Ray-traced tropospheric delays in VLBI analysis

Transcription

1 RADIO SCIENCE, VOL. 47,, doi: /2011rs004918, 2012 Ray-traced tropospheric delays in VLBI analysis Vahab Nafisi, 1,2,3 Matthias Madzak, 2 Johannes Böhm, 2 Alireza A. Ardalan, 1 and Harald Schuh 2 Received 8 November 2011; revised 7 February 2012; accepted 28 February 2012; published 25 April [1] We develop a ray-tracing package for the calculation of path delays of microwave signals in the troposphere based on numerical weather models which we use for the determination of the delays of geodetic Very Long Baseline Interferometry (VLBI) observations. We show results for a two-week campaign of continuous VLBI sessions in 2008 (CONT08), where we apply those ray-traced delays and analyze the repeatability of baseline lengths in comparison to a standard approach with zenith delays and mapping functions. We find improvement in baseline length repeatabilities when no tropospheric gradients are estimated in the analysis. Furthermore, ray-traced delays are applied for Intensive sessions containing the stations Tsukuba (Japan) and Wettzell (Germany) for the determination of Universal Time (UT1). We perform an external validation using GPS-derived length-of-day values and find an improvement for UT1 with ray-traced delays by up to 4.5%. Citation: Nafisi, V., M. Madzak, J. Böhm, A. A. Ardalan, and H. Schuh (2012), Ray-traced tropospheric delays in VLBI analysis, Radio Sci., 47,, doi: /2011rs Introduction [2] The troposphere is a composition of dry gases and water vapor, both parts imposing a time delay on propagating electromagnetic waves. Furthermore, an inhomogeneous medium causes an electromagnetic (EM) wave to propagate along a curved path, which is called the bending effect resulting in the geometric delay. Because of these two effects on space geodetic observations, the observed distances are longer than the straight line between the receiver and the transmitter in vacuum. In this paper, the combination of both effects is called total tropospheric delay or just delay. [3] Tropospheric delay modeling has always been an important issue in space geodetic data analysis. As described in the IERS Conventions 2010 [Petit and Luzum, 2010], a priori zenith hydrostatic delays (ZHD) can be estimated from the surface pressure at the site as suggested by Saastamoinen [1972]. Considering the elevation angle, the ZHD is mapped to the direction of the specific observation using the hydrostatic mapping function [Davis et al., 1985], while zenith wet delays (ZWD) are estimated using the wet mapping function as partial derivative. Furthermore, tropospheric gradients [Chen and Herring, 1997] are usually estimated to consider the azimuthal asymmetry of the delays. 1 Department of Surveying and Geomatics Engineering, College of Engineering, University of Tehran, Tehran, Iran. 2 Institute of Geodesy and Geophysics, Vienna University of Technology, Vienna, Austria. 3 Department of Surveying Engineering, Faculty of Engineering, University of Isfahan, Isfahan, Iran. Copyright 2012 by the American Geophysical Union /12/2011RS [4] Modern mapping functions such as the Vienna Mapping Functions 1 (VMF1) [Böhm et al., 2006a] and the Global Mapping Functions (GMF) [Böhm et al., 2006b] are based on numerical weather models (NWM), which have been continuously improving with regard to their spatial and temporal resolution as well as with regard to advances in data assimilation. One can find a comprehensive and widespread comparison of the models available up to the last decade in the work by Mendes [1999]. Ghoddousi-Fard [2009] has reviewed recent mapping functions based on NWM as well as commonly used gradient mapping functions. [5] Unlike the mapping function method, ray-tracing estimates slant delays for each specific direction. Bean and Thayer [1959] were one of the early proponents of raytracing in a spherically stratified atmosphere and some mathematical models for this purpose were presented. Due to lack of powerful and fast computers and the need of a huge number of calculations to obtain the results, simplifications were used in the equations. [6] Nowadays many fields of science where the propagation of an EM wave through a stratified medium has to be traced can benefit from the ray-tracing technique. The main application of this method in tropospheric research is the determination of the total delay along the trajectory of a signal, which is transmitted from a source. The geometrical optics approximation [Born and Wolf, 1999] can be used to obtain the Eikonal equation, which represents the solution of the Helmholtz equation for an EM wave propagating through a slowly varying medium [Wheelon, 2001]. The Eikonal equation is used to establish a ray-tracing system to determine the raypath and the optical path length. [7] Generally speaking, ray-tracing methods can be categorized in three groups using different assumptions: 1of17

2 CONT08, a continuous two-week VLBI campaign in August For more information about CONT08 we refer to Teke et al. [2011]. Also some validation results from Intensive sessions between Tsukuba and Wettzell are presented in this section. For example, we show the effect of ray-traced delays on baseline length repeatabilities and the estimation of DUT1 values. Outlook and concluding remarks from this research can be found in section Refractive Index of Moist Air [13] For a medium, the refractive index n is defined as the ratio of the velocity of an electromagnetic wave in vacuum to the speed of propagation in this medium. It can be written as n ¼ c v ð1þ Figure 1. Total delays computed by different assumptions at 5 outgoing elevation angle: Horizontal symmetry (black), horizontal asymmetry-3d (blue) and horizontal asymmetry- 2D (red). The sample data set is derived from the epoch 0:00 UTC of August 12th 2008 at the station Tsukuba, Japan. 