Co-location of space geodetic techniques carried out at the Geodetic Observatory Wettzell using a closure in time and a multi-technique reference target

Abstract

The quality of the links between the different space geodetic techniques (VLBI, SLR, GNSS and DORIS) is still one of the major limiting factors for the realization of a unique global terrestrial reference frame that is accurate enough to allow the monitoring of the Earth system, i.e., of processes like sea level change, postglacial rebound and silent earthquakes. According to the specifications of the global geodetic observing system of the International Association of Geodesy, such a reference frame should be accurate to 1 mm over decades, with rates of change stable at the level of 0.1 mm/year. The deficiencies arise from inaccurate or incomplete local ties at many fundamental sites as well as from systematic instrumental biases in the individual space geodetic techniques. Frequently repeated surveys, the continuous monitoring of antenna heights and the geometrical mount stability (Lösler et al. in J Geod 90:467–486, 2016. https://doi.org/10.1007/s00190-016-0887-8) have not provided evidence for insufficient antenna stability. Therefore, we have investigated variations in the respective system delays caused by electronic circuits, which is not adequately captured by the calibration process, either because of subtle differences in the circuitry between geodetic measurement and calibration, high temporal variability or because of lacking resolving bandwidth. The measured system delay variations in the electric chain of both VLBI- and SLR systems reach the order of 100 ps, which is equivalent to 3 cm of path length. Most of this variability is usually removed by the calibrations but by far not all. This paper focuses on the development of new technologies and procedures for co-located geodetic instrumentation in order to identify and remove systematic measurement biases within and between the individual measurement techniques. A closed-loop optical time and frequency distribution system and a common inter-technique reference target provide the possibility to remove variable system delays. The main motivation for the newly established central reference target, locked to the station clock, is the combination of all space geodetic instruments at a single reference point at the observatory. On top of that it provides the unique capability to perform a closure measurement based on the observation of time.

Appendices

Appendix 1: Calibration of the SLR Systems

Here we deal with the SLR station calibration using the WTCO target and the comparison with the local tie coordinates. We compare the difference in range between the WTCO target and a known external target. The range measurements are corrected by the atmosphere group index of refraction ($n_\mathrm{gr}$) computed according to the Marini–Murray model (Marini and Murray 1973) and by the retro-reflector delay correction, which is computed according to

$$\begin{aligned} \mathrm{RR} = \frac{\left( 1-n_\mathrm{RR}\right) }{c} L_\mathrm{RR}, \end{aligned}$$

(1)

where $L_\mathrm{RR}$ is the length of the retro-reflector, $n_\mathrm{RR}$ is the group index of refraction of the retro-reflector and c is the speed of light. The range difference is then compared with the distance computed from the local tie coordinates and is named r. The range difference obtained from the SLR measurements is then the estimated value of the local ties distance and is named ${\widehat{r}}$.

1.1 Calibration of the WLRS system

The SLR station WLRS is calibrated according to the calibration scheme shown in Fig. 18. Since the system is equipped with a monostatic mount, it has a common telescope for the transmit and receive path of the laser pulses. They are mechanically separated by a transmit/receive (T/R) switch, and for the routine SLR observations a real-time calibration procedure is applied. The light leakage of the transmitted laser pulses, taken at a turning mirror, is injected into the receiving path. In this way the most sensitive delays from the transmit and receive electronics ($\tau _\mathrm{start}$, $\tau _\mathrm{stop}$ ) are calibrated. It is important to note that both delays are always calibrated jointly together and they cannot be separated into a transmit and receive path component.

