Abstract
This part describes the effects of the troposphere—strictly speaking the neutral atmosphere—on the propagation delay of space geodetic signals. A theoretical description of this tropospheric propagation delay is given as well as strategies for correcting for it in the data analysis of the space geodetic observations. The differences between the tropospheric effects for microwave techniques, like the Global Navigation Satellite Systems (GNSS) and Very Long Baseline Interferometry (VLBI), and those for optical techniques, like Satellite Laser Ranging (SLR), are discussed. Usually, residual tropospheric delays are estimated in the data analysis, and the parameterization needed to do so is presented. Other possibilities of correcting for the tropospheric delays are their calculation by ray-tracing through the fields of numerical weather models and by utilizing water vapor radiometer measurements. Finally, we shortly discuss how space geodetic techniques can be used in atmospheric analysis in meteorology and climatology.
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Keywords
- Very Long Baseline Interferometry (VLBI)
- Global Navigation Satellite Systems (GNSS)
- Geodetic Space
- Tropospheric Delay
- Numerical Weather Models
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
1 Introduction
After the signals of the space geodetic techniques have passed through the ionosphere (see Part 2 (Alizadeh et al. 2013) for more information about the ionospheric effects) they also need to pass through the neutral atmosphere (primarly the troposphere) before they are observed at the surface of the Earth. In the troposphere the signals experience propagation delays, just as they do in the ionosphere. However, the frequency dependence of the delays is small. For microwave techniques like Global Navigation Satellite Systems (GNSS) and Very Long Baseline Interferometry (VLBI) there is practically no frequency dependence; thus it is impossible to remove the tropospheric delay with a multi-frequency combination like it is in the ionospheric case. For optical techniques like Satellite Lase Ranging (SLR) there exists a small frequency dependence and thus it is possible in principle to remove the tropospheric delays using two frequencies (see Sect. 4.3.2); however due to the amplification of the noise this is currently not practical. Thus the tropospheric delays need to be corrected for by other means.
For this part of the book it is assumed that the reader is familiar with the basic properties of the atmosphere, i.e. what is described in Part 1 (Böhm et al. 2013). We begin here with the basic description of the refractivity of the air in the neutral atmosphere (Sect. 2). Expressions for calculating the refractivity from basic meteorological measurements are presented, as well as the commonly used division of the refractivity into a hydrostatic and a wet part. In Sect. 3 these results are used for calculating the tropospheric path delay, and the properties of the hydrostatic (Sect. 3.1) and wet (Sect. 3.2) delays are discussed. The modeling of the tropospheric delays in the space geodetic data analysis is described in Sect. 4. Either the tropospheric delays are estimated in the data analysis or tropospheric delays obtained by external measurements are used. Two possible sources of external tropospheric information are considered: from ray-tracing through numerical weather models (Sect. 4.1) and inferred from microwave radiometer measurements (Sect. 4.4). The models that are commonly used when estimating the tropospheric delays in the data analysis are given in Sects. 4.2 (microwaves) and 4.3 (optical). However, since the refractivity of the atmosphere is varying randomly due to atmospheric turbulence these models are not perfect. The effects of turbulence are described in Sect. 5. This part of the book concludes with a discussion of the possible use of space geodetic techniques for studies of the atmosphere (Sect. 6).
2 Basics
The propagation of electromagnetic waves are described by Maxwell’s equations (Jackson 1998). For a non-conducting, neutral medium like the troposphere these equations are
where \({\mathbf{{E} }}\) and \({\mathbf{{B} }}\) are the electric field and magnetic field vectors, respectively, \(\varepsilon \) the electric permittivity, \(\mu \) the magnetic permeability. Assuming that the spatial and temporal variations in \(\mu \) and \(\varepsilon \) are small, the equations can be combined into forming a wave equation for the electric field.
where \(c=1/\sqrt{\varepsilon _0\,\mu _0}\) is the speed of light in vacuum and \(n\) is called the refractive index. A similar expression for the magnetic field can also be derived.
It is clear from Eq. (5) that in order to describe the propagation of a radio wave we need to know the refractive index \(n\). In the neutral atmosphere of the Earth \(n\) is very close to one, thus it is more convenient to use the so called refractivity instead. The refractivity \(N\) (in “N-units”, mm/km, or ppm) is related to the refractive index by
In general the refractivity is a complex number. It can be divided into three parts
In case the spatial and temporal variations of \(N\) are small, i.e. the variations over one wavelength or one period are negligible, the effect on the propagation of electro-magnetic waves caused by the real and the imaginary parts of the refractivity can be considered separately. For the signals of space geodetic techniques traveling through the atmosphere, this separation is a reasonable assumption since the wavelengths are shorter than a few decimeters. The real part of the refractivity (\(N_0+N^{\prime }(\nu )\)) causes refraction and propagation delay of signals traveling through the atmosphere. It consists of a frequency-independent (non-dispersive) part \(N_0\) and a frequency-dependent (dispersive) part \(N^{\prime }(\nu )\).
The imaginary part of the refractivity, \(N^{\prime \prime }(\nu )\), causes absorption and is related to the absorption coefficient \(\alpha \)
The power \(W\) of a signal received after propagating along the path \(S\) through the atmosphere will be lower than it would have been (\(W_0\)) in vacuum (i.e. no absorption)
where \(\tau (\infty ,\nu )\) is called the opacity.
Since the observables of space geodetic techniques (e.g. GNSS, VLBI, and SLR) typically are measurements of the travel time of the signals, the absorption is typically not important since it does not affect the propagation delay. Of course, absorption will affect the delay measurements by increasing the noise; higher attenuation will cause the signal-to-noise-ratio to be lower, and thus the accuracy of the measured delay will be worse (in the worst case the signal cannot be detected). However, there is typically no need for modeling this effect in the space geodetic data analysis. Thus, in the following we will concentrate on the real part of the refractive index and the effects caused by it. We will come back to the absorption in Sect. 4.4, where measurements of the absorption by microwave radiometry are used to estimate the atmospheric path delay.
The (real part of) refractivity can be expressed as a function of the densities of the different atmospheric gases and the temperature \(T\) (Debye 1929)
where \(\rho _i\) is the density of the \(i\)th gas, and \(A_i\) and \(B_i\) are constants. The \(B_i\frac{\rho _i}{T}\) term is caused by the permanent dipole moment of the molecules. Since water vapor is the only major atmospheric gas having a permanent dipole moment, we can ignore this term for all other gases. The relative concentrations of the dry atmospheric gases are approximately constant (except carbon dioxide, see Sect. 2.1). Thus we can assume that \(\rho _i=x_i\rho _d\), where \(x_i\) is constant and \(\rho _d\) is the density of dry air. This makes it possible to express the refractivity as a function of pressure, temperature, and humidity (Essen and Froome 1951)
where \(\rho _{lw}\) is the density of liquid water. It is here assumed here that the liquid water droplets are small compared to the wavelength (\(<\)1 mm for microwave techniques), for larger droplets the expression becomes more complicated (Solheim et al. 1999). However, normally the liquid water contribution to the refractivity (\(k_4(\nu )\rho _{lw}\)) is neglected since it is small, especially outside of clouds. The variables \(Z_d\) and \(Z_w\) are compressibility factors for dry air and water vapor, respectively. These describe the deviation of the atmospheric constituents from an ideal gas. The compressibility factor for the \(i\)th constituent of air is given by
where \(M_i\) is the molar mass and \(R\) is the universal gas constant. For an ideal gas we have \(Z=1\). Owens (1967) obtained expressions for \(Z_d^{-1}\) and \(Z_w^{-1}\) by a least squares fitting to thermodynamic data. These expressions are (for \(p_d\) and \(p_w\) in hPa and \(T\) in K)
2.1 Microwaves
Figure 1 shows the total refractivity for frequencies between 0 and 100 GHz for the case when the total pressure is 1013 hPa, the temperature is 300 K, and the relative humidity is 100 % (and for three different values for the concentration of liquid water). The refractivity was calculated using the Millimeter-wave Propagation Model (MPM) (Liebe 1985, 1989; Liebe et al. 1993). As can be seen, the variations in the refractivity as function of frequency are relatively small. The biggest variations are in the range 50–70 GHz, a region where several strong absorption lines exist for oxygen. Below 40 GHz the refractivity is more or less constant. There are small variations around the 22.235 GHz water vapor absorption line, however these can typically be neglected. Since all space geodetic techniques that use microwaves operate at frequencies well below 40 GHz, we can consider the refractivity to be frequency independent for microwaves. Thus the phase (\(c_p=c_0/n\)) and group velocities (\(c_g=c_0/(n+f\,\partial n/\partial f)\)) in the troposphere will be equal.
The total refractivity as function of frequency. The total pressure is 1013 hPa, the temperature 300 K, and the relative humidity is 100 %. Three different cases are shown corresponding to different concentrations of liquid water: 0, 0.05, and 1 g/m\(^3\)
In Fig. 1 three different cases are shown corresponding to different concentrations of liquid water: 0 g/m\(^3\), 0.05 g/m\(^3\) (e.g. fog), and 1 g/m\(^3\) (e.g. inside a cloud). The impact of liquid water on the refractivity is typically neglected since it is relatively small, although in order to achieve highest accuracy in the presence of dense clouds the effect should be considered. The difference between the case with 1 g/m\(^3\) liquid water and the case with no liquid water is about 1.44 mm/km for the frequencies below 10 GHz, and then it decreases slightly with frequency to about 1.35 mm/km at 100 GHz. This agrees with the published values for \(k_4\) in the microwave range, which is generally about 1.45 (Liebe et al. 1993; Solheim et al. 1999).
By ignoring the liquid water term and assuming that the refractivity is frequency independent, Eq. (11) becomes
Several different laboratory measurements have been performed in order to determine the constants \(k_1\), \(k_2\), and \(k_3\) (e.g. Boudouris 1963, Bevis et al. 1994). Thayer (1974) estimated these constants by extrapolating measurements made at optical frequencies to the microwave region. The claimed accuracy was better than what is obtained from other investigations due to more accurate refractivity measurements being available for optical frequencies. However, as pointed out by e.g. Hill et al. (1982) extrapolation of optical measurements is problematic due to several resonance frequencies in the infrared region, thus the values published by Thayer (1974) should not be used. Rüeger (2002a, b) summarized and discussed many of these investigations, and calculated the “best average” values for the constants. These values are given in Table 1. The accuracy of \(k_1\) is 0.015 % and the accuracy of the water vapor part of the refractivity is 0.15 %.
Strictly speaking the constant \(k_1\) is dependent on the relative concentrations of the different dry atmospheric gases, thus if these change \(k_1\) will also change. Most dry atmospheric gases have stable concentrations. Of the major gases only the concentration of carbon dioxide show a significant variation (it is increasing with 1.5–2 ppm/year). Rüeger (2002a) gives a formula for calculating \(k_1\) for different carbon dioxide concentrations. In total the carbon dioxide makes \(k_1\) about 0.03 % larger compared to a carbon dioxide free atmosphere. The \(k_1\) value given by Rüeger (2002a, b) assumed a carbon dioxide concentration of 375 ppm (2004 level). Table 1 also shows \(k_1\) for a carbon dioxide concentration of 392 ppm (2012 level). The concentration of carbon dioxide also shows an annual variation of about 5 ppm, meaning that \(k_1\) will have an annual variation of about \(2.8\cdot 10^{-4}\) K/hPa. This variation is negligible for all practical purposes.
Examples of vertical profiles of the hydrostatic and wet refractivity. The profiles are calulated using radiosonde data from Vienna, Austria
Using Eq. (12) it is possible to rewrite Eq. (15) as
where \(k^{\prime }_2=k_2-k_1\frac{M_w}{M_d}\) and:
\(N_h\) is called the hydrostatic refractivity and \(N_w\) the wet (or non-hydrostatic) refractivity. The hydrostatic refractivity depends only on the total density of air, while the wet part depends only on the partial pressure of water vapor and the temperature. Figure 2 shows examples of vertical profiles of \(N_h\) and \(N_w\). While the hydrostatic part is larger than the wet part, the wet refractivity is much more variable and difficult to model. We will see in Sect. 3.1 that the effect of the hydrostatic refractivity on the propagation of microwaves can be accurately estimated from just the surface pressure, while the modeling of the wet part is more complicated.