3D results are not necessarily larger than those of 2D. The sign of differences between 2D and 3D is dependent on horizontal gradients of the troposphere. [8] Horizontal symmetry: In this case horizontal symmetry of the troposphere is the main assumption. Therefore, the ray direction is sufficiently defined by the elevation angle. Usually, one tropospheric profile, where meteorological parameters such as temperature, pressure and humidity are known, is enough for a specific station. The Vienna Mapping Functions 1 [Böhm et al., 2006a] are based on this concept. [9] Horizontal asymmetry-2d: In this case the azimuthdependent behavior of the troposphere is accounted for. It means that in addition to the elevation angle, the azimuth of the observation becomes a parameter which must be considered for the raypath. However, the ray is not allowed to leave the constant azimuth plane. In other words, any out-ofplane component is neglected. [10] Horizontal asymmetry-3d: In addition to the above mentioned azimuth dependency of the troposphere, it is possible for the raypath to have out-of-plane components. These components are due to horizontal gradients of the refractivity. Obviously this hypothesis is more realistic but, as a drawback, comes along with some difficulties from the computational point of view. [11] To illustrate the differences between the three above mentioned categories of ray-tracers, Figure 1 shows the slant delay variations with respect to azimuth for a sample data set. [12] This paper discusses the applications of the ray-tracing method for calculating total tropospheric delays in Very Long Baseline Interferometry (VLBI) analysis. Section 1 introduces the refractivity of moist air. A ray-tracing system based on Eikonal and Maxwell s equations for total delay computations is developed and presented in section 2. In section 3 we discuss the structure of a typical ray-tracing system. Section 4 deals with the most important computational aspects of a ray-tracing system, which must be considered to improve accuracy and efficiency of the system. In section 5 we show some results for the VLBI campaign where c and v are the velocities in vacuum and in the medium, respectively. [14] In general this index is complex. The imaginary part corresponds to the attenuation of the wave, which is not typically significant for frequencies smaller than 10 GHz [Nilsson, 2008]. Because the refractive index of a signal in moist air is very close to one, n 1 will be very small and it is therefore convenient to introduce and use another parameter, namely refractivity N: N ¼ ðn 1Þ10 6 : ð2þ [15] The total refractivity N of moist air can be expressed as [Davis, 1986] N ¼ N d þ N v ¼ k 1 R d r d þ k 2 R v r v þ k 3 R v r v T : [16] Hereafter k 1, k 2 and k 3 are refractivity coefficients which can be determined experimentally. These coefficients represent the induced dipole moment and the polarization orientation of polar and non-polar molecules that depend on the frequency of an EM wave. However, for microwave signals propagating through the neutral atmosphere the coefficients are practically independent of frequency [Hecht, 2001]. These constants have been estimated using direct measurements made with microwave cavities [Boudouris, 1963] and different coefficients are suggested by various authors, as summarized in Table 1. For detailed reviews and Table 1. Refractivity Coefficients as Realized by Different Authors [Rüeger, 2002a, 2002b] Coefficient k 1 (10 2 K/Pa) k 2 (10 2 K/Pa) ð3þ k 3 (10 3 K 2 /Pa) Essen and Froome [1951] Essen [1953] Smith and Weintraub [1953] Bean [1962] Boudouris [1963] Zhevakin and Naumov [1967] Thayer [1974] Bevis et al. [1994] Rüeger [2002b] best average Rüeger [2002b] best available IUGG of17

3 discussions of the different values we refer to Rüeger [2002a, 2002b] and Healy [2011]. [17] Since NWM provide meteorological parameters, it is useful to express equation (3) in terms of total pressure (p), water vapor pressure (e) and temperature (T). Using the equation of state for non-ideal gases [Kleijer, 2004] p i ¼ Z i r i R i T where i = d, v (R d and R v denote the gas constant for dry air and water vapor, respectively), we can obtain [Davis, 1986] p N ¼ k 1 T þ e k 2 T þ k e 3 T 2 Zv 1 ¼N h þ N nh ð5þ where N h and N nh denote the hydrostatic and non-hydrostatic refractivities, respectively. The separation into these two parts is possible under the assumption of unsaturated conditions for air. In this case the total pressure p can be expressed as a combination of the partial pressure of dry air p d and water vapor e (=p v )[Stull, 2000; Wallace and Hobbs, 2006]. A new refractivity coefficient k 2 introduced in equation (5) is expressed as k 2 ¼ k 2 k 1 R d R v : [18] Z v in equation (5) is the water vapor compressibility factor, which in normal conditions is close to one [Kleijer, 2004]. Ignoring compressibility factors in this equation introduces errors of 0.1 ppm for wet components [Thayer, 1974], which is much lower than the accuracy of this component [Cucurull, 2010]. [19] The water vapor pressure e can be calculated using the following equation [Wallace and Hobbs, 2006] qp e ¼ ɛ þ ð1 ɛþq where ɛ = M v /M d is the ratio of the molar masses of water vapor and dry gasses, and the values of the specific humidity q are provided by NWM Total Tropospheric Delay [20] The total delay can be defined as the difference between the propagation time of a specific wave in a real medium (in our case the troposphere) and in vacuum. In ideal conditions, which means without any dispersion, the path of the ray between the receiver and the source of the wave (a quasar in VLBI) is a straight line. Z S ¼ ds: ð8þ V [21] On the other hand, due to variations in the tropospheric refractive index, the real path of the ray is defined as Z L ¼ T ð4þ ð6þ ð7þ nr; ð q; l; tþds ð9þ where r is the radial distance, q is the co-latitude, and l is the longitude (0 q p, 0 l 2p). n(r, q, l, t) denotes the refractive index at a position defined by r, q and l at time t. Using equations (8) and (9) and refractivity instead of refractive index, the total tropospheric delay reads as 0 1 Z Z Dt ¼ 10 6 Nr; ð q; l; tþds ds SA: ð10þ T [22] The first term of equation (10) represents the signal delay along the path which causes the excess of the path. The second term denotes the so-called geometric delay. The first term inside the brackets is the length of the curved path T. Inserting equation (5) into equation (10), we find Dt ¼ 10 6 Z or T T 0 Z 1 ds SA Z N h ðr; q; l; tþ ds þ 10 6 Dt ¼ Dt h þ Dt nh þ Dt b : T T N nh ðr; q; l; tþ ds ð11þ ð12þ [23] Equation (12) shows the different components of the signal delay due to tropospheric propagation effects, i.e., the hydrostatic (Dt h ) and non-hydrostatic (Dt nh ) parts as well as the geometric delay Dt b which depends on the total refractivity. [24] The geometric delay for elevation angles above 57 is less than 0.1 mm, but it significantly increases with decreasing elevation angles. At 3, 5, and 10 elevation, the effect is about 60, 20, and 3 cm, respectively. 2. Maxwell s Equations and the Eikonal Equation in a Random Media [25] Combining the four Maxwell equations [Fleisch, 2010] allows us to find two important implications. First, we can find the wave equation and an equation for the refractive index can be obtained, which is directly linked to the physical body of the troposphere. Second, the Eikonal equation is obtained using some assumptions and arrangements. This equation plays an important role in the development of our ray-tracing algorithm. With Maxwell s equations and considering a medium free of currents and charges, and a short vacuum wavelength and applying some mathematical operations, we can write the well-known Eikonal equation in vector notation [Born and Wolf, 1999] as rl 2 ¼ n 2 ð * r Þ ð13þ where rl comprises the components of the ray directions, L is the optical path length, n is the refractive index of a medium and * r is the position vector. The surfaces L(r) = constant are called geometrical wave surfaces or geometrical wavefronts Hamiltonian Formalism of the Eikonal Equation [26] Equation (13) is a partial differential equation of the first order for nr ð * iþ and it can be expressed in many alternative forms. In general, we can write the Eikonal equation 3of17