Fig. 18

Calibration schema of WLRS station, comparison with local tie

The WLRS range measurements of the WTCO target can be compared to the range obtained from the local ties. Since the laser beam propagates through the same transmit and receive telescope, a retro-reflector can be applied to reflect the outgoing laser pulses back to the receive path of the telescope. While the T/R switch blocks the transmitted light from damaging the sensitive detector (SPAD), we have modified the T/R switch to allow a tiny percentage of the laser beam to bypass the beam block. We compare the range differences from the WTCO against the external target (ET). Since we are differencing two measurements, there is a part of the calibration path common for both targets. This path is marked in blue color in Fig. 18 and will cancel when the range difference is measured. The actually measured range is marked in red color. From the coordinates of the reference point we can compute the length of the vector connecting the invariant point of the WLRS with the calibration target, ($L_\mathrm{ET}$), and the length of the vector connecting the WLRS with the WTCO reference point ($L_\mathrm{WTCO}$). Since it is not possible to range to the external retro at normal incidence, due to the obscuration caused by the secondary mirror, we must consider the azimuth and elevation offsets between the optical axis of the telescope and the retro-reflector reference point, as it is indicated in Fig. 18. For simplicity we show the azimuth angle only. The azimuth and elevation angles were computed from the telescope pointing, the reference point and the local ties coordinates. The azimuth and elevation angles for external target turned out as $\varphi _\mathrm{ET} = 3.970^{\circ }$ and $\zeta _\mathrm{ET} = 1.905^{\circ }$. For the WTCO target they are $\varphi _\mathrm{WTCO} = 0.014^{\circ }$ and $\zeta _\mathrm{ET} = 1.102^{\circ }$, respectively. The range obtained from the local ties is

$$\begin{aligned} \begin{aligned} r&= r_\mathrm{ET} - r_\mathrm{WTCO}\\&=L_\mathrm{ET} \cos (\varphi _\mathrm{ET}) \cos (\zeta _\mathrm{ET})\\&-L_\mathrm{WTCO} \cos (\varphi _\mathrm{WTCO}) \cos (\zeta _\mathrm{WTCO})\\&= 10.2978-2.4185 = 7.8793~\text {m}. \end{aligned} \end{aligned}$$

(2)

This range value is compared with the ranges obtained from the SLR measurements. They were corrected for the atmospheric index of refraction and the index of refraction of the targets retro-reflector. Both retro-reflectors (ET and WTCO) are made of the same glass BK7 with the group index of refraction of $n_\mathrm{RR} = 1.5206$ for the measurement wavelength $\lambda = 1.064~\mu$m, which is used for the WLRS calibration. The retro-reflector delay of the WLRS external target contributed by as much as $-\,102$ and $-\,69.4$ ps for WTCO, respectively. The measured range difference is then finally estimated to

$$\begin{aligned} {\widehat{r}}= & {} \frac{c}{n_\mathrm{gr}} \left( {\widehat{r}}_\mathrm{WTCO} - {\widehat{r}}_\mathrm{ET} \right) + c \left( \hbox {RR}_\mathrm{WTCO} - \hbox {RR}_\mathrm{ET}\right) . \end{aligned}$$

(3)

Fig. 19

Calibration scheme of SOS-W station. The common calibration path for calibrating WTCO and internal retro-reflector is marked in green. The calibration path measured by SLR and computed from local ties is marked in red

1.2 Calibration of SOS-W system

The SLR station SOS-W must be calibrated according to a different scheme, because the system has a bistatical mount. It uses a separate transmit and receive telescope. Due to the parallax between the transmit and receive telescope we cannot return laser pulses from the transmit to receive path by a single retro-reflector. Therefore, we are using the transmit telescope also for the detection of the laser pulses. The ranging to the WTCO target is compared to the local tie measurement as the range difference between WTCO and the retro-reflector installed inside the transmit telescope. Since the distance of the internal retro $L_{\mathrm{Tx}}$ is not well enough known, it has been established from dedicated measurements, which are shown in Fig. 19a. For that purpose the external calibration target has been employed. While this target is designed to feedback optical pulses from the transmit to the receive telescope, it had to be modified in such a way that a reflective plate was placed in front of the target. From the local tie coordinates (Table 1), the distance of the optical axis of the transmit telescope $T_{x_{L}} = 350$ mm and its depth ($\hbox {ET}_\mathrm{depth} = 0.1735$ m), we can compute a distance from the elevation axis to the front of the external target $h_{\mathrm{Tx}}$ as,

$$\begin{aligned} h_{\mathrm{Tx}} = \sqrt{ L_\mathrm{ET}^2 - \hbox {ET}_L^2 - T_{X_L}^2} - \hbox {ET}_\mathrm{depth} = 2.0115~\text {m}, \end{aligned}$$