It should be noted that in the literature sometimes a division of the refractivity into a dry and a wet part is used (e.g. Perler et al. 2011). The dry refractivity will be the part caused only by the first term of the righthand side of Eq. (15), while the other two terms are designated as the wet part. It is important to remember that the wet refractivity obtained in this case is not the same as the wet (i.e. non-hydrostatic) refractivity obtained when dividing the refractivity into a hydrostatic and wet part (Eq. 18). The division into dry and wet parts makes sense in that it clearly separates the contributions from the dry gases and water vapor (part of the hydrostatic refractivity is caused by water vapor). However, there are practical advantages of using the division into hydrostatic and wet parts, making it more commonly used. As shown in Sect. 3.1 the propagation delay caused by the hydrostatic refractivity can easily be inferred from surface pressure measurements.
2.2 Optical Refractivity of Moist Air
For optical frequencies, the coefficient \(k_3\) in Eq. (11) is very small and can be ignored. However, the frequency dependence of the \(k_1\) and \(k_2\) coefficients needs to be considered. Normally the refractivity is expressed as a function of the density of dry air and water vapor (see Born and Wolf 1999, pp. 95–103)
\(\tilde{k}_d(\nu )\) and \(\tilde{k}_w(\nu )\) are the dispersions of dry air and water vapor components, respectively. \(\rho _d\) and \(\rho _w\) are the density of dry air and water vapor, respectively.
Similarly for microwaves, \(N \) can also be divided into a hydrostatic and a non-hydrostatic (wet) part
where
In the literature, the dispersion formulae for \(\tilde{k}_d(\nu )\) and \(\tilde{k}_w(\nu )\) proposed by various investigators such as Edlén (1966), Barrell and Sears (Jeske 1988, p. 217), Owens (1967), Ciddor (1996), and Ciddor and Hill (1999) can be used. For the accurate calculation of the optical refractivity, IAG has recommended (Rüeger 2002b, Chap. 3) the formulae proposed by Ciddor (1996) and Ciddor and Hill (1999) that are expressed as
where \(\rho _{ds}\) is the density of dry air at 15 \(^{\circ }\)C, 101325 Pa, \(x_w=0\) (where \(x_w=p_w/p\) is the molar fraction of water vapor in moist air). \(\rho _{ws}\) is the density of pure water vapor at 20 \(^{\circ }\)C, 1333 Pa, \(x_w=1\). \(\varUpsilon \) is the wave number (reciprocal of the wavelength, \(\frac{1}{\lambda }=\frac{\nu }{c_0}\)) and \(x_c\) is the fractional carbon dioxide content. A plot of the optical refractivity as function of frequency is shown in Fig. 3.
The total optical refractivity as function of frequency. The total pressure is 1013 hPa, the Temperature 300 K, and the relative humidity is 100 %
3 Definition of Path Delay in the Neutral Atmosphere
In space geodesy normally the travel time (or difference in travel time) between a source in space (a satellite or a quasar) to a receiver on the surface of the Earth is measured. This travel time is then converted to a distance measurement by multiplying with the speed of light in vacuum. The atmosphere will introduce an error in this distance since it will affect the propagation path of the signal and since the propagation speed of the signal in the atmosphere is lower than the speed of light in vacuum.
Path taken by a signal through the atmosphere. The signal will take the path with the shortest propagation time (\(S\)). Since the signal propagates slower in the atmosphere than in vacuum, the geometrical length of \(S\) will be larger than the straight path \(G\)
If the variations in the refractivity over the distance of one wavelength is negligible we can use the geometric optics approximation. This means that the propagation of an electromagnetic wave can be described as a ray. When calculating the propagation time of the electromagnetic wave we thus only have to consider the refractivity along the ray path. For the propagation of the signals used in space geodesy the wavelengths are a few decimeters at most, thus in the Earth’s atmosphere this approximation will normally be valid. The electric path length \(L\) (propagation time divided by the speed of light in vacuum) of a ray propagating along the path \(S\) through the atmosphere will be
The electric path will be longer than the geometric length \(G\) of a straight line between the endpoints of the path for two reasons (see Fig. 4). Firstly, the propagation velocity is lower in the atmosphere than in vacuum. Secondly, the path \(S\) taken by the ray is, according to Fermat’s principle, the path which minimizes \(L\). The atmospheric delay, \(\varDelta L\), is defined as the excess electric path length caused by the atmosphere
where \(S\) is the geometric length of the actual propagation path of the ray. By dividing the refractivity into hydrostatic and wet parts using Eq. (16) we get
where \(\varDelta L_h\) and \(\varDelta L_w\) are called the hydrostatic and wet delay, respectively. Commonly, the effect of bending, \(S-G\), is by convention considered to be part of the hydrostatic delay, i.e. the hydrostatic mapping function (see Sect. 4.2) includes the bending effect.
In space geodesy it is common to refer the slant delays to the delays in the zenith direction (using mapping functions, see Sect. 4.2). The zenith hydrostatic delay \(\varDelta L_h^z\) and the zenith wet delay \(\varDelta L_w^z\) are given by
where \(h_0\) is the altitude of the site.
3.1 Hydrostatic Delay
From Eqs. (17) and (29) we see that the hydrostatic delay only depends on the total density and not on the mixing ratio of wet and dry parts. Following Davis et al. (1985), the hydrostatic delay can be determined by using the hydrostatic equation
where \(g(z)\) is the gravity along the vertical coordinate \(z\), and integration of Eq. (31) yields the pressure \(p_0\) at the height \(h_0\)
Instead of the height-dependent gravity \(g(z)\), we introduce the mean effective gravity \(g_{eff}\)
and the inversion yields the height \(h_{eff}\) which is the height of the center of mass of the atmosphere above the site and can be determined with
Saastamoinen (1972b) used the approximation for the effective height
which holds for all latitudes and all seasons.
3.1.1 Microwaves
With the pressure \(p_0\) at the site, it is now possible to determine the zenith hydrostatic delay
We follow Saastamoinen (1972b) and Davis et al. (1985) to find the appropriate coefficients in Eq. (36). At first, the gravity \(g_{eff}\) at the effective height \(h_{eff}\) is determined with
which combined with Eq. (35) can be written as
with \(g_m=9.7840\) and
where \(\theta \) and \(h_0\) are latitude and orthometric (or ellipsoidal) height of the station. Thus, the zenith hydrostatic delay is
and after substitution of all values we get for the zenith hydrostatic delay in meters
where \(p_0\) is in hPa. The molar masses \(M_d\) and \(M_w\) stay constant up to heights of about 100 km (Davis 1986), which is essential for all troposphere delay models. The errors in the zenith hydrostatic delays are mainly caused by errors in \(k_1\) and in the surface pressure measurements. At typical meteorological conditions the zenith hydrostatic delays are about 2.3 m at sea level. An error in the surface pressure of 1 hPa causes an error of about 2.3 mm. In order to reach an accuracy of 0.1 mm, the pressure has to be measured with an accuracy of 0.05 hPa. The error due to the assumption of hydrostatic equilibrium depends on the wind and is about 0.01 % (0.2 mm path delay). Under severe weather conditions vertical accelerations can reach 1% of the gravity acceleration, which can cause errors in the zenith hydrostatic delays of about 20 mm (Davis et al. 1985).
3.1.2 Optical Zenith Hydrostatic Delays
The optical zenith hydrostatic delay can, just as for microwaves, be expressed as a function of the surface pressure. The derivation is in principle the same and will not be repeated here. The final expression is
The above equation is slightly different from the one developed by Mendes and Pavlis (2004), which is expressed as
Mendes and Pavlis (2004) derived their own dispersion factor \(f_h(\nu )\) based on the modified dispersion formula in Eq. (24) for the wavelength of 0.532 \(\upmu \)m. Equations (42) and (43) produce similar accuracy results if they are applied to real SLR observations.
3.1.3 Pressure Values
For the analysis of space geodetic techniques, there are three possibilities to obtain pressure values at the stations: local pressure recordings at the sites, pressure values from numerical weather models, or empirical models for the pressure (Böhm et al. 2009a). For instance, simple empirical models are the ones by Berg (1948)
with the pressure \(p\) in hPa and orthometric station height \(h\) in m, or the model by Hopfield (1969)
with the atmospheric temperature at sea level \(T_k\) = 293.16 K, the normal lapse rate of temperature with elevation \(\alpha \) = 4.5 K/km, gravity \(g\) at the surface of the Earth (9.7867 m/s\(^2\)) and the gas constant \(R_d\) = 0.287 kJ/K/kg for dry air. More sophisticated empirical models are UNB3m (Leandro et al. 2006) or Global Pressure and Temperature (GPT; Böhm et al. (2007)). UNB3m is based on meteorological parameters (pressure, temperature, humidity, temperature lapse rate, and water vapor pressure height factor) at five latitude bands which are symmetric w.r.t. the equator (similar to the Niell Mapping Functions (Niell 1996)). Input parameters for GPT are station latitude, longitude, height and the day of the year, which is similar to the Global Mapping Functions (GMF; Böhm et al. 2006a) as both, GPT and GMF, are based on spherical harmonics up to degree and order 9.
Table 2 summarizes some properties of the pressure values (or zenith hydrostatic delays) from different sources. Unfortunately, local pressure measurements are usually not available, in particular at GNSS stations. Thus, to get consistent values of a priori zenith hydrostatic delays for global GNSS networks it is preferable to take these values from numerical weather models. For example, the Department of Geodesy and Geoinformation (GEO) at the Vienna University of Technology provides zenith hydrostatic delays calculated from ECMWF data. These are provided on global grids (\(2.5^{\circ }\) times \(2.0^{\circ }\)) and with a temporal resolution of 6 h starting in 1994 (Böhm et al. 2009a). For scientific purposes also forecast values are made available so that they can be used for real-time applications without significant loss of accuracy (Böhm et al. 2009b). Empirical models like GPT are always available for all time epochs, but the spatial resolution is limited as it is represented by spherical harmonics up to degree and order 9 (\(\approx \)20\(^{\circ }\) in latitude/longitude), respectively. The model only includes an annual variation with the zero phase set to 28 January, so it cannot capture short-term and sub-annual weather phenomena. As an example, Fig. 5 shows pressure values at station O’Higgins in Antarctica. It is evident that empirical models like the model by Berg (1948) or GPT cannot describe the short term pressure variations and that the model by Berg (1948) is offset by about 20 hPa.
Pressure values for station O’Higgins in Antarctica from the ECMWF (grey line), local pressure recordings at the radio telescope (red squares), GPT (blue line), and pressure determined with the model by Berg (1948) (black bold line)
We have also compared and validated the empirical models Berg and GPT with pressure values from the ECMWF on global grids (\(10^{\circ }\) in latitude times \(12.5^{\circ }\) in longitude) (Böhm et al. 2009a). The comparison was performed for the year 2005 and the temporal resolution was 10 days (i.e. one global grid was taken every 10 days and consequently 36 grids were used for the statistics). An error in the pressure of 1 hPa corresponds (at sea level) to approximately a 2.3 mm error in the \(L_h^z\). This error will result in an error in the position—especially the vertical component – estimated with a space geodetic technique. In Sect. 4.2 a rule of thumb relating the error in the delay to the error in the vertical coordinate is presented, from this we find that 3 hPa (7 mm zenith delay error) correspond to 1 mm station height difference. It was found that the Berg model has large deficiencies especially around Antarctica, resulting in station height errors of more than 10 mm. The errors for GPT for this region are smaller, and these might completely disappear with an increased degree and order of the spherical harmonic expansion. There are almost no biases for the rest of the Earth (see Böhm et al. (2009b)).
Simulated station height standard deviations in mm for GPT with respect to the pressure values provided by the ECMWF (based on 36 epochs in 2005 for a 2\(^{\circ }\) latitude times 2.5\(^{\circ }\) longitude grid) (modified from Böhm et al. (2009b))
GPT only accounts for an annual variation of the pressure with rather small amplitudes compared to the other (e.g. weekly) variations of the pressure. Thus, the standard deviations of the differences to the grid values from the ECMWF are almost the same for the Berg model (constant pressure per site) and GPT. In Fig. 6 the simulated station height standard deviations are plotted for the case of using GPT compared pressure values from the ECMWF. There is an increase of the standard deviations towards higher latitudes (with maximum values at around \(\pm 60^{\circ }\) latitude), which is due to the larger pressure variations in these regions compared to equatorial regions (Böhm et al. 2009a).
3.2 Wet Delay
3.2.1 Microwave Zenith Wet Delays
From Eq. (30) the zenith wet delay is
The first term in Eq. (46) is about 1.6 % of the second term.