4 in the Hamiltonian canonical formalism as follows [Born and Wolf, 1999; Cerveny, 2005]: Hr ð * ; rlþ 1 n o a ðrl:rlþa 2 nr ð * Þ a ¼ 0 dr * i du ¼ H rl i drl i du ¼ H r * i dl du ¼rL H i : rl i ð14þ ð15þ ð16þ ð17þ [27] The scalar value a determines the parameter of interest u (see Table A1 in Appendix A). In general it is a real number but in applications we must consider it to be an integer number. Hr ð * i; rl i Þ is called the Hamiltonian function or just Hamiltonian. [28] In a 3D space this system consists of seven equations. Six equations are obtained from equations (15) and (16) and must be solved altogether. The result of these six equations is r i = r i (u), which describes the trajectory of a 3D curve. The seventh equation, i.e., equation (17), can be solved independently and yields the optical path length. Details of a ray-tracing system for practical purposes based on the Hamiltonian formalism are presented in Appendix A. In the following section we discuss different aspects of this problem. 3. Structure of the Ray-Tracing System [29] If we want to describe our ray-tracing system, we can distinguish four main properties. Since these parts are very general, one can find different realizations. [30] Inputs: Inputs of the system can be divided into three different groups [31] - Numerical weather model data set, which provides basic meteorological parameters needed for the estimation of refractivity like temperature, water vapor pressure and total pressure. [32] - Coordinates of the stations (receivers) and time of observation, which provide basic information for horizontal and temporal interpolation. [33] - Elevation angles and azimuth angles of the raypath, which provide information for the direction of the ray and also the stopping criteria for the iterations. [34] Pre-calculations: For speeding up the ray-tracing calculations, some pre-calculations should be performed before the main ray-tracing computations. Outputs of this step are vertical inter- and extrapolated refractivity profiles. In addition, we need to set some parameters such as the interpolation and extrapolation method, transformation between geodetic and meteorological systems, auxiliary atmospheric data sets, height of upper limit of the troposphere and size of increments. [35] Ray-tracing: The solution of the Eikonal equation is the main result of this part. Based on wave propagation theory and optics approximation, which is explained in section 2, we can construct seven partial differential equations for a 3D ray-tracing system. Six of them must be solved simultaneously and the seventh equation yields the optical path length. [36] Outputs: Important outputs of this partial differential system are the coordinates of the points along the trajectory. Combining these coordinates and refractivity information, the total delay for our observation is determined. The total delay can be converted to a slant factor, which is slant delay divided by zenith delay. Because the curved raypath is not known in advance, a few iterations are needed to complete the work, i.e., to find the initial angles corresponding to the final ray tangent directions. A criterion for stopping the iterations is when successive outgoing elevation and azimuth angles do not differ by more than 0.1 mm (delay) or 60 milliarcseconds. The number of iterations depends mainly on elevation angle and the meteorological conditions of the troposphere. A priori bending angle models can help to reduce the number of iterations [Hobiger et al., 2008]. 4. Computational Aspects of a Ray-Tracing System [37] In the following subsections we discuss the most important details and computational aspects of a tropospheric ray-tracing system. Although the effect on the delay of some of them is negligible, it can be useful for drawing a general perspective about the ray-tracing package. The influence of errors in the slant delay on station heights is determined using a rule of thumb: The error in the station height is approximately 1/5 of the delay error for a minimum elevation angle of 5 [MacMillan and Ma, 1994]. Although this rule of thumb has been developed for mapping functions which are independent of azimuth, we will also use it for individual azimuths. This is not fully correct, but it provides the possibility to better understand and assess our findings. [38] Different numerical weather models as input for raytracing systems can be used. A direct outcome of different characteristics of these NWM like uppermost pressure level, number of pressure levels, spatial resolution and assimilation method, are discrepancies in the final ray-traced results. Some examples are given by Nafisi et al. [2012], where results of a comparison campaign are presented and analyzed Horizontal Resolution and Interpolation [39] NWM provide data sets with different resolutions. The resolution of a data set can affect the final results in two ways. First, a finer resolution can provide more reasonable and more accurate information for a ray-tracing system. In other words, we can get a more realistic image of the troposphere. Second, to obtain meteorological parameters for a specific station, it is necessary to interpolate using some grid points around the point of interest. [40] On the other hand there are also disadvantages with finer data sets, e.g., the huge amount of data, which could reduce the computational efficiency of the software. For an application of fine-mesh numerical weather model data in space geodesy we refer to Hobiger et al. [2010]. As we want to find the best resolution for our system we have to assess the size of the differences in total delays due to different 4of17