(4)

where $L_\mathrm{ET}$ is the length of vector connecting invariant point of the telescope and the external target. The position of the internal transmit telescope retro-reflector is computed according to

$$\begin{aligned} {\widehat{L}}_{\mathrm{Tx}} = h_{\mathrm{Tx}} - \frac{{\widehat{r}}_{hs} c }{n_\mathrm{gr}} = 0.0548~\text {m}, \end{aligned}$$

(5)

where ${\widehat{r}}_{hs}$ is the measured range difference between the front of the external target and the range to the internal retro. We do not need to take into account the index of refraction of the lenses installed in the transmit telescope, since we will use $L_{\mathrm{Tx}}$ to establish range differences, where the refractive index will cancel out. The range ${\widehat{r}}_{hs}$ was measured in several independent experiments performed on different days, and the mean of the range value was used for the calculation. The range measurements of the SOS-W to the WTCO target correspond to the range difference of WTCO and the transmit telescope internal retro, see Fig. 19b. The common calibration path is marked in blue color again. From the local ties one can compute the range, which we obtain as

$$\begin{aligned} r = h_\mathrm{WTCO} - {\widehat{L}}_{\mathrm{Tx}} = 65.8793~\text {m}, \end{aligned}$$

(6)

where the distance $h_\mathrm{WTCO}$ is computed from the local ties coordinates of SOS-W and WTCO. The range from WTCO to the telescope internal retro is estimated from the SOS-W measurements according to

$$\begin{aligned} {\widehat{r}} = \frac{c}{n_\mathrm{gr}} \left( {\widehat{r}}_\mathrm{WTCO} - {\widehat{r}}_{{\mathrm{IT}}} \right) + c \hbox {RR}~ , \end{aligned}$$

(7)

where ${\widehat{r}}_\mathrm{WTCO} - {\widehat{r}}_{{\mathrm{IT}}}$ is the range difference between WTCO and the transmit telescope internal retro, $\mathrm{RR} = -\,69.9$ ps is the WTCO retro-reflector correction computed according to (1), where $n_\mathrm{CCR} = 1.5249$ for the SOS-W wavelength $\lambda = 850$ nm.

Appendix 2: Calibration of the VLBI System

In this section we will deal with the VLBI delay model with respect to the calibration by the WTCO target, which was inspired by Thomas (1978). We will discuss the required power level of the emitted calibration signal and while the VLBI antenna is not irradiated from the far field, we will show that an unambiguous signal propagation path exists.

1.1 VLBI delay model

The schematic of the delay model is shown in Fig. 20.

Fig. 20

Schematic of the VLBI delay model; ADC denotes analog to digital converter

The following definitions of some symbols have been used in the analysis:

t :

true time,

$t_\mathrm{c}$ :

station time,

$\omega _{0}$ :

true oscillator frequency (in terms of true time); reference frequency of the H-maser,

$\omega _\mathrm{c}$ :

nominal oscillator frequency in the observatory; reference frequency of the H-maser.

