Gravitational deformation of ring-focus antennas for VGOS: first investigations at the Onsala twin telescopes project
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Abstract
The receiving properties of radio telescopes used in geodetic and astrometric very long baseline interferometry (VLBI) depend on the surface quality and stability of the main reflector. Deformations of the main reflector as well as changes in the sub-reflector position affect the geometrical ray path length significantly. The deformation pattern and its impact on the VLBI results of conventional radio telescopes have been studied by several research groups using holography, laser tracker, close-range photogrammetry and laser scanner methods. Signal path variations (SPV) of up to 1 cm were reported, which cause, when unaccounted for, systematic biases of the estimated vertical positions of the radio telescopes in the geodetic VLBI analysis and potentially even affect the estimated scale of derived global geodetic reference frames. As a result of the realization of the VLBI 2010 agenda, the geodetic VLBI network is currently extended by several new radio telescopes, which are of a more compact and stiffer design and are able to move faster than conventional radio telescopes. These new telescopes will form the backbone of the next generation geodetic VLBI system, often referred to as VGOS (VLBI Global Observing System). In this investigation, for the first time the deformation pattern of this new generation of radio telescopes for VGOS is studied. ONSA13NE, one of the Onsala twin telescopes at the Onsala Space Observatory, was observed in several elevation angles using close-range photogrammetry. In general, these methods require a crane for preparing the reflector as well as for the data collection. To reduce the observation time and the technical effort during the measurement process, an unmanned aircraft system (UAS) was used for the first time. Using this system, the measurement campaign per elevation angle took less than 30 min. The collected data were used to model the geometrical ray path and its variations. Depending on the distance from the optical axis, the ray path length varies in a range of about \(\pm \,1\,\hbox {mm}\). To combine the ray path variations, an illumination function was introduced as weighting function. The resulting total SPV is about \(- \,0.5\) mm. A simple elevation-dependent SPV model is presented that can easily be used and implemented in VLBI data analysis software packages to correct for gravitational deformation in VGOS radio telescopes. The uncertainty is almost \(200\,\upmu \hbox {m}\) (\(2\sigma \)) and is derived by Monte Carlo simulations applied to the entire analysis process.
Keywords
Ring-focus paraboloid Radio telescope Antenna deformation VLBI VGOS Signal path variation SQP Reverse engineering Photogrammetry Unmanned aircraft system1 Introduction
Radio telescopes are large technical facilities, that are, among other possible applications such as radio astronomy and astrophysics, deep space tracking or astrometry, used as space geodetic instruments for geodetic very long baseline interferometry (VLBI) observations. Due to various effects, the geometry of these space geodetic instruments can deviate significantly from their ideal geometric representation. These deviations restrict the reliability of the derived geodetic VLBI products, e.g., the estimated station coordinates. For instance, variations of the environmental temperature cause thermal expansion of the telescope structure (e.g., Haas et al. 1999), thus influencing the dimension of the instruments. Automated one-dimensional measuring systems yield seasonal variations of the vertical component of several millimeters caused by changes in temperature (e.g., Johansson et al. 1996; Zernecke 1999). According to Wresnik et al. (2007), the thermal expansion can be modeled, if representative expansion coefficients of the structure are known and the relevant temperature is recorded. Moreover, strategies for continual estimation of the geometrical reference point have been derived (e.g., Lösler 2009; Kallio and Poutanen 2012; Ning et al. 2015; Lösler et al. 2018a) and allow for monitoring the spatial position of the telescope. Mähler et al. (2018) recently report seasonal horizontal variations of about 0.2–0.4 mm. Neidhardt et al. (2010) derived daily variations of the telescope tower caused by the path of the Sun and corresponding unidirectional warming.
Besides external influences, the geometry of the main reflector surface may deform by the reflector’s structural weight. During a geodetic VLBI observation session, a radio telescope points sequentially to several radio sources in different directions. Depending on the pointing direction, different structural loads may occur. For large conventional radio telescopes, variations of the focal length between several millimeters up to the centimeter level have been detected (e.g., Sarti et al. 2009; Nothnagel et al. 2013; Bergstrand et al. 2019). The Global Geodetic Observing System (GGOS) calls for high-accuracy station positions at the millimeter level for deriving a reliable global geodetic reference frame (e.g., Rothacher et al. 2009). Deviations in the signal path affect the accuracy of VLBI and result in systematic biases of the estimated vertical positions of the radio telescopes and, therefore, potentially even affect the estimated scale of derived global geodetic reference frames such as the International Terrestrial Reference Frame (ITRF) (cf. Sarti et al. 2011). To reach the GGOS requirements, these signal path variations (SPV) should be taken into account in the analysis of geodetic VLBI data. For that reason, the VLBI community has been encouraged to include structural gravitational deformation models for VLBI radio telescopes for the upcoming ITRF2020 (cf. Altamimi et al. 2018). Therefore, the International VLBI Service for Geodesy and Astrometry as well as the IERS Working Group on Site Survey and Co-location focus on measuring and modeling gravitational deformations for as many VLBI radio telescopes as possible (cf. Bergstrand 2018; Gross and Herring 2018). An important point in this context is that the telescopes in general show individual deformation behavior (cf. Sarti et al. 2009; Nothnagel et al. 2013; Bergstrand et al. 2019), and thus each type of telescope has to be measured and modeled individually.