5 Figure 2. Differences between ray-traced slant factors determined from data sets with different resolutions (0.5 and 1.0 ) for Tsukuba station, August 12th 26th Slant factors are multiplied by a nominal total zenith delay of 2.5 m. resolutions. Figure 2 shows the differences between raytraced slant factors computed from two different data sets with a spatial resolution of 0.5 and 1.0, the latter being a down-sampled version of the high-resolution data set. Differences reach about 1 dm for lowest elevations. Therefore, we use a spatial resolution of 0.5 in VLBI analysis. [41] The original input of a ray-tracing system based on numerical weather models is a gridded data set containing meteorological values for each node point at a vertical layer. To obtain the meteorological parameters for a specific point, i.e., the location of the receiver or an arbitrary integration point, a horizontal interpolation is necessary. For our raytracing system, we can consider different methods of horizontal interpolations. The influence of spline, arithmetic mean, weighted mean and bilinear interpolation is investigated in this study. [42] For an evaluation of the interpolation methods, a comparison is carried out using gridded refractivity profiles obtained from ECMWF meteorological parameters. The goal of this procedure is to retrieve the refractivity for one point of interest (Figure 3) by spline and mean interpolation. Three versions of spline interpolation are applied: (1) Spline1: with 16 (green) points around the point of interest, (2) Spline2: with 8 points along the meridian and (3) Spline3: with 8 points along the parallel. For the method of mean interpolation, we have used all 16 points (mean2) or only the 4 nearest points (mean1). These methods are applied for all vertical layers from the station height to about 76 km. ECMWF data are used for five stations (Tsukuba, Medicina, Wettzell, Kokee and HartRAO) at four epochs per day from August 12th 26th [43] We can estimate the uncertainty of the total zenith delay using real and interpolated refractivity profiles. Our results are presented in Figure 4, and we use the different strategies in VLBI analysis in section 5.1. [44] Another aspect concerning horizontal interpolation is the map projection used to produce various NWM. If the model grid is not aligned with the geodetic coordinate system defined by longitude and latitude, resampling might be Figure 3. A method for evaluating various horizontal interpolation methods, using a real refractivity data set. The point of interest is the red point in the center. The goal of the procedure is to retrieve the refractivity at the red square from other grid points (green points) and different interpolation methods. Brown solid and violet dashed rectangles show the chosen points for spline2 and spline3, respectively. needed, because the horizontal components of the gradient are expressed in spherical coordinates. Hobiger et al. [2008] state that such an irregularity can significantly increase the computation time required to solve the Eikonal equation. If such resampling is necessary, it should be performed in a pre-calculation step. For ECMWF data sets, for example, the transformation is not needed, because the map projection is already based on a rectangular grid Refractivity Constants [45] As shown in section 1, the refractivity (or equivalent refractive index) of the troposphere is a very important input Figure 4. Zenith total delay errors for different methods (Spline1: black, Spline2: brown, Spline3: orange, Mean1: yellow, and Mean2: white) and five VLBI stations (Wett: Wettzell, Koke: Kokee Park, Tsuk: Tsukuba, Hart: HartRAO, Medi: Medicina), August 12th 26th of17

6 Figure 5. Slant factor differences computed for a 5 elevation angle for different refractivity constants with respect to Rüeger best average: Rüeger best available (blue-solid), IUGG (green-solid), Bevis (magenta-solid), Smith and Weintraub (red-bold), Essen (magenta-dashed), and Essen and Froome (black-bold) at station Tsukuba, 18th August 2008, 0 h. Slant factors (dimensionless) are multiplied by 2500 mm to get the results in mm. for the ray-tracing system. This parameter can be determined using meteorological parameters (temperature, pressure, and water vapor pressure) and refractivity coefficients. Figure 5 shows differences in slant factors and slant delays computed from some of the refractivity constants tabulated in Table 1. The reference for Figure 5 (Rüeger best average) is used in the VLBI analysis. This model was suggested by the IAG Working Group for the ray-tracing comparison campaign [Nafisi et al., 2012] Radius of the Earth [46] The radius of curvature of the Earth has a significant impact on slant delays. When using simplified geometrical figures of the Earth such as a sphere of constant radius, errors in the delay can be expected which are a function of latitude. Even when adopting a sphere of constant radius, such a sphere has its radial direction aligned with the ellipsoidal normal. Some of the more common representations for computing the radius of the Earth which can be used by ray-tracing methods are Gaussian curvature or Euler s formula[nafisi et al., 2012]. Figure 6 shows slant factor differences computed using Gaussian and constant radii with respect to Euler s formula, the latter radii being azimuth-dependent. Typically, the differences for the constant Earth radius are larger than for the Gaussian radius which is latitude-dependent. [47] The formula for the radius by Euler is more realistic because the effect of the direction (azimuth) is included. The maximum difference between Euler and the constant radius is about 7.5 mm, corresponding to the station height error of around 1.5 mm. We use Gaussian curvature in VLBI analysis which can be seen as arithmetic mean of Euler s formula over all azimuths. Figure 7 shows the latitude-dependency of the results. Delays based on the Gaussian radius have no latitude-dependent errors at azimuths of odd multiples of 45 (45, 135, 225, 315 ), while a constant radius yields latitude-dependent errors. Figure 6. Slant factor differences computed using different radii with respect to Euler s formula: Gaussian curvature (red) and constant radius (black). The outgoing elevation angle is equal to 5. Results are multiplied by 2500 mm to get the results in mm. Values are shown for Tsukuba on August 12th Vertical Interpolation [48] Since NWM are usually based on a geopotential height system we need to transform them into a global geodetic coordinate system, which is defined using geometrical (ellipsoidal) heights [Hobiger et al., 2008]. Thus, in order to use the information from NWM for a ray-tracing calculation, it is necessary to transform heights of the profiles before they can be introduced in the new reference system. The geopotential height (V) is defined as the ratio of the geopotential value F and a constant standard value of the gravity acceleration g 0, which is usually assigned m/s 2 in meteorological applications. In contrast, the orthometric height H 0 is defined as F H o ¼ g ðq; l; H o Þ ¼ V g o g ðq; l; H o Þ ð18þ Figure 7. Slant factor differences computed using different radii with respect to Euler s formula: Gaussian curvature (black triangles) and constant radius (red circles). The outgoing elevation angle is equal to 5 and the results are for azimuth equal to 0. Results are multiplied by 2500 mm to get the results in mm for all CONT08 stations on August 12th, of17