The derivation of a model to account for the zenith wet delay (\(\varDelta L_w^z\)) is by far more challenging than the one for the hydrostatic delay. This is due to high spatial and temporal variability and unpredictability of the amount of water vapor. Thus, the temperature and the water vapor content at the Earth surface are not representative for the air masses above. This is the reason why numerous models have been developed over the past few decades for the wet delay, while preserving Saastamoinen’s model (with slight modifications) for determining the hydrostatic delay. The zenith wet delay varies between a few mm at the poles and about 40 cm above the equatorial regions. In order to keep millimeter accuracy in space geodetic techniques, the \(\varDelta L_w^z\) is nowadays estimated as an additional parameter within the data analysis. Nevertheless, some models are listed below and can be used as an initial value in the data analysis or for applications not requiring high accuracy.
Saastamoinen (1972b) proposes the calculation of the zenith wet delay \(\varDelta L^z_w\) based on ideal gas laws using a simple relation
where \(p_{w0}\) is the water vapor pressure and \(T_0\) is the temperature at the surface. Similar to the hydrostatic delay, Hopfield (1969) proposes an expression for \(\varDelta L_w^z\) as follows
with \(N_{w}(h_0)\) the refractivity of wet air at the surface (located at height \(h_0\)) and a mean value \(h_w=11000\) m for the height of the troposphere up to which water vapor exists. Ifadis (1986) proposes to model the zenith wet delay as a function of surface pressure, partial water vapor pressure and temperature. Mendes and Langley (1998) derived a linear relation between \(\varDelta L_w^z\) and partial water vapor pressure. Some other models are being described by Mendes (1999). An approximate relation between water vapor pressure and delay reads
Assuming an isothermal atmosphere with exponential decrease of water vapor pressure \(p_w\), and assuming that water vapor exists until a height of 2 km, we get an approximation for the wet delay as a function of water vapor pressure at the Earth’s surface \(p_{w0}\)
An even simpler way is a rule of thumb that suggests that the wet zenith delay in cm equals the water vapor pressure in hPa at the Earth’s surface
In any case, information of water vapor pressure and/or temperature at the surface has to be known. If no surface meteorological observation is available, we can use the simple model of the standard atmosphere where \(p_w\) can be calculated as a function of the relative humidity \(f\), i.e.
3.2.2 Conversion of Zenith Wet Delays to Precipitable Water
The zenith wet delay can be related to the amount of integrated water vapor above the station. Following Eq. (46) and using the expression for integrated mean temperature \(T_m\) (Bevis et al. 1992)
we can write
Applying the ideal gas laws, Eq. (54) can be reformulated as
To be able to calculate the mean temperature \(T_m\) the vertical profiles of the water vapor and temperature have to be known. Such data can be obtained from radiosonde measurements or calculated (and predicted) from operational meteorological models (Wang et al. 2005). In absence of this data, the empirically derived model by e.g. Bevis et al. (1992) and Emardson and Derks (2000) can be used. The determination of the mean temperature \(T_m\) from Eq. (53) is based on the weighting with water vapor pressure in the atmosphere. Since water vapor is mainly located near the Earth surface the mean temperature \(T_m\) will be highly correlated with the temperature at the Earth surface \(T_0\). Using 8718 profiles of radiosonde launches at 13 stations in the United States between 27 and 65\(^{\circ }\) northern latitude, between 0 and 1600 m height, and over a time span of 2 years Bevis et al. (1992) found
with a standard deviation of \(\pm \) 4.74 K. If \(\varDelta L_w^z\) and surface temperature are known without error, the integrated water vapor can be computed with an average error of less than 4 %.
It is clear from Eq. (55) that the wet delay is proportional to the integrated water vapor content IWV (\(IWV=\int _0^{\infty }\rho _w\,dz\)). Since IWV is a variable that can be easily obtained from numerical weather prediction models or measured by other techniques, it is of great interest to have a simple expression for calculating the wet delay from IWV, and vice versa. Thus we define the proportionality constant \(\varPi \) such that
where \(\varDelta L^{z}_{w}\) is the wet tropospheric delay in the zenith direction. By comparing Eqs. (55) and (57), we find that \(\varPi \) can be related to \(T_m\) by
The integrated water vapor in zenith direction can also be provided as precipitable water (PW) which corresponds to the height of the equivalent water column above the station
where \(\rho _{w,fl}\) is the density of liquid water in \(\mathrm{kg/m}^{3}\). With a dimensionless quantity \(\kappa \) we can relate the \(\varDelta L_w^z\) and PW
with \(\kappa \) defined as
The precipitable water is roughly 0.16 of \(\varDelta L_w^z\). This value can vary by more than 15 %, mainly as a function of latitude and season (Fölsche 1999).
Errors in the factor \(\varPi \) are mainly caused by errors in \(T_m\) and the constants in Eq. (58). According to Fölsche (1999) the influence of errors in the mean temperature is at least one order of magnitude larger than errors in the constants. It is shown in the following how accurate \(T_m\) has to be determined to get the amount of water vapor with a certain accuracy.
The partial derivative of \(\varPi \) with respect to the mean temperature yields
This means that \(\varPi \) is changed by about 20 \(\mathrm{kg/m}^{3}\) if \(T_m (=270~\mathrm{K})\) is changed by 4 K. Assuming a zenith wet delay of 200 mm this corresponds to an error in the precipitable water of about 4 mm. Requirements for better accuracies of PW (1 mm or better) indicate that the real weather data should be used to derive the mean temperature instead of approximations such as provided in Eq. (56).
3.2.3 Optical Zenith Wet Delays
Substituting Eq. (22) into Eq. (30) yields
Following Saastamoinen (1972a), the integral \(\int \limits _{h_0}^{\infty }\rho _w(h)\,\mathrm{d }h\) can be approximated by
where \(p_{w0}\) is the surface pressure of water vapor. Substituting Eq. (64) into Eq. (30), the zenith wet delay of optical measurements can be modeled as
Equation (65) is slightly different from the one developed by Mendes and Pavlis (2004), which is expressed as
Mendes and Pavlis (2004) derived their own dispersion factors \(f_h(\lambda )\) and \(f_{nh}(\lambda )\) based on the modified dispersion formula in Eq. (24) for the 0.532 \(\upmu \)m wavelength. Equations (65) and (66) produce similar accuracy results if they are applied to real SLR observations.
4 Modeling Delays in the Neutral Atmosphere
There are basically two ways to handle the atmospheric delays when analyzing space geodetic data; either external measurements of the atmospheric delays are used to correct the measurements, or the atmospheric delays are parameterized and estimated in the data analysis. As seen in Sect. 3.1 the hydrostatic delay can be accurately determined from surface pressure measurements. However, the wet delay cannot be estimated that accurately from meteorological measurements at the surface. Thus it is common when analyzing space geodetic data to use surface pressure measurements to model the hydrostatic delay, while the wet delay is estimated in the data analysis. In the data analysis the tropospheric delays are modeled using mapping functions and gradients (see Sect. 4.2). An alternative strategy is to also use external estimates of the hydrostatic and wet delays. Such estimates could for example be obtained from ray-tracing though numerical weather models (Sect. 4.1), a technique also commonly applied for deriving expression for the tropospheric mapping functions. Another possibility is to infer the tropospheric delay from measurements by external instruments such as Water Vapor Radiometers (WVR) (Sect. 4.4). The numerical values etc. given in these sections are for microwaves, although the general principles could of course also be applied in the case of optical techniques. Tropospheric modeling for optical frequencies (e.g. SLR) is discussed in Sect. 4.3.
4.1 Ray-Tracing
From e.g. radiosonde data or numerical weather prediction models we can calculate the refractivity field of the atmosphere. This could be used to estimate the atmospheric delay simply by integrating the refractivity along the propagation path of the signal. However, the problem is that we normally do not exactly know the propagation path. To discover it we can apply the so-called ray-tracing technique. The ray-tracing technique has been used in many fields of science where the propagation of an electro-magnetic wave through a stratified medium has to be quantified. The ray-tracing is based on the so-called Eikonal equation, which represents the solution of the Helmholtz equation for an electro-magnetic wave propagating through a slowly varying medium (Iizuka 2008; Wheelon 2001). Using the Eikonal equation, we can determine the ray path and the optical path length. In the following sub-sections, we present a ray-tracing system that can be applied for tropospheric modeling and discuss some basic elements in tropospheric ray-tracing calculations through Numerical Weather Prediction Models.
4.1.1 Eikonal Equation and Ray Path
To derive the Eikonal equation we start from Maxwell’s equations (Eqs. 1–4). It is convenient to use the H-field \({\mathbf{{H} }}\) instead of the magnetic field \({\mathbf{{B} }}\), where \({\mathbf{{H} }}\) is defined by
We can consider a general time-harmonic field
where \(L({\mathbf{{r} }})\) is the optical path, and \({\mathbf{{e} }}({\mathbf{{r} }})\) and \({\mathbf{{h} }}({\mathbf{{r} }})\) are (complex) vector functions. The wave number for vacuum (\(k_0\)) is defined as
where \(\lambda _0\) is the vacuum wavelength. By inserting these equations into Maxwell’s equations (Eqs. 1–4) and assume that there are no free charges and zero conductivity, we get after some calculations (Born and Wolf 1999)
For small vacuum wavelength, and therefore large wave number for vacuum (\(k_0\))
By eliminating \({\mathbf{{h} }}\) between Eqs. (75) and (76), and considering Eq. (77), the following differential equation is obtained which is independent of the amplitude vectors \({\mathbf{{e} }}\) and \({\mathbf{{h} }}\)
where \(\varvec{\nabla } L\) comprises the components of the ray directions, \(L\) is the optical path length, \(n=c\sqrt{\varepsilon \mu }\) is the refractivity index of a medium, and \({\mathbf{{r} }}\) is the position vector. This equation is the well-known Eikonal equation. The surfaces \(L({\mathbf{{r} }}) = constant\) are called geometrical wave surfaces or the geometrical wave-fronts.
4.1.2 Hamiltonian Formalism of Eikonal Equation
The above mentioned Eikonal equation is a partial differential equation of the first order for \(n \left( {\mathbf{{r} }}\right) \) and it is possible to express it in many alternative forms. In general, the Eikonal equation can be written in the Hamiltonian canonical formalism as follows (Born and Wolf 1999; Cerveny 2005; Nafisi et al. 2012a)
Here \(\alpha \) is a scalar value related to the parameter of interest \(u\) (see Table 3). In general it is a real number but in our applications we can consider it to be an integer. \({\fancyscript{H}} \left( {\mathbf{{r} }}, \varvec{\nabla } L \right) \) is called Hamiltonian function or just the Hamiltonian. In a 3D space this system consists of seven equations. Six equations are obtained from Eqs. (81) and (82) must be solved together. The result of these six equations is \({\mathbf{{r} }}={\mathbf{{r} }}(u)\) which is the trajectory of the signal in space. The seventh equation, i.e. Eq. (83), can be solved independently and yields the optical path length.
4.1.3 Ray-Tracing System for Tropospheric Modeling
Equations (80)–(82) can be used for constructing a ray-tracing system for any specific application by simply selecting a correct value for the scalar \(\alpha \) (Nafisi et al. 2012a). For \(\alpha =0\), the parameter \(u\) represents travel time \(t\) along the ray. If \(\alpha =1\), the parameter \(u\) represents the arc-length \(s\) along the ray. In case of \(\alpha =2\), the parameter \(u\) is equal to \(dt/dn\), which represents the natural variables along the ray (Cerveny 2005).