The nominal frequency is related to the true frequency by

$$\begin{aligned} \omega _\mathrm{c} = \omega _{0} + \Delta \omega _\mathrm{c}, \end{aligned}$$

(8)

where $\Delta \omega _\mathrm{c}$ is the frequency offset and the station time is related to the true time by

$$\begin{aligned} t_\mathrm{c} = t + \tau _\mathrm{c}, \end{aligned}$$

(9)

where $\tau _\mathrm{c}$ is a clock error. Since we are analyzing a digitized signal, it is useful to define bit time $t_\mathrm{b}$, which is marking the bits in the recording medium. While the recording system (ADC) is connected with the H-maser through some frequency distribution, the recorder is delayed with respect to the station time by the delay $\tau _\mathrm{r}$, or $t_\mathrm{b} = t_\mathrm{c} - \tau _\mathrm{r} = t + \tau _\mathrm{c} - \tau _\mathrm{r}$. The delay reference point is defined as an output from a frequency standard (H-maser in our case). The phase of the reference frequency signal at the reference point is given in terms of true time and the nominal frequency by

$$\begin{aligned} \phi _\mathrm{c} = \omega _\mathrm{c} (t+\tau _\mathrm{c}) . \end{aligned}$$

(10)

The reference phase is chosen in this way since the true time t is the common scale for all stations. The clock error $\tau _\mathrm{c}$ is explicitly shown, and the calibrating signal frequency generated by the phase calibration (pcal) instrument is calculated as the nominal frequency.

In the calibration scheme we are using two tone signal generators, which are injected into the receiver chain. They are generating a set of tones, which we are calling pcal ($\hbox {pcal}_1$ and $\hbox {pcal}_2$). The $\hbox {pcal}_1$ generator is generating a signal from the nominal frequency, whose phase is delayed by the cable delay $\tau _u$ and the delay inside the pcal electronics $\tau _\mathrm{pcal1}$

$$\begin{aligned} \phi _n = \omega _n (t + \tau _\mathrm{c} - \tau _u + \tau _\mathrm{pcal1}), \end{aligned}$$

(11)

which is in radians. Since the unit radians is hard to visualize, we can divide the phase by $\omega _n$ to obtain the phase in units of time. Therefore, in all the figures we are using $\phi _n$ in the units of ps. The effects on the phase of all receiver chain components can be divided into three major categories. All group delays, including the effective group delay through the filters, will be represented by a total delay $\tau _{I}$. All phase shifts (excepts those from the group delays) will be represented by a total shift $\phi _{n}(\omega _{n})$. The overall effect of the mixing signals can be described as one combined mixing signal, which is represented by $\Psi _h = \omega _h t + \phi _h$, where $\omega _h$ is the total mixing frequency and $\phi _h$ is the phase shift in the mixing process. The overall phase of the $\hbox {pcal}_1$ tones digitized at time $t_\mathrm{b}$ using the ADC is

$$\begin{aligned} \phi _n^{p1}(t_\mathrm{b})= & {} \left( \omega _n-\omega _h \right) t + \omega _n \left( \tau _{t1} - \tau _{I} \right) \nonumber \\&-\, \phi _I(\omega _n) + \phi _h, \end{aligned}$$

(12)

where $\tau _{t1} = \tau _\mathrm{c} - \tau _\mathrm{r} - \tau _u-\tau _\mathrm{pcal1}$. In the same way we can express the relation of the second $\hbox {pcal}_2$ system, which is installed at the WTCO target. The tones are generated from the same H-maser, which has a phase $\phi _\mathrm{c} = \omega _\mathrm{c} (t + \tau _\mathrm{c} - \tau _\mathrm{wtco})$. The overall phase of the digitized $\hbox {pcal}_2$ tones generated from the target is

$$\begin{aligned} \phi _n^{p2}(t_\mathrm{b})= & {} \left( \omega _n-\omega _h \right) t \nonumber \\&+\, \omega _n \left( \tau _{t2} - \tau _{I} \right) - \phi _I(\omega _n) + \phi _h, \end{aligned}$$

(13)

where $\tau _{t2} = \tau _\mathrm{c} - \tau _\mathrm{r} - \tau _\mathrm{wtco} - \tau _\mathrm{pcal1} - \tau _\mathrm{FS}$ and $\tau _\mathrm{FS}$ is the free space signal delay. It is proportional to the local ties.