7 where g ðq; l; H o Þ is the mean acceleration due to gravity between the geoid and the point. By introducing the geoid undulation H u, the ellipsoidal height H e can be calculated using H e ¼ H o þ H u : ð19þ [49] Different values of gravity in equation (18) can be used, such as normal gravity, normal gravity without centrifugal contribution, or an EGM96 expansion of the actual gravity or the Office of the Federal Coordinator for Meteorology [2007] model value. Investigations show that the impact of different gravity values on the total delay is insignificant if the formula considers the effect of centrifugal acceleration due to Earth rotation [Urquhart, 2011; Nievinski, 2009]. [50] After the transformation of geopotential heights into heights above the reference ellipsoid, the meteorological parameters (total pressure, water vapor pressure and temperature) must be interpolated in order to obtain a finer resolution for the ray-tracing calculation. Because of the nonlinearity of total pressure, water vapor pressure and temperature for the calculation of the refractivity (equation (5)) it is more adequate to interpolate these parameters separately and determine the refractivity in a later step. In the case of temperature, a simple linear interpolation is sufficient to gain a higher spatial resolution in the vertical domain. Considering the total pressure and water vapor pressure behavior the most straight forward approach is to use a logarithmic interpolation. To find the interpolated pressure p int at height h,wecanusethe pressure and height of layer i (h i and p i, respectively) and calculate p int according to Wallace and Hobbs [2006] ð p int ¼ p i exp h h iþg m ð20þ R d T v where T v is the virtual temperature, which can be obtained from the following equation [Wallace and Hobbs, 2006] T:p T v ¼ p 1 M v M d : ð21þ e [51] The virtual temperature is the temperature that dry air at pressure p would have when its density is equal to that of moist air at temperature T and pressure p [Haltnier and Martin, 1957; Kleijer, 2004]. M v and M d represent the molecular weight of water and dry air, respectively. Gravity in equation (20) can be calculated using different formulae. For g m we can use a formula, which is a function of latitude and height, for more details see Kraus [2004]. The interpolation of water vapor pressure is performed using the formula [Böhm and Schuh, 2003] e int ¼ e i exp h int h i ð22þ C where C ¼ h iþ1 h i log e ð23þ iþ1 e i and (i+1) and (i) refer to the levels above and below the point of interest (int). [52] NWM can provide data only up to a certain pressure level. For ray-tracing computations it is necessary to consider refractivity values above this maximum height. For this purpose standard atmosphere models play an important role and can be used as reasonable supplementary data sets. In this step the accuracy of the results will mainly depend on four important factors: [53] Extrapolation technique: For the extrapolation the same methods as those for the interpolation can be applied, but it is worth to mention that in this part of the troposphere the humidity components of air disappear and therefore water vapor pressure will be equal to zero. This makes calculations simpler for the upper part of the troposphere. [54] Height of the uppermost pressure level: The height of the uppermost level is dependent on the numerical weather model and can vary between e.g., 50 and 0.1 hpa. For a comparison, showing the effect of these different heights on final results see Urquhart [2011]. [55] Upper limit of the troposphere: The upper limit of the troposphere defines the end point of the extrapolation. The selection of this value implicitly can control computational efficiency and accuracy of the ray-tracing system. An optimum selection for this parameter can reduce the number of vertically interpolated layers, and therefore the computation time. In theory, this point is defined as where the air density is practically equal to zero or, in other words, the refractive index is approximately equal to one. In this study we have considered three different heights for the upper limit of the troposphere. The computed slant delays are compared with those using an upper limit of 100 km. We have used 5 as lowest elevation angle and 86, 76 and 66 km as upper limits of the troposphere. The maximum differences are 0.09, 0.2 and 1.1 mm, respectively. We have chosen 76 km as the upper limit of the troposphere, which can impose a maximum error of 0.04 mm on station height. [56] Optimum number of height levels: The number of height levels chosen in ray-tracing systems controls both the accuracy of the ray-traced delay and the computational speed. Therefore, a proper choice of the interpolated heights can significantly impact the overall performance of the raytracer. Rocken et al. [2001] suggested an integration step size of 10 m between 0 and 2 km, 20 m between 2 and 6 km, 50 m between 6 and 16 km, 100 m between 16 and 36 km and 500 m between 36 and 136 km. They confirmed that this selection yields nearly identical results (sub-millimeter agreement at 1 elevation) to a 5 m integration for the entire height range. Urquhart [2011] utilized these integration step sizes along the raypath, whereas Böhm and Schuh [2003] used them for vertical profiles. Nievinski [2009] employed an adaptive scheme by which the spacing is smaller or larger depending on the smoothness of the refractivity field. Hobiger et al. [2008] proposed an iterative procedure to estimate the number of height levels. He suggested generating 320 layers from each data set. This enables achieving millimeter-level accuracy in ray-tracing even at very low elevation angles. [57] Differences between slant delays using a constant 10-m step and the one suggested by Rocken et al. [2001] for a sample data set are shown in Figure 8. Maximum discrepancies are seen as expected for low elevation angles. For elevation angles above 10, these differences are practically equal to zero. It is important to mention that the number of 7of17

