occultation data

The existing radiosonde stations cover most of the land areas but rare oceanic areas or polar regions; the number of radiosonde stations is less in the Southern Hemisphere than in the Northern Hemisphere; and low time resolution of radiosonde data leads to flaws in verifying the accuracy and stability of GTm-III on a global scale. In view of the above-mentioned issues related to radiosonde data, later we use the 2012 COSMIC occultation data to test the accuracy and stability of each model. T_m calculated from COSMIC occultation data is taken as the reference value, and T_m estimates obtained with GTm-III, GTm-II and Bevis T_m–T_s relationship are tested. The results in a whole year period are shown in Table 4.

Table 4.

Open in new tab

Statistical results of different T_m models tested by COSMIC occultation data.

Model	Bias (in K)	MAE (in K)	RMS (in K)
GTm-III	−0.5	2.9	3.9
GTm-II	−1.4	3.9	5.0
Bevis	0.9	3.4	4.4

Table 4.

Open in new tab

Statistical results of different T_m models tested by COSMIC occultation data.

Model	Bias (in K)	MAE (in K)	RMS (in K)
GTm-III	−0.5	2.9	3.9
GTm-II	−1.4	3.9	5.0
Bevis	0.9	3.4	4.4

Table 4 shows that the accuracy and stability of GTm-III are the highest globally (Bias, −0.5 K; MAE, 2.9 K; RMS, 3.9 K), obviously better than those of GTm-II and Bevis T_m–T_s relationship (T_s is the measured surface temperature). This is because GTm-III takes into account the characteristics of T_m of semi-annual and diurnal periodicity, and uses globally high-accuracy T_m gird data for modelling.

To test the accuracy of the models at different time points, we make daily statistics. The MAE and RMS of each model varying with DOY are shown in Fig. 7.

Figure 7.

MBE, MAE and RMS of different models at different days of year tested by COSMIC data.

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Fig. 7 shows that the accuracy and stability of each model have no obvious changes with season, there are minor differences between results from radiosonde data and those from COSMIC occultation data, because COSMIC data can reflect comprehensive global changes at every moment of every day while radiosonde data only reflect the changes at two moments of every day on the continent of the Northern Hemisphere. As a result, the test with radiosonde data shows significant seasonal differences, but this phenomenon does not appear in the test with COSMIC data. When compared with GTm-II, GTm-III has high accuracy and stability on a daily scale. This observation reflects that the T_m estimated by GTm-III at each time of every day is closer to the reference value, proving the advantages of GTm-III in model and data source. Similar results are obtained when comparing GTm-III with the Bevis T_m–T_s relationship. Though, the Bevis T_m–T_s relationship considers changes of T_m in 1 d by using observed temperatures it is a locally applicable formula, which cannot represent global relationships between T_m and the Earth's surface temperature.

Finally, the accuracy of the models at different latitudes is tested by integrating data from stations at similar latitudes (Fig. 8).

Figure 8.

MBE, MAE and RMS of different models tested by COSMIC data in different latitudes.

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Fig. 8 shows that among the three models, GTm-III provides T_m estimates with the highest precision; the accuracy and stability of GTm-III are best at low latitudes and better in the Southern Hemisphere than in the Northern Hemisphere. The precision of T_m estimated by GTm-III is higher than that of GTm-II at all latitudes. Compared with Bevis, GTm-III has relative low precision in the north of N40º, relative high precision between N40º and S35º, and highest in the south of S55º. In the south of 66ºS, the precision of T_m estimated by Bevis decreases with increasing latitude, with the maximal MAE close to 10 K and maximal RMS greater than 10 K. T_m–T_s relationship changes with station locations and seasons, and climate in the Arctic and Antarctic regions are significantly different, so precisions of T_m estimated by the Bevis T_m–T_s relationship in the two regions are quite different.

5 CONCLUSIONS

By considering the actual variation characteristics of T_m, an accurate global empirical T_m model, GTm-III, is constructed using high-precision GGOS Atmosphere T_m grid data. The characteristics of T_m in semi-annual and diurnal periodicity are added into GTm-III based on the existing conventional model, and the initial phase of each cycle is estimated. The GTm-III, which can obtain T_m at any time and place without measured T_s data, is validated with a complete set of external data: T_m grids data for testing the accuracy and stability of a model at four moments of the day on a global scale; radiosonde data for testing the accuracy and stability of a model at two moments of the day in partial land areas; and COSMIC occultation data for testing the accuracy and stability of a model at most times of the day worldwide. Results show that GTm-III is superior to GTm-II and the former represents the most stable and a truly globally applicable model for estimating T_m.

The authors would like to thank ‘GGOS Atmosphere’ for providing grids of T_m and COSMIC for the occultation data. This research was supported by the National Natural Science Foundation of China (41174012 and 41274022) and The National High Technology Research and Development Program of China (2013AA122502).

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SUPPORTING INFORMATION

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Appendix (Supplementary Data).

Please note: Oxford University Press is not responsible for the content or functionality of any supporting materials supplied by the authors. Any queries (other than missing material) should be directed to the corresponding author for the article.

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Article Contents

Abstract

1 INTRODUCTION

2 METHODS FOR COMPUTING T_m

2.1 Computing T_m based on atmospheric profile information

2.2 Computing T_m based on linear regression formula

2.3 Existing global T_m models: GTm-I and GTm-II

3 CONSTRUCTION OF A NEW GLOBAL T_m MODEL: GTm-III

3.1 Expression of GTm-III

3.2 Data source for constructing GTm-III

3.3 Calculate of GTm-III coefficients

4 VALIDATION OF GTm-III

4.1 T_m grid data from GGOS Atmosphere

4.2 Radiosonde data

4.3 COSMIC occultation data

5 CONCLUSIONS

REFERENCES

SUPPORTING INFORMATION

Supplementary data

Citations

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Article Contents

GTm-III: a new global empirical model for mapping zenith wet delays onto precipitable water vapour

Abstract

1 INTRODUCTION

2 METHODS FOR COMPUTING Tm

2.1 Computing Tm based on atmospheric profile information

2.2 Computing Tm based on linear regression formula

2.3 Existing global Tm models: GTm-I and GTm-II

3 CONSTRUCTION OF A NEW GLOBAL Tm MODEL: GTm-III

3.1 Expression of GTm-III

3.2 Data source for constructing GTm-III

3.3 Calculate of GTm-III coefficients

4 VALIDATION OF GTm-III

4.1 Tm grid data from GGOS Atmosphere

4.2 Radiosonde data

4.3 COSMIC occultation data

5 CONCLUSIONS

REFERENCES

SUPPORTING INFORMATION

Supplementary data

Citations

Views

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This Feature Is Available To Subscribers Only

2 METHODS FOR COMPUTING T_m

2.1 Computing T_m based on atmospheric profile information

2.2 Computing T_m based on linear regression formula

2.3 Existing global T_m models: GTm-I and GTm-II

3 CONSTRUCTION OF A NEW GLOBAL T_m MODEL: GTm-III

4.1 T_m grid data from GGOS Atmosphere