The goal of the WTCO target calibration is the comparison of the free space signal delay $\tau _\mathrm{FS}$ against the survey and the control of the instrumental delay. Therefore, we compute the phase difference of the tones generated in the antenna and at the WTCO target:

$$\begin{aligned} \Delta \phi _n(t_\mathrm{b})&= \phi _n^{p2} - \phi _n^{p1} \nonumber \\&= -\omega _n\tau _\mathrm{FS}+\omega _n \left( \tau _u-\tau _\mathrm{wtco} +\tau _\mathrm{pcal1} - \tau _\mathrm{pcal2} \right) .\nonumber \\ \end{aligned}$$

(14)

One can see that the recorded phase difference is directly proportional to the free space delay (i.e., local ties) and is biased by the delay difference accumulated in the distribution of the reference frequency and the delays of the pcal generators.

In the current installation we could not stabilize the delay $\tau _\mathrm{wtco}$ and $\tau _\mathrm{pcal2}$; therefore, we used two radio telescopes (RTW and TTW1) to build up the phase difference. TTW1 was recently put into routine operation (Schüler et al. 2015); however, it was not connected to the same frequency reference as the RTW. Therefore, TTW1 received a signal generated from another H-maser. Due to the high stability of the H-masers involved, we can treat this situation as a signal delay in TTW1. It amounts to $\tau _\mathrm{TTWO}$ at short time scales with respect to the RTW. Therefore, we can write $\tau _\mathrm{c}^\mathrm{TTW1} = \tau _\mathrm{c}^\mathrm{RTW} + \tau _\mathrm{TWOTT}$ and the phase in TTW1 is given by

$$\begin{aligned} \Delta \phi _n^\mathrm{TTW1}(t_\mathrm{b})&= -\omega _n\tau _\mathrm{FS}\nonumber \\&\quad +\,\omega _n \left( \tau _u-\tau _\mathrm{wtco} \right. \nonumber \\&\quad \left. + \,\tau _\mathrm{pcal1}-\tau _\mathrm{pcal2} - \tau _\mathrm{TWOTT}\right) . \end{aligned}$$

(15)

The phase difference between RTW and TTW1 is

$$\begin{aligned} \Delta \Theta _n(t_\mathrm{b})&= \Delta \phi _n^\mathrm{TTW1} - \Delta \phi _n^\mathrm{RTW} \nonumber \\&=-\omega _n \left( \tau _\mathrm{FS}^\mathrm{TTW1} - \tau _\mathrm{FS}^\mathrm{RTW} \right) \nonumber \\&\quad +\,\omega _n \left( \tau _u^\mathrm{TTW1} - \tau _u^\mathrm{RTW} + \tau _\mathrm{pcal1}^\mathrm{TTW1}\right. \nonumber \\&\left. \quad - \,\tau _\mathrm{pcal1}^\mathrm{RTW} - \tau _\mathrm{TWOTT} \right) . \end{aligned}$$

(16)

The clock offset $\tau _\mathrm{TWOTT}$ was measured using a two-way time transfer technique (Kodet et al. 2016).

The phase difference between the radio telescopes is now no longer dependent on the delays in the WTCO target, and it is directly proportional to the local tie distance biased by the cable delay difference, which brings the pcal reference signal to the antenna feeds and the electronic delay difference in the pcal systems used in the radio telescopes.

Now we will discuss the effect of the signal strength of the generated tones with respect to the precision of the calibration. The pcal phase of a single tone is defined as $\phi _n(t_\mathrm{b})=\omega _n t + \alpha _n(t_\mathrm{b})$ and the digitized signal has the form

$$\begin{aligned} y = \sum _n{A_n} \cos {\left( \omega _n t_\mathrm{b} + \alpha (t_\mathrm{b}) \right) } + w(t_\mathrm{b}) , \end{aligned}$$