When applying ray-tracing for the determination of total delays along the ray path the natural choice is \(\alpha =1\). However, it is also possible to use \(\alpha =2\) to construct a tropospheric ray-tracing system (see Gegout et al. (2011)). Ray-tracing systems can be expressed and solved in any curvilinear coordinate system, including spherical coordinates. Selecting \(\alpha =1\) and representing the function \({\fancyscript{H}}\) in a spherical polar coordinate system \((r,\theta ,\lambda )\), Eq. (80) can be rewritten as
where \(r\) is the radial distance, \(\theta \) is the co-latitude, and \(\lambda \) is the longitude (\(0 \le \theta \le \pi , 0 \le \lambda \le 2 \pi \)). \(L_r= \frac{\partial L}{\partial r}\), \(L_\theta =\frac{\partial L}{\partial \theta }\) and \(L_\lambda =\frac{\partial L}{\partial \lambda }\) are the elements of ray directions. Now, by substituting Eq. (84) into Eqs. (81) and (82), we obtain
where
This system of equations is a direct result of Eqs. (81) and (82) in a 3D medium, and we need to solve all six differential equations simultaneously. The final output is the positions of the points along the trajectory of the ray. For solving the above ray-tracing system, the following initial conditions at the starting point (station) can be used (Nafisi et al. 2012a)
where \(a_0\) and \(z_0\) are the initial geodetic azimuth and zenith angle, respectively. In the 3D case, gradients are important factors which can affect the bending of the ray path and therefore of the total ray-traced delay. The gradient can be rewritten as
From a practical point of view, we must find a sophisticated technique for computing the gradients of the refractive index in Eqs. (88), (89) and (90), which are
Taking the effects of the gradients on the ray-traced delay into account is important for a ray-tracing algorithm. In particular in the case of symmetries, ray-tracing in curvilinear coordinates system would be easier. However, in general the ray-tracing systems in a curvilinear coordinates are more complex and sometimes fail. A typical example in atmospheric ray-tracing is the solution for ray-tracing in spherical polar coordinates when \( \theta \approx 0^{\circ }\) or \( \theta \approx 180^{\circ }\) . According to Cerveny (2005) a general solution for removing such singularities is the use of transformations between ray-tracing systems in various forms. Using standard transformation relations, a ray-tracing system represented in curvilinear coordinates can be transformed into the universal Cartesian coordinates, and after ray-tracing computations in this coordinate system the results can be again transformed back to the curvilinear coordinates. In another method suggested by Alkhalifah and Fomel (2001) a small constant parameter (\(\delta \)) is added in the denominator of fractions in the ray-tracing equations to provide numerical stability. The value of this constant must be chosen carefully; otherwise we may get unreasonable results. Other stability solutions have been suggested by Schneider (1993) and Fowler (1994). In addition, the variation of the size of the grid cells in polar coordinates can causes a stability problems, especially in presence of small-scale inhomogeneities (Fowler 1994).
By substituting Eq. (84) into Eq. (83) and solving the equation, we can obtain this well-known equation to calculate the optical path length \(L\)
Once the position of a point along the ray path has been determined by Eqs. (85)–(90), the refractivity index \(n\) and the optical path length \(L\) can be calculated using Eq. (102). As mentioned before, the total delay is defined as the difference between the optical path length \(L\) and the straight line distance \(G\)
3D ray-tracing can be easily reduced to 2D ray-tracing by substituting \(\frac{\partial n}{\partial \theta }=0\) and \(\frac{\partial n}{\partial \lambda }=0\) into Eqs. (85)–(90). In this case, we assume that the ray does not leave a plane of constant azimuth angle. For the 2D system, the coupled partial derivatives in Eqs. (85)–(90) reduce to four equations
Equation (91) remains valid also for 2D ray-tracing systems. For a horizontally stratified atmosphere, further simplifications can be applied to improve the calculation speed (Thayer 1967). According to Böhm (2004), we can develop a 1D ray-tracing. Figure 7 shows the geometry of this method for a troposphere with \(k\) different refractivity layers. The geocentric distances can be estimated by adding the radius of the Earth (\(R_e\)) to the heights of each layer
In this method, the elevation angles (\(e_i\)) are with respect to a horizontal plane of the station, whereas \(\theta _i\) show the angles between the ray path and the tangents to the layers in each intersection point of the trajectory. At the first point (receiver) these two are same, i.e. \(e_1=\theta _1\) .
Geometry of a 1D ray-tracing method, for a receiver located at \(P_1\) and the upper limit of the troposphere at \(P_k\). Points \(P_2\) and \(P_3\) show two sample points of the ray path. The y- and z-axis of the Cartesian coordinate system are parallel to horizon and zenith direction at the site, respectively. \(S_2=\Vert {\mathbf{{P} }}_3-{\mathbf{{P} }}_2\Vert \) is the distance between two successive points along the path
In this coordinate system, the \(z\) and \(y\) components are equal to \(r_1\) and zero, respectively. Following the geometrical relation in Fig. 7 we find
These parameters must be calculated for all layers, and then it is possible to find the total slant delay (\(\varDelta L_d\)) as follow
This equation can be divided into two terms for finding hydrostatic and non-hydrostatic components of the delay separately. By inserting \(h_i\) (along the zenith direction) instead of \(S_i\) (along the slant path) an equation for total zenith delay will be derived. As mentioned before, the bending effect is a part of the total delay and must be estimated separately. This parameter is zero for the zenith direction and
for a slant direction. Implementation of the above ray-tracing system can vary substantially, with different degrees of complexity and accuracy. Examples of ray-tracing algorithms in atmospheric studies are given by Bean and Thayer (1959), Thayer (1967), Budden (1985), Davis (1986), Mendes (1999), Pany (2002), Böhm and Schuh (2003), Thessin (2005), Hulley (2007), Hobiger et al. (2008), Nievinski (2009), Wijaya (2010), Gegout et al. (2011) and Nafisi et al. (2012a). Several ray-tracing algorithms were compared by Nafisi et al. (2012b). Also it is possible to express the Eikonal equation and the ray-tracing system in curvilinear non-orthogonal coordinates systems. For details see Cerveny et al. (1988).
4.2 Mapping Functions and Gradients
In the analysis of space geodetic data the troposphere path delay \(\varDelta L(e)\) at the elevation angle \(e\) is usually represented as the product of the zenith delay \(\varDelta L^z\) and an elevation-dependent mapping function \(mf(e)\) with
This concept is not only used to determine a priori slant delays for the observations, but the mapping function is also the partial derivative to estimate residual zenith delays. Typically, the zenith delay is estimated with a temporal resolution of 20–60 min in VLBI and GPS analysis. In VLBI analysis—when there is only one observation at a time at a station—this allows the zenith delays to be estimated in a least-squares adjustment. In the analysis of space geodetic observations not only the zenith delays are estimated, but also other parameters like the station clocks and the stations heights (Fig. 8). The partial derivatives of the observed delays w.r.t. the station heights are dependent on the elevation angle only, but whereas the partial derivatives w.r.t. the clocks (= 1) and the station heights (=\(\sin (e)\)) are exactly known, the partial derivatives for the zenith delays (i.e. the mapping functions) are only known with a limited accuracy. Consequently—via the correlations between zenith delays, station heights, and clocks—any imperfection of the mapping function will also result in errors in the station height estimates (and clock errors).
Different elevation-dependence of the tropospheric delays (red), clock values (green), and height components (blue). Whereas the partial derivatives for the estimation of clocks and height components are strictly 1 and \(\sin (e)\), respectively, the partial derivative for the estimation of the zenith delay is the mapping function which is approximately \(1/\sin (e)\), but is not perfectly known (modified from Rothacher et al. (1998))
Considering Eq. (118) we find the following relationship: If the mapping function is in error (too large), then the estimated zenith delay \(\varDelta L^z\) is too small, because the observed tropospheric delay \(\varDelta L(e)\) does not change. Consequently, the estimated station height goes up. Niell et al. (2001) set up a rule of thumb specifying that the error in the station height is approximately one third of the delay error at a cutoff elevation angle of 7\(^\circ \). Böhm (2004) revisited this rule of thumb for VLBI analysis (and a cutoff elevation angle of \(5^\circ \)) specifying that the station height error is about one fifth of the delay error at the lowest elevation angle. This is close to the value 0.22 found by MacMillan and Ma (1994). The corresponding decrease of the zenith delay is about one half of the station height increase.
4.2.1 Azimuthal Symmetry: Mapping Functions
Assuming azimuthal symmetry of the neutral atmosphere around the station (i.e. at a constant elevation angle the delay is not dependent on the azimuth angle of the observation), the approach as described in Eq. (119) is used (e.g. Davis et al. 1985)
\(\varDelta L(e)\) is the total path delay of the microwaves in the neutral atmosphere and \(e\) is the elevation angle of the observation to the satellite or the quasar (vacuum or geometric elevation angle). \(\varDelta L_h^z\) and \(\varDelta L_w^z\) are the a priori zenith hydrostatic and the estimated wet delays, and \(mf_h(e)\) and \(mf_w(e)\) are the so-called mapping functions which provide the ratio of the delay to the delay in zenith direction. The input to both mapping functions is the vacuum elevation angle \(e\), because the bending effect is accounted for by the hydrostatic mapping function.
Errors in the zenith hydrostatic delays or the mapping functions have an influence on station height errors as described with the rules of thumb by Niell et al. (2001) or Böhm (2004) mentioned above. The following two examples illustrate this rule of thumb, which—holding for both GPS and VLBI—depends on the actual distribution of elevation angels and on whether elevation angle-dependent weighting is used: The zenith hydrostatic and wet delays shall be 2000 and 200 mm, respectively, the minimum elevation angle is \(5^{\circ }\), and the corresponding values for the true hydrostatic and wet mapping functions are 10.15 (\(mf_h(5^{\circ })\)) and 10.75 (\(mf_w(5^{\circ })\)) (Böhm et al. 2006b).
-
We assume an error in the total pressure at the station of +10 hPa, e.g. when using the “mean” pressure from GPT during a time of low pressure. + 10 hPa correspond to \(\approx \)+20 mm zenith hydrostatic delay (Saastamoinen 1972b), which is then mapped down to \(5^{\circ }\) elevation angle using the wrong mapping function (wet instead of hydrostatic, factor \(-\)0.6 = 10.15–10.75). At \(5^{\circ }\) elevation angle the mapping function error is \(-\)12 mm, and one fifth of it, i.e. \(-\)2.4 mm, would be the resulting station height error. This results in a kind of atmosphere loading correction (see Part 4, Wijaya et al. (2013)), because during a pressure low the station heights go up (Tregoning and Herring 2006; Steigenberger et al. 2009).
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We consider an error in the wet mapping function of 0.1 (\(mf_w(5^{\circ })\) = 10.85 instead of 10.75) or in the hydrostatic mapping function of 0.01 (\(mf_h(5^{\circ })\) = 10.16 instead of 10.15). The error at \(5^{\circ }\) elevation angle is in both cases 20 mm, i.e. the resulting error in the station height would be approximately +4 mm.
The scale height of the wet part of the troposphere is about 2 km, the scale height of the hydrostatic part is about 8 km. The mapping functions describe the ratio of the paths \((AB)/(B_0B)\) (wet) and \((AC)/(C_0C)\) (hydrostatic). The wet mapping function is larger than the hydrostatic mapping function
The scale height of the wet part in the troposphere is about 2 km, whereas the scale height of the hydrostatic part is about 8 km (cf. Fig. 9). The mapping functions describe the ratio \((AB)/(B_0B)\) (wet) and \((AC)/(C_0C)\) (hydrostatic). Due to the curvature of the Earth and the smaller scale height of the wet part, the hydrostatic mapping function is smaller than the wet mapping function. Exceptions are mapping functions for observations at very low elevation angles where the geometric bending effect, which is attributed to the hydrostatic mapping function, is increasing considerably. Thus, the mapping functions are a measure for the thickness of the atmosphere compared to the Earth radius (Niell et al. 2001). If the thickness of the atmosphere gets smaller, it appears to be flatter, and the mapping function approaches \(1/\sin (e)\). Assuming a flat and evenly stratified atmosphere the mapping function is \(1/\sin (e)\). For higher elevation angles (>\(20^{\circ }\) elevation) these mapping functions are sufficiently accurate. Marini (1972) showed that the dependence on the elevation angle of the mapping functions for any horizontally stratified atmosphere can be described with continued fractions, when \(a\), \(b\), \(c\), etc. are constants (Eq. 120). For verification Marini (1972) used standard atmosphere data but no real weather data.
This concept was first used in a model for the refraction of the hydrostatic atmosphere (Marini and Murray 1973) which has since then been applied in the analysis of geodetic and astrometric VLBI observations for a long time. The zenith delay corresponds to that by Saastamoinen (1972b), and the mapping functions are represented by a continued fraction form with two coefficients \(a\) and \(b\). The first mapping functions for space geodetic applications with different coefficients for the hydrostatic and wet parts were published by Chao (1974) who replaced the second \(\sin (e)\) by \(\tan (e)\) to get unity in zenith direction.
Davis et al. (1985) developed the mapping function CfA2.2 for the hydrostatic delays down to \(5^{\circ }\) elevation; it is based on the approach by Chao (1974) but extended by an additional constant \(c\). Based on ray-tracing through various standard atmospheres with elevation angles between 5 and \(90^{\circ }\), the coefficients \(a\), \(b\), and \(c\) were determined as functions of pressure, water vapor pressure, and temperature at the Earth surface, and from vertical temperature gradients and the height of the troposphere. Herring (1992) developed coefficients for the mapping function MTT (MIT Temperature) as functions of latitude, height, and the temperature at the site. Unlike Davis et al. (1985) he did not use standard atmospheres but radiosonde data. The MTT mapping functions are based on a slightly changed continued fraction form which is widely accepted nowadays
The factor in the denominator ensures that the mapping function is equal to one in the zenith direction. The strong dependence of the MTT mapping function (Herring 1992) on surface temperature induced Niell (1996) to develop the New Mapping Functions (NMF, now often called Niell Mapping Functions). The NMF do not use meteorological parameters at the sites, but only the day of the year (doy), station latitude, and station height as input parameters. Thus, they can be easily applied at stations without meteorological sensors, which often is the case for GNSS stations. Niell (1996) used standard atmosphere data at various latitudes to determine hydrostatic and wet mapping functions down to \(3^{\circ }\) elevation. Similar to Davis et al. (1985) and Herring (1992), he used ray-tracing methods to determine the coefficients \(a\), \(b\), and \(c\) of the continued fraction form in Eq. (121). NMF is based on sine functions to describe the temporal variation of the coefficients. The period is 365.25 days and the maximum/minimum is set to January 28 (doy 28). There is also a height correction for the hydrostatic NMF (NMFh) which describes that mapping functions increase with increasing height, i.e. the atmosphere above the site becomes flatter.