8 Figure 8. Differences between slant delays using constant step sizes of 10 m and those suggested by Rocken et al. [2001] for station Tsukuba, August 12th, vertically interpolated and extrapolated layers using Rocken et al. s [2001] suggestion is 872 whereas using constant increments the number increases to This illustrates the disadvantages of the constant increments from a computational efficiency point of view. In this case both the pre-calculation and the main ray-tracing computations will be very time consuming without any considerable improvement of the final results. Maximum absolute and mean values of differences between slant delays of these two selections are 0.7 mm and 0.08 mm respectively for an elevation angle of 5, which corresponds to a station height error of about 0.1 mm Time Interpolation [58] For geodetic applications we need to compute the total delay for each observation epoch. The observation rate is dependent on the space geodetic technique. Since NWM provide data every 3 or 6 h, time interpolation between two epochs has to be carried out. Weighted averaging (linear interpolation) can be used to calculate the meteorological parameters for the time of interest Solutions of the Partial Differential Equations [59] As mentioned before, finding solutions of the partial differential equations is an important part of solving the Eikonal equation. In fact, the Eikonal equation is nothing but a system of equations containing seven (in 3D case) or five (in 2D case) differential equations. [60] Runge Kutta methods, as an important family of implicit and explicit methods for the approximation of solutions of ordinary differential equations, can be used for this task. Equations (A9) to (A14) (see Appendix A) show the initial values for the unknowns, which are the coordinates (r, q, l) of each point along the raypath in a specific coordinate system (which is in our case a spherical coordinate system), and the elements of the ray direction in these points (L r, L q, L l ). [61] To find the components of the gradient, which are needed for the Eikonal equation, different methods may be used. One possibility is presented by Hobiger et al. [2008], where four points around the point of interest in the above and below layers are involved and a bilinear interpolation is performed to find the final result. This method can be improved if we use more than four points in a layer and a Figure 9. Horizontal gradient values (sum of the two horizontal components of the gradient) in mm for stations Wettzell (red) and Tsukuba (black). different interpolation method. In this paper we have used 8 points along the meridian and a spline interpolation. This choice is based on our investigation shown in the subsection on horizontal interpolation. However, this part of the work can be reduced considerably if we restrict our calculations only to the 2D case, where the horizontal components of the gradient are set to zero Gradient Components of the Refractive Index [62] An important question is whether a 2D or 3D raytracing algorithm should be used. We know that 2D methods are faster and therefore more efficient than 3D methods in terms of time. In this subsection we discuss this challenge from different viewpoints and try to quantify these effects. Figure 9 shows horizontal gradient values for stations Wettzell and Tsukuba. [63] The most important difference between a 2D and 3D ray-tracing method is the negligence of out of plane components of the trajectory in the 2D case. In other words, in the 2D case the ray is not allowed to leave the plane of constant azimuth. In fact, we have assumed azimuthal Figure 10. Delay differences between 2D and 3D at 5 elevation for station Wettzell at epoch of first available ECMWF data set on August 13th Red: Difference in total delay, black: Difference in total delay without geometric delay, blue: Difference in geometric delay. 8of17

9 Figure 11. Delay differences between 2D and 3D at 5 elevation for station Tsukuba at epoch of first available ECMWF data set on August 13th Red: Difference in total delay, black: Difference in total delay without geometric delay, blue: Difference in geometric delay. asymmetry of the troposphere but ignored the effect of gradients, which is due to the out of plane components. Figures 10 and 11 show the differences between 2D and 3D ray-traced delays for the VLBI stations Tsukuba and Wettzell, respectively. In Table 2, these differences are summarized in terms of mean and maximum absolute values for nine VLBI stations participating in CONT08. [64] As by-products of our ray-tracing system we can calculate the refractivity gradient components and therefore find the effect of these components along the raypath (for i = 1 to m vertical layers) as follows DSrðr;l;qÞ ¼ m X nðr; l; qþ 1 nðr; l; qþ þ r r q i i i¼1 1 nðr; l; qþ þ dsi : r sin q l ð24þ i [65] In Figure 9, the last two terms in equation (24) are referred to as gradient. We see a very similar pattern for geometric delay differences and gradient effects. Gu and Brunner [1990] have developed a formula, which shows analytically the dependency of the curvature (geometric delay) on the gradient components. Their original equation is developed for refractivity in a dispersive medium, which depends on frequency in addition to position. For the troposphere, it is Figure 12. Differences between 3D and 2D ray-traced delays with respect to the elevation angle, calculated for all observations of CONT08. possible to modify this equation to be a function of refractive index in a non-dispersive medium [Wijaya, 2010]. Since slant delays usually increase at lower elevation angles, also the difference between 2D and 3D ray-traced delays is largest close to the horizon. To illustrate this difference with respect to elevation angle, we calculated 2D and 3D ray-traced delays for all CONT08 observations, and the difference is shown in Figure Application in VLBI Analysis [66] The principle observable in geodetic VLBI is the difference in arrival times at two or more radio telescopes of signals emitted by distant extragalactic radio sources. Due to the very long baselines (up to one diameter of the Earth) between the telescopes, the delay in the neutral atmosphere differs from station to station significantly, and an error in the modeled delay enters into geodetic parameters like baseline lengths or Earth orientation parameters. Recent studies based on turbulence [e.g., Pany et al., 2011; Nilsson and Haas, 2010] have shown that troposphere delay modeling is and will continue to be the most critical error source in the analysis of VLBI observations. However, compared to observations with the Global Navigation Satellite Systems (GNSS), VLBI is very well suited for the accuracy assessment of ray-traced delays at low elevation angles because it is not subject to effects like phase center variations or multipath. Table 2. Differences Between 2D and 3D Ray-Traced Delays (2D Minus 3D) in mm for Nine CONT08 Stations Computed From the First Available ECMWF Data Each Day From August 12th 16th, 2008 at 5 Elevation VLBI Station Mean (Hyd. + Non-hyd.) Maximum Absolute (Hyd. + Non-hyd.) Mean (Bending Only) Maximum Absolute (Bending Only) Tsukuba Wettzell HartRAO Kokee Medicina Onsala TIGOCONC Westford Zelenchk of 17