(17)

where $w(t_\mathrm{b})$ is white stationary noise with a normal distribution and a spectral power density $C_0$. In the data analysis, we are interested in the phase of the interferometer $\alpha (t_\mathrm{b})$. While the VLBI system is a quantization system (i.e., one- or two-bit sampling), the pcal power level cannot saturate the receiver chain. The variance of the extracted phase with respect to the integration time is (Hrdina 1996)

$$\begin{aligned} \sigma _{\alpha _n}^2 \simeq \frac{2C_0}{A_n^2 T}=\frac{C_0}{P_n T} . \end{aligned}$$

(18)

T is the integration time and $P_n$ is the tone power. In the experiments we set the tone power to $-\,45$ dBm in the baseband, while the noise power spectral density $C_0$ is $-\,65$ dBm/Hz. For an integration time of 1 s, the standard deviation of the extracted tones is $5^{\circ }$. Supposing that the mean measurement frequency is 8.4 GHz we will get the root mean square scatter in time to be 1.7 ps. That means, we can estimate from 1 second data the tone phase with 1.7 ps standard deviation, by averaging it is even feasible to get below 1 ps rms.

1.2 RTW geometry model of near-field propagation

Another problem is that the VLBI telescopes are parabolic antennas, which are designed to reflect plane waves into the focus plane, where the feed horn is located. Since we irradiate these antennas from a relatively short distance, we cannot consider the wave front as a plane wave. Therefore, we have analyzed what the consequences are, when the signal is generated from a point source about 100 meters away from the main parabolic reflector. At this point in time we have only looked in detail at the 20-m RTW telescope. Since the problem is rotation symmetrical, we have limited the analysis to a 2D model. The RTW has a parabolic antenna with a hyperbolic sub-reflector. The main reflector can be written as

$$\begin{aligned} z_\mathrm{r} = \frac{1}{4f_\mathrm{r}}x_\mathrm{r}^2, \quad x_\mathrm{r} \in \left[ -\frac{D}{2}, \frac{D}{2} \right] \end{aligned}$$

(19)

where $f_\mathrm{r} = 9$ m is the prime focus of the main reflector and $D = 20$ m is the diameter of the telescope. The hyperbolic sub-reflector is described by the equation

$$\begin{aligned} z_\mathrm{sub} = a_\mathrm{sub} \sqrt{1+\frac{x_\mathrm{sub}^2}{b_\mathrm{sub}^2}} + c_\mathrm{sub}, \quad x_\mathrm{r} \in \left[ -\frac{D_\mathrm{sub}}{2}, \frac{D_\mathrm{sub}}{2} \right] \end{aligned}$$

(20)

where $a_\mathrm{sub} = 2.113224,b_\mathrm{sub} = 2.436167,c_\mathrm{sub} = 5.775$ and $D_\mathrm{sub} = 2.7$ m. We have analyzed the problem using geometric wave propagation. We have integrated the entire propagation path of the microwave beam from the transmitting antenna to the antenna feed horn, which we have defined as an aperture of constant diameter. Using the analytical definition of the antenna, we can find the exact reflection points on the surface of the telescope and therefore the length of the propagation path from the transmit antenna to the RTW telescope. The spatial distortion is then defined as the difference of the longest and the shortest beam propagating path. The transmitter was modeled as a point source radiating isotropically. When the source is at infinity, the spatial distortion is zero. In our case the transmitting antenna has the coordinates $S=[S_x, S_z]$ as illustrated in Fig. 21. The beam propagates from the source S to the sub-reflector and hits the reflector at point $P_\mathrm{r} = [x_{r-i}, z_{r-i}] = [x_{r-i}, \frac{1}{4f_\mathrm{r}} x_{r-i}]$ before it is reflected from the tangent of the reflector. The angle of the reflected beam is

$$\begin{aligned} \varphi _{P_{r-\mathrm{ref}}} = \tan ^{-1} \frac{S_{z} - z_{r-i}}{S_x - x_{r-i}} + 2\tan ^{-1}\frac{1}{2f_\mathrm{r}}x_{r-i} . \end{aligned}$$