Niell (2000) was the first to determine mapping functions from numerical weather models which are often available with a time resolution of 6 h. For the Isobaric Mapping Functions (IMF), he used empirical functions for \(b\) and \(c\), and he determined the coefficients \(a\) of the continued fraction in Eq. (121) from re-analysis data of the Goddard Space Flight Center Data Assimilation Office (DAO) (Schubert et al. 1993). For the hydrostatic IMF he used the height of the 200 hPa pressure level which is readily available with most numerical weather models and which is describing the thickness of the atmosphere well. For the wet IMF Niell suggested to use a coarse ray-trace at \(3.3^{\circ }\) initial elevation angle through numerical weather models. However, some practical and conceptual limitations in the computation of the wet IMF induced Böhm and Schuh (2004) to develop the Vienna Mapping Functions (VMF).
Thus, the VMF are characterized by the removal of some weaknesses of the IMFw, e.g. the coarse vertical resolution of weather model data is improved by vertical interpolation and the bending effect is taken into account rigorously. The same approach is applied for the wet and the hydrostatic mapping function, i.e. a ray-tracing is performed at an initial elevation angle of \(3.3^{\circ }\) for the hydrostatic and wet components yielding the hydrostatic delay, the wet delay, as well as the bending effect and the outgoing (vacuum) elevation angle which is \({\approx }3^{\circ }\). (Please notice that always the refractivity profile above the site vertical is used, which makes the 1D ray-tracing simple and causes the delays to be symmetric with azimuth.) Together with the zenith hydrostatic and wet delays which are also determined by ray-tracing, the hydrostatic and wet mapping functions (Eq. (121)) at the outgoing elevation angle are calculated. The geometric bending effect is added to the hydrostatic mapping function. Similar to IMF, empirical functions are used for the \(b\) and \(c\) coefficients, which allows the determination of \(a\) in Eq. (121) by simple inversion. Since the coefficients \(a\), \(b\), and \(c\) are highly correlated, small errors in \(b\) and \(c\) can easily be compensated with the \(a\) coefficients. However, Böhm et al. (2006b) improved the \(b\) and \(c\) coefficients, and consequently the \(a\) coefficients had to be re-calculated. The coefficients of the so-called VMF1 are \(b_h\) = 0.0029, \(b_w\) = 0.00146, \(c_w\) = 0.04391, and the coefficient \(c_h\) is provided with Eq. (122) and Table 4
The VMF1 are valid (tuned) for elevation angles above \(3^{\circ }\), and the largest deviations from ray-traces at other elevations show up at about \(5^{\circ }\) elevation angle. The Vienna Mapping Function 1 is realized as discrete time series (resolution 6 h) of coefficients \(a\), either on a global grid or at certain geodetic sites (see http://ggosatm.hg.tuwien.ac.at/). Mind that with the gridded version of the VMF1, the height correction of Niell (1996) has to be applied.
Instead of separating the delays into a hydrostatic and a wet part, an alternative concept of total mapping functions has also been investigated for troposphere delay modeling (Böhm et al. 2006b), i.e. the use of a single total mapping function \(mf_t\) (Eq. 123) both for mapping down the a priori zenith total delays \(\varDelta L_{t,0}^z\) and as partial derivative for the estimation of the residual total delays \(\varDelta L_{t,res}^z\) (Eqs. 124 and 125). Table 5 summarizes the parameters for the \(c_t\) coefficient which have been determined with the same approach as the \(c_h\) coefficients of the hydrotstatic VMF1. The \(b\) coefficient of the total VMF1 is also \(b_t\) = 0.0029.
Although a priori zenith total delays are required in the data analysis, a priori zenith hydrostatic delays can also be applied because the mapping function for the a priori zenith delays is the same as for the residual zenith delays (this only holds if there are no constraints on the estimated zenith delays). With the classical separation into a hydrostatic and a wet part, errors of the a priori zenith hydrostatic delays cannot be fully compensated by the estimation of the remaining wet part since the hydrostatic and wet mapping functions are not identical, especially at low elevation angles. The advantage of the concept of total mapping functions is that the results are not affected by poor a priori zenith hydrostatic delays (Böhm et al. 2006b). On the other hand, the value of total mapping function is close to that of the hydrostatic mapping function, so it cannot account for the rapid variation of the wet zenith delays which occurs even between the 6 hourly epochs of the total VMF1. Thus, it is preferable to keep the separation into a hydrostatic and a wet mapping function, at least as long as the time resolution is not 3 h or better.
The goal of the Global Mapping Functions (GMF; Böhm et al. 2006a) is to make available mapping functions which can be used globally and implemented easily in existing geodetic data analysis softwares and which are consistent with NWM-based mapping functions, in particular with the VMF1 (Böhm et al. 2006b). Compared to the NMF (Niell 1996), the parameterization of the coefficients in the three-term continued fraction (Eq. 121) has been refined to include also a longitude dependence. Using global grids of monthly mean profiles for pressure, temperature, and humidity from the ECMWF 40 years reanalysis data (ERA40), the coefficients \(a_h\) and \(a_w\) were determined using data from the period September 1999 to August 2002 applying the same strategy and the same \(b\) and \(c\) coefficients used for VMF1. Thus, at each of the 312 grid points, 36 monthly \(a\) values were obtained for the hydrostatic and wet mapping functions, respectively. The hydrostatic coefficients were reduced to mean sea level by applying the height correction given by Niell (1996). The mean values, \(a_0\), and the annual amplitudes, \(A\), of the sinusoidal function (Eq. 126) were fitted to the \(a\) parameter time series of each grid point, with the phases referred to January 28, corresponding to the NMF. The standard deviations of the monthly values at the single grid points with respect to the values obtained from Eq. (126) increase from the equator towards larger latitudes, with a maximum value of 8 mm (expressed as equivalent station height error) in Siberia. For the wet component, the standard deviations are generally smaller, with the maximum values being about 3 mm at the equator Böhm et al. (2006a).
Then, the global grid of the mean values \(a_0\) and that of the amplitudes \(A\) for both the hydrostatic and wet coefficients of the continued fraction form were expanded into spatial spherical harmonic coefficients up to degree and order 9 (according to Eq. (127) for \(a_0\)) in a least-squares adjustment. The residuals of the global grid of \(a_0\) and \(A\) values to the spherical harmonics are in the sub-millimeter range (in terms of station height). The hydrostatic and wet coefficients \(a\) for any site coordinates and day of year can then be determined using Eq. (126).
In Fig. 10 VMF1 and GMF hydrostatic mapping functions at \(5^{\circ }\) elevation angle are plotted for Fortaleza, Brazil. The GMF reflects a seasonal variability and, in this respect, agrees well with the VMF1. However, a deficiency is evident in the empirical mapping function compared to the VMF1 because GMF does not reveal the special meteorological conditions described by the VMF1 during the El Niño event in 2009/10.
Hydrostatic mapping functions VMF1 and GMF at \(5^{\circ }\) elevation at Fortaleza, Brazil. Phenomena such as the El Niño event in 2009 cannot be captured with empirical mapping functions like GMF that contain only average seasonal terms
Niell (2006) compared the mapping functions VMF1, IMF, GMF, and NMF with mapping functions derived from ray-tracing of radiosonde data in 1992, which were assumed to be the most accurate reference possible. The standard deviation was converted to station height scatter with the rule of thumb by Niell et al. (2001) (one third of the delay at 7\(^{\circ }\) elevation angle). The best agreement was found for the VMF1, both for the hydrostatic and the wet mapping function. All hydrostatic mapping functions show the lowest scatter at the equator, because there are only small pressure and temperature variations. The scatter increases with station latitude, in particular for the empirical GMF and NMF, which of course cannot account for the variations at synoptic time scales (\(\sim \)10 days). The situation is different for the wet mapping functions where the scatter is largest at the equator. This is due to the fact that the zenith wet delays are largest over the equator (up to 40 cm). At the poles, there is hardly any humidity; thus, errors in the wet mapping functions are not critical for the estimation of station heights (and zenith wet delays). Again, the performance of the wet VMF1 is best. However, it has to be mentioned here that the radiosonde data might have been assimilated in the NWM which are used to determine the VMF1 and IMF. Consequently the station height scatter for these two mapping functions may be too optimistic.
There have been many investigations comparing the application of different mapping functions in VLBI or GPS analysis. For example, Böhm et al. (2007) and Steigenberger et al. (2009) compared various mapping functions in GPS analysis, and Tesmer et al. (2007) compared them in VLBI analysis. Moreover, Kouba (2008) compared the gridded VMF1 against the VMF1 determined for specific sites, or Böhm et al. (2009b) assessed the accuracy of forecast VMF1 for the application in real time analysis.
4.2.2 Azimuthal Asymmetry: Gradients
Mapping functions as described above allow for the modeling of path delays under the assumption of azimuthal symmetry of the neutral atmosphere around the station. Consequently, vertical refractivity profiles above the sites are sufficient to determine the path delays at arbitrary elevation angles or the mapping functions, respectively, because the refractivity is always taken from the vertical profile as it is the case for the VMF1. However, due to certain climatic and weather phenomena path delays will not be constant at varying azimuths. For example, at sites at northern latitudes the path delay towards south will be systematically larger than towards north, because the height of the troposphere above the equator is larger than above the poles.
The refractivity in the vicinity of the vertical profile above the site can be determined with the linear horizontal gradients of refractivity \(\xi \)
In the following the derivation of linear horizontal gradients is shown following Davis et al. (1993). The Taylor series up to degree one for the refractivity at a station is (Fig. 11)
\(N_0(z)\) is the refractivity above the site, \({\mathbf{{x} }}\) is the horizontal position vector (origin is placed at the site), and \(\varvec{\xi }(z)\) is the linear horizontal gradient vector of refractivity at height \(z\). The index \(i\) refers to the i-th component of \({\mathbf{{x} }}\): 1 towards north and 2 towards east. The path delay (hydrostatic or wet) at an arbitrary direction can be found by integration of Eq. (128) along the path \(s\). If expressed with elevation angle \(e\) and azimuth angle \(a\), we get
where \(\varDelta L_0\) is the delay without gradients. In Eq. (131) the difference between the paths with and without gradients has been neglected. Davis et al. (1993) state that this difference is as large as 1 mm for \(N = 300\), \(\partial N/\partial x_i = 1\) km\(^{-1}\), and \(e = 20^{\circ }\). If the concept of an azimuth-dependent mapping function is used, Eq. (131) can be written as
with
where
and \(mf_0\) and \(\varDelta L^z\) are the mapping function and the path delay in zenith direction for the symmetric case. Thus, the gradients cause a change in the mapping function which can be described by an additional term \(\delta mf\). With
and normalized gradients of refractivity
and when \(\hat{{\mathbf{{n} }}}\) and \(\hat{{\mathbf{{e} }}}\) refer to the unit vectors in north and east direction and \(e^{\prime }\) is the refracted elevation angle (which only differs from the geometric elevation angle at low elevations), we get
and
when
Equation (138) shows that the elevation-dependence of the azimuth-dependent mapping function \(\delta mf(a,e)\) consists of two parts: the dependence on \(mf_0\) and on the factor \(\cot (e^{\prime })\). As already mentioned the mapping functions are dependent on the geometric elevation angle \(e\), whereas the cotangent depends on the refracted elevation angle \(e^{\prime }\), because close to the site this angle determines the refraction. It holds that
where \(\delta e\) can be described with (Davis et al. 1993)
\(N_S\) is the refractivity at the Earth surface, and for \(N_S = 300\), \(e = 5^{\circ }\) we get \(\delta e \approx 0.2^{\circ }\). Since \(\delta e\) is small, \(\cot (e^{\prime })\) can be expanded into a series, and we get for the deviation from the symmetric mapping function
With the delay gradients (or just gradients) \({\mathbf{{G} }}\)
we get for the path delay
The equations above can be used to determine gradients by integrating over the horizontal gradients of refractivity along the site vertical (see e.g. MacMillan and Ma (1997), Böhm and Schuh (2007))
where \(a\) denotes the azimuth direction (e.g. \(e\) or \(n\)). Figure 12 shows refractivity gradients at the site vertical for the station Fortaleza.