10 Table 3. Details of the Standard Ray-Tracing System Parameter Option Horizontal interpolation spline Number of points around 8 points, along the meridian Refractivity constants Rüeger best average Horizontal resolution 0.5 of data set Type 2D Supplementary atmosphere U.S.76 Radius of curvature Gaussian curvature of the Earth Gravity National Imagery and Mapping Agency [2000] Vertical interpolation (extrapolation) temperature and specific humidity: linear, pressure: logarithmically Height of upper limit of the 76 Km troposphere zenith delays but not estimating gradients, and (3) estimating neither zenith delays nor gradients. The setup of the raytracing system used in this investigation (called standard version ) is summarized in Table 3. [71] In general, if gradients and residual zenith delays are estimated, the application of ray-traced delays yields similar 5.1. CONT08 Baseline Length Repeatabilities [67] Ray-traced tropospheric delays are included in the VLBI analysis of the CONT08 experiment. [68] For this validation we used the Vienna VLBI Software (VieVS) [Böhm et al., 2011], which has been modified to read external ray-traced delays. Station coordinates were estimated using the No Net Translation (NNT) and the No Net Rotation (NNR) condition on VTRF2008 [Böckmann et al., 2010]. Atmospheric loading [Petrov and Boy, 2004] and tidal ocean loading [Scherneck, 1991] based on the ocean model FES2004 [Lyard et al., 2006] were considered. Source coordinates were fixed to ICRF2. Five constant Earth orientation parameters were estimated in addition to IERS EOP 05 C04 [Gambis, 2004; Bizouard and Gambis, 2009]. For a quality assessment we use baseline length repeatabilities, i.e., standard deviations of baseline lengths from the 15 CONT08 sessions. [69] The results are compared to those of a standard approach where a priori total delays are set up as the sum of hydrostatic and wet slant delays, each of them being the product of the zenith delay derived from data of the ECMWF and the respective mapping function (VMF1) [Böhm et al., 2006a]. Both models, ray-tracing and ECMWF/VMF1, include the wet part, and, if residual zenith delays are estimated, the wet mapping function is used as partial derivative. [70] We have analyzed three approaches: (1) Estimating zenith delays and gradients in the analysis, (2) estimating Figure 13. Baseline length repeatabilities using ray-traced delays (red) and ECMWF/VMF1 (black); gradients and zenith wet delays are estimated in both cases. Figure 14. Baseline length repeatability differences for various baselines: (a) Gradients and zenith delays estimated, (b) no gradients but zenith delays estimated, and (c) no gradients and no zenith delays estimated. Red: Tsukuba-baselines, black: Wettzell-baselines, blue: TIGOCONC-baselines, green: Tsukuba-Wettzell, orange: Wettzell-TIGOCONC, yellow: Tsukuba-TIGOCONC. Positive differences show improvements for ray-tracing w.r.t the approach using the Vienna Mapping Function 1. Mind that the vertical scale is different in each of the three plots. 10 of 17

11 Table 4. Mean Improvement of Baseline Length Repeatabilities Using Ray-Traced Delays With Respect to ECMWF/VMF1 and Percentage of Improved Baselines in Brackets a Case Model Zenith Delays, Gradients (mm) Zenith Delays, No Gradients (mm) No Gradients, No ZWD (mm) 1 Spline, 8 points, 0.5 resolution, 3D 0.07 (45%) 1.38 (84%) (100%) 2 Spline, 4 points, 0.5 resolution, 2D 0.10 (47%) 1.35 (84%) (100%) 3 Spline, 8 points, 0.5 resolution, 2D 0.10 (45%) 1.35 (82%) (100%) 4 Bilinear,4 points, 0.5 resolution, 2D 0.13 (44%) 1.32 (82%) (100%) 5 Spline, 16 points, 0.5 resolution, 2D 0.14 (45%) 1.31 (82%) (100%) 6 Spline, 8 points, 1 resolution, 2D 0.17 (45%) 1.29 (84%) (100%) 7 Spline, 4 points, 1 resolution, 2D 0.24 (40%) 1.21 (82%) (100%) 8 Nearest, 0.1 resolution, 2D 031 (40%) 1.14 (78%) (100%) 9 WM, 4 points, 0.5 resolution, 2D 0.47 (20%) 0.98 (76%) (100%) 10 Spline, 8 points, 0.1 resolution, 2D 0.51 (33%) 0.95 (80%) (100%) 11 Nearest, 0.5 resolution, 2D 0.80 (29%) 0.65 (78%) (100%) a For this comparison three different cases (gradients and ZWD, no gradients, neither gradients nor ZWD were estimated) are considered. Rüeger best average constants are used for the refractivity estimation. baseline length repeatabilities as the standard approach with mapping functions (Figure 13). It should be mentioned here that the estimation of gradients also accounts for other azimuth-dependent non-tropospheric error sources (e.g., systematic cable stretching in case of VLBI; multipath and unmodeled phase center variations in case of GNSS). [72] However, taking a closer look at the stations, raytraced delays provide better tropospheric corrections at some stations, whereas at other stations the corrections are worse. The dependency on particular baselines is shown in Figure 14, where the results for three stations (Tsukuba, Wettzell, and TIGOCONC) are highlighted. According to Figure 14, it can be clearly seen that ray-traced delays degrade baseline length repeatabilities for baselines with Tsukuba if zenith delays and gradients are estimated. On the other hand, baselines including TIGOCONC are mostly improved. Furthermore, for the two other approaches where gradients and/or zenith delays are not estimated, we can find similar clusters of these three stations. Figure 15. Mean improvement of baseline length repeatabilities using ray-traced delays and different refractivity constants, with respect to ECMWF/VMF1. 