(21)

The reflected beam from the reflector can be expressed by $z_{r-r} = a_{r-r} x_{r-r} + b_{r-r}$, where $a_{r-r} = \tan ^{-1} \varphi _{P_{r-\mathrm{ref}}}$ and $b_{r-r} = z_{r-i} + \tan ^{-1} \varphi _{P_{r-\mathrm{ref}}} x_{r-i}$. This beam hits the sub-reflector at point $P_s = [x_{s-i}, z_{s-i}]$, where

$$\begin{aligned} \begin{aligned} x_{s-i}&= \frac{a_{r-r} \pm \sqrt{a_{r-r}^2 - 4a_\mathrm{sub} (f_\mathrm{sub} - b_\mathrm{sub}) }}{2a_\mathrm{sub}}, \\ z_{s-i}&= a_{r-r}x_{r-r}+b_{r-r}. \end{aligned} \end{aligned}$$

(22)

The beam is further reflected from the tangent of the sub-reflector, which is described by the equation

$$\begin{aligned} z_{\mathrm{sub-T}} = \frac{\mathrm{d}z_\mathrm{sub}}{\mathrm{d}x_\mathrm{sub}} = \frac{a_\mathrm{sub}}{b_\mathrm{sub}^2}\frac{x_\mathrm{sub}}{\sqrt{1+\frac{x_\mathrm{sub}^2}{b_\mathrm{sub}^2}}}. \end{aligned}$$

(23)

Fig. 21

Schematic of RTW geometry model of near-field propagation

The beam is reflected from sub-reflector under the angle

$$\begin{aligned} \varphi _{P_s - \mathrm{ref}} = \pi - \varphi _{P_R \mathrm{-ref}} + 2\tan ^{-1} z_{\mathrm{sub-T}}. \end{aligned}$$

(24)

In this calculation the antenna feed is modeled as an aperture with diameter of $D_{\mathrm{feed}} = 0.6$ m and we assume that all beams entering the feed horn will not suffer any further degradation of the phase relationship. The wave front propagating from the point source is spatially distorted by the different propagation paths. The RTW feed is defined as

$$\begin{aligned} z_{\mathrm{feed}} = f_{\mathrm{feed}}, \quad x_\mathrm{r} \in \left[ -\frac{D_{\mathrm{feed}}}{2}, \frac{D_{\mathrm{feed}}}{2} \right] \end{aligned}$$

(25)

where $f_{\mathrm{feed}}$=2.55 m is the focus of the sub-reflector. The reflected beam from sub-reflector is described by equation

$$\begin{aligned} z_{s-r} = a_{s-r}x_{s-r}+b_{s-r}, \end{aligned}$$

(26)

where $a_{s-r} = \tan ^{-1} \varphi _{P_{s-\mathrm{ref}}}$ and $b_{s-r} = z_{s-r} - \tan ^{-1} \varphi _{P_{s-\mathrm{ref}}}$. The beam hits the radio telescope feed at the point

$$\begin{aligned} x_{\mathrm{feed}} = \frac{f_{\mathrm{feed}} - b_{s-r}}{a_{s-r}}. \end{aligned}$$

(27)

The overall propagation path is defined by the individual refraction points

$$\begin{aligned} \begin{aligned} L^2&= (S_x - x_{r-i})^2 + (S_z - z_{r-i})^2 \\&\quad +\, (x_{r-i} - x_{s-i})^2 + (z_{r-i} - z_{s-i})^2 \\&\quad +\,(x_{s-i} - x_{\mathrm{feed}})^2+ (z_{s-i} - z_{\mathrm{feed}})^2. \end{aligned} \end{aligned}$$

(28)

Only a very small section of the wave front from the point source is entering the feed horn from the mean propagation distance of $L = 64.7924$ m, where the difference in phase between the longest and the shortest distance amounts to less than 1 mm. In other words, the phase of the received tones will be distorted about 2.1 ps at maximum.