Weighted (with height) refractivity gradients (\(dN(z)\cdot z\)) towards east at Fortaleza (Brazil) on 21 November 2011 at 0:00 UT. The black line shows the hydrostatic, the red line the wet gradients
In the following it is shown how gradients can be interpreted. Equation (128) can be written in the form
with the constant vector \(\varvec{\alpha }\). This means that the relative gradient of refractivity is constant, and we get for the gradient of the delay
Supposing that refractivity decreases exponentially with height, i.e.
and \(H\) is the scale height, we get for the integral in Eq. (149) the expression
and for the gradient of refractivity
The scale height \(H\) is the height of the neutral atmosphere (or of a part of it) if the density is constant with height and the total mass is conserved. For the gradients of refractivity at the Earth surface we get
This shows that for a given value of \({\mathbf{{G} }}\) the gradients of refractivity \(\varvec{\xi }\) are inversely proportional to the squared scale heights \(H\). A typical value for the gradients \({\mathbf{{G} }}\) is 1 mm. This corresponds to a path delay of \(\approx \)65 mm at \(7^{\circ }\) elevation. Assuming a scale height \(H\) of 1 km the gradient of refractivity \(\varvec{\xi }\) is 1 km\(^{-1}\). With a scale height of 8 km \(|\varvec{\xi }|\) = 0.015 km\(^{-1}\). Hydrostatic atmospheric gradients which are caused by pressure and temperature gradients, have a large spatial resolution of about 100 km (Gardner 1976) and a temporal resolution of hours to days. Wet gradients have a small spatial resolution (\({<}10\) km) and they can vary at hourly time scales or faster, although longer time scales are also possible (e.g. at coastal regions). Wet gradients are functions of water vapor pressure and temperature.
Presently, two models for the gradients are used in the analysis of space geodetic techniques. These are the model by MacMillan (1995) that follows Davis et al. (1993) and the model by Chen and Herring (1997). Both concepts will be described below. Furthermore, it is shown that the concept of horizontal gradients corresponds to a tilting of the mapping functions. This is used by Niell (2001) who uses the tilting of the 200 hPa pressure level to get hydrostatic gradients. MacMillan (1995) proposes to use the simple model
i.e. the difference between \(e\) and \(e^{\prime }\) is neglected. Chen and Herring (1997) use the gradient model
and
After integration they get for the coefficient \(C\)
For scale heights of 6.5 km for the hydrostatic part and 1.5 km for the wet part of the neutral atmosphere, they find the values 0.0031 and 0.0007 for the hydrostatic and wet coefficients \(C\), respectively. For the estimation of total gradients, Herring (1992) suggests using 0.0032.
Tilting of the mapping function by the angle \(\beta \) assuming a horizontally stratified atmosphere
Gradients can also be interpreted by a tilting of the mapping function (Rothacher et al. 1998) see Fig. 13. The basic relations are shown below assuming that the atmosphere is flat (mapping function \(1/\sin (e)\)) and that the path delay in zenith direction is not changed by the tilting. If the gradient \(G\) is the deflection of the path delay in zenith direction due to the tilting angle \(\beta \), it is shown that the path delay at the elevation angle \(e\) due to the gradients is \(\cot (e)mf(e)G\), which is exactly what is proposed by MacMillan (1995). Starting with
and using \(1/\sin (e)\) as mapping function and with
we get
A gradient \(G\) of 1 mm corresponds to tilting angles of 1.5’ (hydrostatic) and 17’ (wet). The effects on the zenith delays are \(2\cdot 10^{-4}\) and \(2\cdot 10^{-5}\) mm, respectively, and can be neglected.
It is important to estimate gradients in the analysis of space geodetic observations, in particular when observing at low elevation angles. On the other hand, there is no need to apply a priori gradients if no constraints are applied on the estimation of gradients.
4.3 Atmospheric Delays for SLR
4.3.1 Single-Color SLR Observations
The accuracy of the results obtained from Satellite Laser Ranging (SLR) measurements is ultimately limited by the atmospheric propagation effects (as well as by by the hardware system). In the 1970s, Marini and Murray (1973) developed a model to correct the atmospheric delays in SLR measurements and their model became a standard correction model at that time. In the early 2000s, Mendes et al. (2002) pointed out some limitation in that model, namely as regards the modeling of the elevation dependency of the zenith atmospheric delays (the mapping function component of the model). Thus, Mendes et al. (2002) developed their mapping functions (FCULa and FCULb) that represents significant improvement over the mapping functions of the Marini-Murray model. Of particular interest is the ability of the new mapping functions to be used in combination with any zenith delay model, i.e. Mendes and Pavlis (2004) model, used to predict the atmosphere delay in the zenith direction.
Similar to the microwave frequencies, the mapping functions are normally modeled using Eq. (121). New mapping functions have been developed based on ray-tracing through one full year (1999) of radiosonde data from 180 stations, globally distributed, with a variable number of balloon launches per day. Two different parameterizations were proposed by Mendes et al. (2002) with the coefficients in Eq. (121) written as functions of the selected parameters. One parameterization of the mapping function (FCULa) requires both site location and meteorological (surface temperature) data. The coefficients of the mapping function have the following mathematical formulation
where \(T_0\) is the temperature at the station in degrees centigrade, \(\theta \) is the station latitude, and \(h_0\) is the orthometric height of the station, in meters. The coefficients \(b\) and \(c\) are modeled similarly.
The second parameterization (FCULb) does not depend on any meteorological data, i.e. similar to the model developed by Niell (1996) for radio wavelengths. For this function, the coefficients have the following form
where \(\theta _d\) is the latitude of the station, in degrees, and doy is the decimal day of year (UTC day since the beginning of the year). The coefficients in Eq. (163) are different with those in Eq. (122) as the later one is derived based on microwave refractivity index, which is independent on frequency.
These mapping functions along with the zenith delay model of Mendes and Pavlis (2004) have become the standard model for correcting SLR measurements. Comparing to the previously used mapping functions, the advantages of the new mapping functions are obvious. They represent simpler expressions than those proposed by the Marini and Murray (1973) model and allow the use of better zenith delay models. The coefficients of the mapping functions are presented in Table 6.
The latest progress in atmospheric corrections for single-color SLR is provided by Hulley and Pavlis (2007) who applied a ray-tracing technique to calculate propagation effects, including the effects of horizontal refractivity gradients. The use of ray-tracing trough numerical weather models has been shown to improve the accuracy of the SLR results.
4.3.2 Two-Color SLR Observations
The alternative to modeling is the application of two-color (i.e. two-frequency) SLR measurements for the direct computation of the propagation delay by utilizing the fact that the neutral atmosphere is dispersive for optical frequencies. The dispersion causes the optical path lengths at two different frequencies to differ. This difference depends on the two frequencies and is proportional to the path integrated atmospheric density. Thus, the difference between the two optical paths can be used for calculating the propagation delays (Wijaya and Brunner 2011). This method has the potential to improve the accuracy of SLR results (Abshire and Gardner 1985).
Based on the previous works of Prilepin (1957) and Bender and Owens (1965), Abshire and Gardner (1985) developed an atmospheric correction formula for the two-color SLR measurements (the 2C-SLR formula). This formula was later studied by several investigators (Greene and Herring 1986).
From the two-color SLR measurements theoretical path lengths \({\fancyscript{R}}_1\) and \({\fancyscript{R}}_2\) are obtained (assume that the system measures simultaneously individual optical paths). With the 2C-SLR formula (Abshire and Gardner 1985), the chord (straight line) distance \(\sigma \) can be calculated by adding the term \(\mu ({\fancyscript{R}}_2 - {\fancyscript{R}}_1)\) to \({\fancyscript{R}}_1\)
where
with \(k(\nu _i)\) being the dispersion factor. This factor depends on frequency and can be calculated using (Edlén 1966; Abshire and Gardner 1985)
where \(\lambda _i\) is the \(i\):th wavelength in meters.
The 2C-SLR formula presented in Eq. (164) remove the largest part of the total propagation delay, namely that associated with the dry atmospheric density. However, the water vapor density and curvature effects still remain (Abshire and Gardner 1985; Greene and Herring 1986). At optical frequencies, water vapor contributes only about 1 % of the refractivity, however, since it is highly variable and it can introduce substantial errors. By using ray-tracing through atmospheric profiles, Wijaya and Brunner (2011) showed that the zenith wet delay can be several millimeters for SLR measurements. Furthermore, the magnitude of the bending effects could reach a few centimeters for measurements taken at an elevation angle of \(10^{\circ }\).
The precision required for the range difference \(({\fancyscript{R}}_1 - {\fancyscript{R}}_2)\) measurements is very stringent (few micrometers), which is mainly due to amplification of the measurement noise by the scaling factor \(\mu \) (Abshire and Gardner 1985; Greene and Herring 1986). This requirement cannot currently be achieved. However, if in the future the range difference measurements could be improved to reach the required precision, the two-color SLR system would be an interesting way of reduce atmospheric propagation effects.
In order to anticipate possible future development of the two-color SLR systems, Wijaya and Brunner (2011) have developed a new correction formula that can be seen as an extension of the 2C-SLR formula
The power of dispersion \(\mu \) is given by
and the water vapor factor is
where \(K=\left( \frac{\tilde{k}_w^\star (\nu _2)}{\tilde{k}_w^\star (\nu _1)} - \frac{\tilde{k}_d(\nu _2)}{\tilde{k}_d(\nu _1)}\right) \). The slant integrated water vapor (SIWV) is
The formula in Eq. (167) eliminates the total atmospheric density effect including its gradient and provides two terms to calculate the water vapor and the curvature effects. The dispersion effect (contained in the second term in Eq. (167)) can be obtained from the observed optical path length difference \(({\fancyscript{R}}_1-{\fancyscript{R}}_2)\). The third term represents the curvature effect of the ray path \(S_1\) and the propagation corrections from \(S_2\) to \(S_1\). The last term represents the effects of the water vapor density. The constant \(\mu \) represent the power of the dispersion effects and the constant \(H_{21}\) is the scaling factor for the wet delays. Both of these coefficients only dependent on the frequencies of the optical signals and can be calculated using Eqs. (168) and (169). The new formula, Eq. (167), is a general expression for the atmospheric correction of two-color SLR measurements. When the water vapor and the curvature effects are neglected, the new formula reduces to the 2C-SLR formula, Eq. (164). Further detail of this new formula can be found in Wijaya and Brunner (2011).
4.4 Water Vapor Radiometry
A Water Vapor Radiometer (WVR) is an instrument that can be used to estimate the atmospheric wet delay. It does this by measuring the thermal radiation from the sky at microwave frequencies where the atmospheric attenuation due to water vapor is relatively high. These measurements can then be related to the wet delay.
Equation (9) describes the attenuation of radio signals caused by the atmosphere. The attenuation will depend upon frequency, as well as on temperature, pressure, humidity, and liquid water content. For details about how \(\alpha \) can be modeled as function of these quantities, see e.g. Liebe et al. (1993) and Rosenkranz (1998).
Normally the radiation power from the sky is expressed using the brightness temperature \(T_B\), which is defined as the physical temperature a black body would have if it radiates the same amount of power as the sky. For low frequencies \(T_B\) is related to \(W_{sky}\) by
where \(k_b\) is Bolzmann’s constant and \(B\) the observed bandwidth. This relation is valid for frequencies \(\nu \ll k_bT/h\), where \(h\) is Planck’s constant. This requirement is fulfilled at microwave frequencies for normal atmospheric conditions.
The brightness temperature can be calculated from the equation of radiative transfer
where \(T_{bg}\) is the brightness temperature of the cosmic microwave background (\(\approx \)2.7 K). The opacity \(\tau (s,\nu )\) is given by
\(T_B\) depends upon the pressure, temperature, humidity, and liquid water density profiles. The dependencies on these quantities will vary with frequency; for some frequencies (e.g. close to the center of a water vapor absorption line like 22.235 or 183 GHz) there will be a high sensitivity to water vapor while other frequencies (e.g. close to the center of the oxygen absorption lines around 60 and 120 GHz) are more sensitive to the pressure and temperature. For good sensitivity to these parameters, the attenuation needs to be high but not too high. If the attenuation is too high the brightness temperature will approach the physical temperature of the atmosphere, thus the sensitivity to the atmospheric properties will be lost. For retrieval of the water vapor content the frequencies close to the 22.235 GHz line are normally the most suitable ones since this line is not too strong. For very dry conditions (e.g. high altitudes) higher frequencies (e.g. 183 GHz) will give higher sensitivity, however for normal conditions the attenuation is too high.