1, Zhevakin Naumov; 2, Boudouris; 3, IUGG; 4, Thayer; 5, Rüeger best average; 6, Rüeger best available; 7, Smith Weintraub; 8, Essen Froome; 9, Bevis. For this comparison both gradients and ZWD are estimated. The resolution of the data set is 0.5 and the spline method with 8 points is used for horizontal interpolation. [73] The advantages and possibilities of the ray-tracing method can be seen in those cases where no additional gradients are estimated. On average we can find an improvement of more than 1.2 mm in baseline length repeatabilities, which reaches more than 25 mm if no residual zenith delays are estimated. For more details, we refer to Nafisi et al. [2011]. [74] Next we present statistical values, showing how important the selection of elements of a ray-tracing system is. The changing elements and their effects on baseline length repeatabilities are shown in Table 4. Again, baseline length repeatabilities are estimated from the analyses of the VLBI CONT08 campaign. Figure 15 shows the effect of different refractivity constants on the final results. Since there are different constants for the hydrostatic and wet refractivity involved, the ray-traced delays derived with different sets of coefficients cannot be transformed to each other by a single scaling factor. [75] Although differences and improvements are mostly small and not significant, we sum up the main points and statements according to the above mentioned results as follows: [76] - 3D methods show a better performance compared to 2D. Most likely it is because of the out of plane components, which are ignored in all 2D methods. On the other hand a 3D ray-tracing program is more time consuming, i.e., by a factor of about ten in our implementation. [77] - Regarding horizontal interpolation methods, results from spline methods are better than bilinear and weighted mean interpolation. These results support our findings from section 4.1, where we discussed different horizontal interpolation methods. The weighted mean method is the fastest but yields worse results. [78] - In general, results from 0.5 resolution data sets are more accurate. Only when we use the nearest neighborhood method for horizontal interpolation, 0.1 data sets should be Table 5. Number of Scans for Various Cases Minimum Maximum Mean Tsukuba + Wettzell (+Ny Ålesund) Wettzell + Kokee Park Tsukuba + Westford of 17

12 Figure 16. Differences in DUT1 estimates in ms for Intensive sessions including Tsukuba or Wettzell (for Tsukuba and Wettzell ray-traced delays are applied). All estimates below 0.05 ms and above 0.10 ms are removed as outliers. Blue crosses: Surface pressure model minus ECMWF/ VMF1, red plus signs: Ray-tracing minus ECMWF/VMF1. used instead of 0.5. It should be mentioned that the size of the data sets also affects the efficiency of processing. For example, in a pre-calculation step of our method, meteorological parameters from gridded ECMWF data sets are converted to refractivity index profiles and saved in a binary Matlab-internal format. The file size for one epoch is on average 6 and 127 MB for resolutions 0.5 and 0.1, respectively. This means that during the main processing step more memory is needed for loading the high resolution profiles and the computation time increases. [79] - Regarding the no gradient case, 76% of all baselines are improved when using ray-traced delays instead of the standard approach ECMWF/VMF1. The average improvement over all baselines is 1.18 mm. [80] - In the no gradient, no zenith delay case, improvements compared to ECMWF/VMF1 can be seen for all baselines, on average by mm. However, the repeatability for both methods (ECMWF/VMF1 and ray-tracing) is significantly degraded compared to the no gradient case. [81] - Furthermore, we want to mention that even in the case of applying ray-traced delays and estimating zenith delays, the additional estimation of gradients improves baseline length repeatabilities significantly (not shown here) Universal Time (UT1) Estimation From Intensive Sessions [82] Another test for the validity of ray-traced delays is the estimation of UT1-UTC from VLBI Intensive sessions. Since usually only two or three stations are observing for one hour in these sessions, they are very sensitive to asymmetries in the troposphere delays around the stations, because those asymmetries are not averaged out over time and over stations as is the case with 24-h sessions containing five or more stations. Böhm et al. [2010] found improvement in UT1 values for ray-traced delays at Tsukuba that were determined from data of the Japan Meteorological Agency (JMA). [83] The IVS Intensive sessions are usually separated into three groups: (1) Intensive 1 (INT1) sessions operated from Monday through Friday at 18:30, where Kokee Park and Wettzell are involved. Tsukuba and Wettzell observe together in (2) Intensive 2 (INT2) sessions, which run on Saturday and Sunday at 7:30. Finally there are (3) Intensive 3 (INT3) sessions every Monday at 7:00 containing Tsukuba, Wettzell and Ny Ålesund. [84] For our investigation, we used Intensive sessions containing Tsukuba and/or Wettzell from July 15th, 2010 to March 30th, 2011 to estimate UT1-UTC (DUT1). To correct the tropospheric delay, three different strategies are applied: ray-traced delays as derived from data of the ECMWF, ECMWF/VMF1 delays, and delays estimated from surface pressure measurements and the model of Saastamoinen [1972]. For the latter we used the VMF1 as mapping functions. We estimated one linear clock function and constant zenith delays at each station. Again we used Lagrange interpolated IERS EOP 05 C04 model values as a priori values for the Earth orientation parameters plus the IERS model accounting for ocean tidal effects on polar motion and DUT1. Station coordinates, gradients and other Earth orientation parameters (apart from DUT1) were fixed in our solutions. We analyzed 29 Intensive sessions on the baseline Tsukuba-Wettzell, and 147 Intensive sessions which contain either Tsukuba (together with Westford) or Wettzell (together with Kokee Park). The number of scans for these three cases is shown in Table 5. Ray-traced delays were available for stations Wettzell and Tsukuba. Figure 16 shows the differences of DUT1 estimates in ms w.r.t. a priori models. [85] According to the results presented in section 5.1, where we have investigated the station-dependency of baseline length repeatabilities, we expected the results for DUT1 estimation for Wettzell-Kokee sessions to be better than for Tsukuba-Westford sessions. However, Table 6 shows a different behavior. Table 6 shows statistics (standard deviation and mean formal uncertainties) for estimated DUT1 values. A potential reason for this is the number of scans for Wettzell-Kokee sessions, which is considerably smaller than the one of Tsukuba-Westford sessions (Table 5). Both the Tsukuba Westford sessions and the Tsukuba Wettzell sessions show smaller standard deviations of DUT1 values and a larger number of scans per session. Furthermore it should be pointed out that for Tsukuba-and-Wettzell sessions where ray-traced delays are Table 6. Standard Deviation (Std) and Mean Formal Uncertainty (Mean Error) of DUT1 Estimation in ms Tsukuba + Wettzell (29 Sessions) Wettzell + Kokee (125 Sessions) Tsukuba + Westford (22 Sessions) Std Mean Error Std Mean Error Std Mean Error Surface pressure model ECMWF/VMF Ray-tracing of 17