The sensitivity to the different atmospheric quantities also varies with height, e.g. some frequencies have a higher sensitivity to humidity close to the ground (frequencies on the edge of a water vapor absorption line) while others are more sensitive to the humidity at high altitudes (frequencies close to a water vapor absorption line). Thus it is in principle possible to estimate the humidity profile using radiometer measurements at several different frequencies having different sensitivity to humidity with height, a so-called radiometric profiler (Askne and Westwater 1986; Scheve and Swift 1999). This humidity profile could then be taken to calculate the tropospheric delay. However, the need for using many channels makes the radiometric profilers expensive, and it is difficult to find a set of frequencies from which the humidity profile can be estimated without running into any singularity problems. Furthermore, if only the integrated amount of water vapor—or the wet delay—is of interest we do not necessarily need to know the profile. If a frequency can be found where the sensitivity to the refractivity is constant with height, this is sufficient.
Normally the brightness temperature is not used directly to estimate the wet delay. Instead the opacity \(\tau (\infty ,\nu )\) is calculated from the brightness temperature, which is then used for the wet delay estimation. By introducing the effective temperature of the atmosphere \(T_{eff}\)
we can write \(T_B\) as
Thus the opacity can be estimated by
Some WVR retrieval algorithms to estimate the wet delay from \(\tau \) directly (Westwater et al. 1989; Bosisio and Mallet 1998). However, this requires that the effective temperature \(T_{eff}\) is accurately estimated. An alternative way is to use the linearized brightness temperature \(T^{\prime }_B\) (Wu 1979)
The linearized brightness temperature can be calculated from the opacity by
where the linearized effective temperature \(T^{\prime }_{eff}\) is given by
If \(T_{eff}\) and \(T^{\prime }_{eff}\) are consistently modeled, the error in \(T^{\prime }_B\) caused by an error in \(T_{eff}\) will be approximately canceled by the error in \(T^{\prime }_{eff}\). Thus the linearized brightness temperature can normally be estimated with higher accuracy than the opacity.
The linearized brightness temperature can be divided into four parts
where \(T^{\prime }_{wv}\), \(T^{\prime }_{lw}\), and \(T^{\prime }_{ox}\) are the contributions from water vapor, liquid water, and oxygen, respectively. \(T_{bg}\) is constant and well known. The oxygen part can be accurately modeled using measurements of the surface pressure and temperature (Jarlemark 1997), while the liquid water contribution is approximately proportional to the frequency squared (if the water droplets are much smaller than the wavelength). If the frequencies are properly chosen, \(T^{\prime }_{wv}\) is proportional to the wet delay. Thus the wet delay can be estimated by a combination of measurements at two different frequencies \(\nu _1\) and \(\nu _2\)
where \(T_{bg,ox}\) is the contribution from oxygen and the cosmic microwave background.
For the estimation of the wet delay we need to know the retrieval coefficient \(c_b\), as well as \(T_{bg,ox}\), \(T_{eff}\), and \(T^{\prime }_{eff}\). Normally these parameters are modeled as functions of the surface pressure, temperature, and humidity. To model \(c_b\), one can use WVR measurements and simultaneous observations of the wet delay made by another instruments. Then the model coefficients can be obtained by fitting the radiometer observations to the wet delay observations. The disadvantage of this method is that it requires a long time series of measurements, ideally longer than one year in order to be able to take seasonal variations into account. Furthermore, any systematic error in the wet delay measurements will cause systematic errors in the retrieval coefficient. A more commonly used method is to use profiles of pressure, temperature and humidity obtained from e.g. radiosondes to calculate the theoretical values of \(T^{\prime }_B\), \(T_{bg,ox}\), \(T_{eff}\), \(T^{\prime }_{eff}\), and \(\varDelta L_w\). These can then be used to estimate appropriate models for the parameters. For details, see e.g. Elgered (1993) and Jarlemark (1997).
Several studies have been performed where WVRs have been used to correct for the wet tropospheric delays in space geodetic (mostly VLBI) data analysis. Examples of such studies are Elgered et al. (1991), Kuehn et al. (1991), Ware et al. (1993), Emardson et al. (1999) and Nothnagel et al. (2007). The results indicate that the accuracy of the VLBI estimates can be improved if WVRs are used to calibrate the tropospheric delay compared to the normal method of estimating the tropospheric delay as function in the data analysis (see Sect. 4.2), although the results are inconclusive. It should be noted that the WVR calibration was only applied at a few sites since most VLBI stations are not equipped with radiometers. One problem with radiometers is that they cannot be used during rain for several reasons (liquid water on radiometer antenna, saturation problems, water droplets may not be “small”, ...). Furthermore, most radiometer antennas have relatively large beam-widths (several degrees), and thus the observations are limited to high elevation angles (\(>\)15–20\(^{\circ }\)) in order to avoid picking up radiation from the ground. Hence, either VLBI observations made at low elevation angles have to be excluded, or the WVR measurements need to be extrapolated to low elevation angles, which is a process that can introduce errors.
5 Atmospheric Turbulence
The normal modeling of atmospheric delays in space geodesy, i.e. using mapping functions and horizontal gradients assumes that the spatial variations in the refractivity are linear, and that the temporal variations can be described by e.g. piece-wise linear functions. For the large-scale variations this is an adequate approximation, however at small scales there are non-linear variations caused by atmospheric turbulence. Although it is normally impossible to correct for these random fluctuations, it can be important to model them in order to minimize their effect on the results.
Atmospheric turbulence occurs when energy from e.g. wind shears and temperature gradients creates turbulent eddies. These eddies then break down into smaller eddies until at very small scales the energy of the eddies are dissipated into heat. Inside each eddy the air is mixed, and thus large-scale variations in any atmospheric quantity, e.g. refractivity, will be mixed to create random small-scale variations.
A turbulent eddy with a size \(R\) will have a characteristic wind velocity \(v\). Kolmogorov (1941a, b) assumed that the rate of which kinetic energy (per unit mass) of an eddy is transferred to smaller eddies is only dependent on \(R\) and \(v\). By dimensional analysis it is clear that this rate must be proportional to \(v^3/R\). For stationary turbulence the kinetic energy for the eddies of a specific size will be constant, i.e. the kinetic energy received from larger eddies must be equal to the energy lost to smaller scale eddies (assuming no dissipation into heat at larger scale). At small scales the kinetic energy is dissipated into heat with a dissipation rate \(\varepsilon \), which thus must be equal to the kinetic energy rate of all larger eddies. Thus \(v\propto \varepsilon ^{1/3}R^{1/3}\), or equivalently that the structure function for the velocity fluctuations between \({\mathbf{{r} }}\) and \({\mathbf{{r} }}+{\mathbf{{R} }}\) is given by
where \(C_v\) is the velocity structure constant.
It can be shown that the structure functions for other atmospheric variables, like the temperature and the refractive index, are similar (i.e. also proportional to \(\Vert {\mathbf{{R} }}\Vert ^{2/3}\)). Thus the fluctuations in refractive index between two locations, \({\mathbf{{r} }}\) and \({\mathbf{{r} }}+{\mathbf{{R} }}\), can be described by the structure function \(D_n({{\mathbf{{R} }}})\)
The constant \(C_n^2\) is called the refractive index structure constant. This equation is however not valid for large scales since it becomes infinite when the distance approaches infinity, which is unrealistic. In order to fix this problem, Treuhaft and Lanyi (1987) modified the expression by introducing a saturation length scale \(L\)
With this expression \(D_n({{\mathbf{{R} }}})\) will converge to \(C_n^2L^{2/3}\) as \(\left\| {\mathbf{{R} }}\right\| \) goes to infinity.
Turbulence does not only cause spatial variations in the refractive index, but also temporal variations. One way to describe the temporal variations is to assume Taylor’s frozen flow hypothesis (Taylor 1938). In this hypothesis the turbulent variations in the refractive index in the atmosphere are frozen and move with the wind velocity \({\mathbf{{v} }}\), i.e. it is assumed that \(n({\mathbf{{r} }},t)=n({\mathbf{{r} }}-{\mathbf{{v} }}(t-t_0),t_0)\). This is an approximation which works well over shorter time periods but may not be valid over longer time periods (hours, days). By using Taylor’s frozen flow hypothesis the temporal variations in the refractive index over a time period \(T\) can be d escribed by the structure function \(D_n(T)\)
By combining Eqs. (184) and (185) we get a general expression for the structure function for the fluctuations in the refractive index between \({\mathbf{{r} }}_1\) at time \(t_1\) and \({\mathbf{{r} }}_2\) at time \(t_2\)
5.1 Turbulence Effects on Tropospheric Delays
For the modeling of the effects of turbulence on tropospheric delays we make two assumptions. First that the geometric optics approximation is still valid, i.e. that the turbulent variation within one wavelength can be ignored. Furthermore, we assume that there is no effect on the propagation path of the signal. Both these assumption can be considered approximately valid for weak turbulence. For the case of very strong turbulence more advanced considerations need to be made, see e.g. Tatarskii (1971) and Ishimaru (1978).
The fluctuations in the refractive index will cause variations in the atmospheric delays of radio signals; both in time and as function of direction and location. When describing these fluctuations it is convenient to work with the Equivalent Zenith Tropospheric Delay (EZTD) instead of the slant delays. The EZTD \(\varDelta L^z\) is the slant tropospheric delay divided by a symmetric mapping function \(mf\)
where \(e\) is the elevation angle, \(a\) is the azimuth angle, and \(\mathbf{r }(z)\) denotes the position of the ray at height \(z\).
The structure function between two EZTD, observed at two different times in two different directions and possibly from two different locations, can be calculated by:
Using the relation \(A\cdot B=0.5(A^2+B^2-(A-B)^2\), this simplifies to
With the expression for \(D_n\) given by Eq. (186), the structure function \(D_L\) can be calculated.
The spatial structure function \(D_L\) (Eq. 186) for the zenith wet delay
Figure 14 shows the structure function \(D_L(\rho )\) for spatial variations in the zenith wet delay as function of the distance \(\rho \). In the calculations it was assumed that \(C_n^2=1\cdot 10^{-14}\) m\(^{-2/3}\) up to a height of \(H=2\) km and zero above, and \(L=3000\) km. We can here note three regions: for \(\rho \ll H\) we have \(D_L(\rho )\propto \rho ^{5/3}\), for \(H\ll \rho \ll L\) we have \(D_L(\rho )\propto \rho ^{2/3}\), and for \(\rho \gg L\) we \(D_L(\rho )\approx C_n^2\,H^2\,L^{2/3}\) (i.e. constant).
From he structure function \(D_L\) the covariance matrix for the turbulent fluctuations in the tropospheric delay can be calculated. This can then be applied to weight the observations in the estimation procedure. For more details, see e.g. Treuhaft and Lanyi (1987). It is also possible to use \(D_L\) for simulating tropospheric delays, see e.g. Nilsson and Haas (2010).
5.2 Estimating \(C_n^2\)
In order to calculate the structure function \(D_L\) the profile of the structure constant \(C_n^2\) needs to be known. This parameter is highly variable, both in time and between different locations. Several methods to determine \(C_n^2\) exist, see Nilsson and Haas (2010) for a review of some of them.
One way to determine \(C_n^2\) is to use observations of variations in the tropospheric delay. For example, \(C_n^2\) can be estimated from the observed variations of the tropospheric delay between different directions (Nilsson et al. 2005) or between different locations (Treuhaft and Lanyi 1987). Here we describe how to estimate \(C_n^2\) from observations of the zenith delay variance over a time period \(T\)
After some calculations, it can be shown that (Treuhaft and Lanyi 1987)
In order to estimate \(C_n^2\) using observations of \(\sigma _L^2\) we need to know the shape of the \(C_n^2\) profile. When we want to estimate a \(C_n^2\) that can be used for calculating \(D_L\) it is however not critical to know the exact shape of the profile, most important is then that the integrated value of \(C_n^2\) is correct. Thus we can for example make the approximation that \(C_n^2\) has an exponential profile or, even more simple, assume that \(C_n^2\) is constant up to a height \(H\) and zero above as done by Treuhaft and Lanyi (1987). Using the latter approximation, \(C_n^2\) at heights lower than \(H\) can be calculated from
The height \(H\) should be chosen such that the largest fluctuations in the refractive index occur below \(H\). For microwaves where the fluctuations in the wet delay is dominating an appropriate choice is the scale height of water vapor (approximately 2 km).
Another possible way to obtain \(C_n^2\) is to use vertical profiles of pressure, temperature, and humidity obtained from e.g. radiosonde measurements. As discussed above, when turbulence is present, large-scale variations (gradients) in the atmosphere are mixed and create small-scale fluctuations. Thus there will be variations caused by both the large-scale gradients as well as turbulence. At large scales the variations due to gradients will dominate, and at small scales turbulence. At some scale in between the magnitudes of the large-scale gradients and by turbulence will be equal. This scale is proportional to the so-called outer scale of turbulence \(L_0\). Typically values of \(L_0\) range between a few meters to several hundreds of meters. The value can vary with time, but for the calculations of \(C_n^2\) typically a mean value is used (d’Auria et al. 1993). Hence
Thus \(C_n^2\) could be calculated from the gradient in \(n\). Since the gradient of the refractive index is typically more than one order of magnitude larger in the vertical direction than in the horizontal direction, we can approximate the refractive index gradient by its vertical component. However, one problem that needs to be considered is that the refractive index is not conserved in adiabatic motion in the atmosphere. When an air parcel is moved up or down in the atmosphere (e.g. due to turbulence) its size will change due to the change of atmospheric pressure with height. This in turn will cause the temperature and partial pressure of water vapor—and thus also the refractivity—to change. This needs to be corrected for when calculating \(C_n^2\). The way to do this is to consider the vertical gradient in the refractive index caused by vertical gradients in quantities conserved under adiabatic motion in the atmosphere, e.g. the potential temperature \(\theta \) and the specific humidity \(q\)
where \(p_0=\)1000 hPa. Thus \(C_n^2\) can be calculated as
where \(a^2\) is a constant (\(a^2\approx \) 2.8 (d’Auria et al. 1993)). The vertical gradients should not contain any variations due to turbulence, thus they should be evaluated over height intervals larger than \(L_0\).
Equation (196) is however only valid if there is turbulence present. When no turbulence is present \(C_n^2\) should be (close to) zero. One way to determine if the air is turbulent or not is to use the Richardson number \(Ri\) (Richardson 1920)
The atmosphere is turbulent when \(Ri\) is larger than a critical Richardson number \(Ri_c\), typically \(Ri_c\approx 0.25\). Thus Eq. (196) modifies to
where \(F=1\) if \(Ri<Ri_c\) and zero otherwise. This is however still a bit too simplistic since this assumes a very sharp transition between a turbulent and a non-turbulent state. d’Auria et al. (1993) presented a model for \(F\) giving a more smooth transition between 0 and 1 around \(Ri_c\).
6 Applications of Space Geodetic Techniques for Atmospheric Studies
As discussed earlier, it is important to have a good model of the delay in the neutral atmosphere in order to obtain the highest accuracy in the space geodetic results (e.g. station positions). Since external estimates of the wet delay with high enough accuracy are typically not available (at least for microwave techniques), the common way of handling the wet delay in the data analysis is to estimate it, i.e. by modeling it using mapping functions and gradients as described in Sect. 4.2. Thus the results of the data analysis will also contain information about the tropospheric delay, which is closely related to the IWV (see Sect. 3.2.2). Several studies investigated the accuracy of the zenith wet delays and IWV estimated from VLBI and GNSS (e.g. Herring et al. 1990, Tralli and Lichten 1990, Bevis et al. 1992, Snajdrova et al. 2006, Teke et al. 2011), and these show that it is on the same level or better then that of other techniques. Thus there exists a large interest in applying space geodetic techniques, especially GNSS, for atmospheric studies. For example, zenith wet delays can be used to study climate trends (Sect. 6.1), or assimilated into numerical weather prediction models to improve weather forecasts (Sect. 6.2). With wet delays estimated from a local GNSS network one can even attempt to estimate the 3D structure of the atmospheric water vapor by applying tomographic methods (Sect. 6.3).
6.1 Long-Term Water Vapor Trends
Since the zenith wet delay is closely related to the integrated water vapor content (see Sect. 3.2.2), we can analyze \(\varDelta L_w^z\) estimated from space geodetic techniques to study the variations of the atmospheric water vapor content in time. For example, it is possible to study diurnal and seasonal variations as well as long term trends in the water vapor content. Such information is of great interest in climatology since the water vapor content is closely related to the temperature. Climate models typically predict that the average relative humidity remains constant as the temperature changes (Trenberth et al. 2003). Since the saturation water vapor pressure depends approximately exponentially on the temperature, this means that a a change in the temperature will cause a corresponding change in the water vapor content. It is predicted that an increase in temperature of 1 K will increase the water vapor content by 6–7 % (Trenberth et al. 2003; Bengtsson et al. 2004). It is important to monitor the water vapor content since water vapor is a greenhouse gas, in fact the most important one. Additionally, higher water vapor content can also indicate an intensified hydrological cycle, including increased precipitation.
Zenith wet delay trends in mm/decade estimated from GPS data in Sweden and Finland, 1997–2006. Modified from Nilsson and Elgered (2008)
Several studies have calculated long-term trends in \(\varDelta L_w^z\) (or IWV) estimated from GNSS and VLBI, e.g. Gradinarsky et al. (2002); Jin et al. (2007); Steigenberger et al. (2007); Heinkelmann et al. (2007); Ning and Elgered (2012). An example of \(\varDelta L_w^z\) trends calculated from ten years of GPS data in Sweden and Finland is shown in Fig. 15. For more details, see Nilsson and Elgered (2008).
6.2 GNSS Meteorology
Water vapor is a very important parameter in meteorology and in order to get accurate weather forecasts it is very important to have accurate measurements of the water vapor content. A problem is that the water vapor content is highly variable in both space and time, and traditional instruments (e.g. radiosondes) do not provide the water vapor content with high enough spatial and temporal resolution. With the establishment of relative dense GNSS networks the meteorological community has started to be interested in assimilating zenith total delays or zenith wet delays estimated from these GNSS networks in the numerical weather prediction models (Gutman and Benjamin 2001; Poli et al. 2007).
Several investigations of assimilating GNSS tropospheric delays in numerical weather prediction models have been performed. For example, in Europe this has been investigated in the projects COST-716 (Elgered et al. 2005), TOUGH (Vedel 2006), and E-GVAP (http://egvap.dmi.dk/). It has been shown that the quality of the forecasts improve if GNSS data are assimilated, especially in cases of high precipitation (Vedel and Huang 2004; Karabatić et al. 2011).
6.3 GNSS Tropospheric Tomography
Tomography is a method which can be used to estimate the 2D or 3D structure of a quantity from measurements of the integral of the quantity along different paths. It is a method commonly used in medicine, seismology, material science, and a number of other fields. Tomography can also be applied to atmospheric delay measurements in order to reconstruct the 3D structure of the wet (or total) refractivity. This requires that the slant wet delays are measured by several stations in a local (inter-station distance maximum a few km) network. The only space geodetic technique for which such dense networks are available is GNSS.
Example of a GNSS tomography scenario. Note that for better visibility the relative size of the troposphere have been enlarged; in reality the troposphere (\(=\) top of highest voxel) is about 10 km heigh, the inter-station distances a few km, while the satellites are are at about 20000 km altitude
A picture of a GNSS tomography scenario is shown in Fig. 16. In order to estimate the wet refractivity field from the observed wet delays, the atmosphere above the GNSS network is parameterized. The most commonly used parameterization is voxels, although other parameterizations are also possible (Perler et al. 2011). Voxel parameterization means that the atmosphere is divided into a number of boxes (called voxels, volume pixels) in which the refractivity is assumed constant. Thus the wet tropospheric delays along the rays of the observed GNSS signals can be described by a linear combination of the voxel refractivities
\(\varDelta L_i\) is the wet tropospheric delay along the \(i\)th ray, \(n_{vox}\) is the number of voxels, \(N_j\) is the refractivity of the \(j\)th voxel, and \(D_{ij}\) is the distance traveled by ray \(i\) in voxel \(j\). Since the station and satellite coordinates are normally known, \(D_{ij}\) can be calculated. Having observations of the tropospheric delays along several different rays, a linear system of equations is obtained
\(\varDelta {\mathbf{{ L} }}\) and \({\mathbf{{N} }}\) are vectors containing the wet delays and the refractivities, respectively, and \(\mathbf{D }\) is a matrix containing the \(D_{ij}\) values. By inverting the system, the voxel refractivities are estimated.
There are however a few problems with this method. One is how to estimate the slant wet delays along the GNSS signal rays. One way is to estimate the zenith wet delay and gradients in a normal GNSS analysis, then use these to calculate the slant wet delays. This is for example done by Champollion et al. (2005). However, this method will limit the accuracy since it assumes that all horizontal variations in the refractivity above a station are linear, something which is not always realistic. In order to improve the slant wet delays, it is often assumed that the post-fit residuals of the GNSS analysis will contain the unmodeled parts of the slant wet delays, and thus adding these to the calculated slant wet delays will give the true delays (Alber et al. 2000; Troller et al. 2006). This is not true, the residuals will contain also other errors of the GNSS measurements (e.g. multi-path). Another approach is to model the slant delays by using Eq. (199) in the GNSS data analysis instead of zenith delays and gradients. First results using this approach are presented by Nilsson and Gradinarsky (2006) and Nilsson et al. (2007).
Another problem is the normally weak geometry since tomography ideally requires that there are rays crossing the investigated volume in all possible directions. In GNSS tomography, however, all rays are going between the top of the troposphere to the stations on the surface of the Earth, while there are no rays entering and/or leaving the voxel grid at the sides. This makes the sensitivity to the vertical refractivity profile very low and as a result the equation system (200) will be ill-conditioned. This problem can be solved by either constraining the refractivity field to some a priori field obtained either by models or external measurements like radiosondes. The problem is not as big if the GNSS stations are placed at very different altitudes (e.g. if there are differences of several kilometers between highest and lowest stations) (Nilsson and Gradinarsky 2006).
Furthermore, since the satellite geometry will change during the day, some voxels may at times have no or only a few rays passing through it. Thus, in order to avoid singularity problems, constraints need to be applied. Simple constraints are for example inter-voxel constraints which constrain the refractivity of a voxel to the mean refractivity of the neighboring voxels (Flores et al. 2000). A more advanced approach is to use a Kalman filter with a covariance matrix for the voxel refractivity calculated from turbulence theory using Eq. (186) (Gradinarsky and Jarlemark 2004; Nilsson and Gradinarsky 2006).
Simulation results of tomographic reconstruction of the refractivity field. The upper plot shows the result obtained when the simulated refractivity field was 20 mm/km in the fifth layer and zero elsewhere. In the lower plot the simulated refractivity field is 20 mm/km only in the middle voxel of the fifth layer, and zero elsewhere. These results are from Nilsson (2007)
Figure 17 shows the results of two simulations demonstrating the strengths and weaknesses of GNSS tomography. In the upper plot the case where the refractivity is 20 mm/km in the layer between 3 and 4 km altitude and zero elsewhere is simulated. As seen the tomographic reconstruction is not working well. This is because of the weak geometry in the vertical direction, resulting in a very low sensitivity to the height of the layer with non-zero refractivity. Thus the refractivity is spread out over all layers in the tomographic reconstruction. Most refractivity is put in the lower layers simply because the tomographic software is set up to allow for a higher variability in the refractivity at lower altitudes than at higher altitudes. The estimated refractivity in the lowest layer is however correctly estimated to zero. Here the sensitivity is higher since the stations are at slightly different altitudes. In the lower plot of Fig. 17 the simulated refractivity field is non-zero just in one voxel: the middle voxel (of 25) of the layer between 3 and 4 km. In this case the tomographic reconstruction is working better. This demonstrates that GNSS tomography can have a good sensitivity to horizontal fluctuations in the refractivity field, and is even able to determine at which height these fluctuations are occurring. For more details about the simulation setup and other results from similar simulations, see Nilsson (2007).
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Acknowledgments
First of all we would like to thank the reviewer, Gunnar Elgered, for providing very valuable suggestions to improve the quality of this part of the book. We are grateful for the financial support from the German Science Foundation (DFG, SCHU 1103/3-2), and from the Austrian Science Fund (FWF, P20902-N10).
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Nilsson, T., Böhm, J., Wijaya, D.D., Tresch, A., Nafisi, V., Schuh, H. (2013). Path Delays in the Neutral Atmosphere. In: Böhm, J., Schuh, H. (eds) Atmospheric Effects in Space Geodesy. Springer Atmospheric Sciences. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-36932-2_3
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