In the framework of the VLBI 2010 agenda (Niell et al. 2006), several new radio telescopes have been planned, are under construction, or have already been installed. The goal is to improve the existing network of the International VLBI Service for Geodesy and Astrometry (e.g., Schlüter and Behrend 2007; Nothnagel et al. 2017) and to reach the accuracy requirements that are specified by GGOS. VGOS radio telescopes are new generation VLBI instruments with a more compact design, stiffer mechanical structure, and faster drive systems than conventional radio telescopes. Usually, the diameter of the main reflector is about 12–13 m (e.g., Haas 2013). So far, the deformation of the reflector system and its possible impact on the optical ray path have not yet been investigated, except from finite element analysis performed by the manufacturers. The investigation presented in this manuscript focuses therefore on observations of the deformational behavior of a VGOS-specified radio telescope, exemplified at the Onsala twin telescopes (OTT) project of the Onsala Space Observatory (OSO). The OTT are equipped with ring-focus paraboloids as the main reflectors and rotational-symmetric Gregorian sub-reflectors. Often this kind of telescope type is referred simply to as ring-focus telescope.
Section 2 discusses methods for measuring the main reflector surface of VLBI radio telescopes in terms of spatial restriction, expected uncertainties, and the accuracy requirements that shall be achieved. Moreover, the UAS-based data collection and the data preparation are described. The mathematical model of the ring-focus paraboloid is derived in Sect. 3. Similarities of this model to the simplified rotationally symmetric paraboloid are highlighted. Section 4 introduces the applied SPV model of a ring-focus paraboloid. The data analysis and the results are explained in Sect. 5. Finally, Sect. 6 concludes the paper.
2 Surface measurement methods
The goal in the framework of reverse engineering is to derive model-specific geometric parameters, e.g., dimension, curvature or orientation, by measuring an object. To quantify the parameters of the signal path of VLBI radio telescopes and their possible variations, several measurement methods with various limitations exist. The choice of the measurement method depends on the surrounding conditions, the accuracy requirements, and the size of the expected deformations. The measurement effort increases, if the main reflector is observed in several elevation angles, i.e., from \(0^{\circ }\hbox { to }90^{\circ }\). For instance, the use of tactile observation methods such as theodolite measurement systems or laser trackers is not suitable because of the time-consuming single-point measurement mode. Moreover, a stable platform is needed to carry out measurements in several elevation positions. Close-range photogrammetric methods usually require a crane for mounting coded targets on the reflector surface and for the data acquisition itself (e.g., Shankar et al. 2009; Süß et al. 2012). Furthermore, the spatial restrictions around the radio telescope limit this method, especially if the radio telescope is enclosed by a protecting radome like for the 20 m telescope at Onsala.
As an alternative to close-range photogrammetry, the capability of terrestrial laser scanners as a targetless method was investigated in recent years (e.g., Dutescu et al. 2009; Sarti et al. 2009). Technical innovations and a deeper understanding of instrument-dependent systematic errors were introduced to the measurement and analysis process in order to achieve reliable results (e.g., Lichti 2007, 2010; Holst et al. 2017). The possible mounting points of the laser scanner are limited by the construction parts of the radio telescope. In general, a single position close to the sub-reflector is chosen, which allows observing nearly the full reflector surface and avoiding glancing intersections (cf. Holst et al. 2017; Bergstrand et al. 2019).
The OTT are free-standing radio telescopes designed as ring-focus paraboloids with fixed rotational-symmetric Gregorian sub-reflectors (e.g., Haas 2013). The system focal point is located close to the sub-reflector and enclosed inside the receiver cone where the feed horn and receiver are located. Due to shading effects of this construction, measurements taken from a single position are incomplete. For this reason, in case laser scanning should be performed, several laser scanner positions would be needed to achieve a complete 3D-point cloud that covers the entire surface of the main reflector.
Photogrammetric methods on the other hand are unaffected by such constructional restrictions, because the observed points are derived from a large number of images taken at different camera positions. Advantageous camera positions induce additional reliable intersection conditions during image processing. Moreover, the camera positions vary without changing the radio telescope configuration.
As pointed out before, besides the limiting constructional conditions, there are further aspects, i.e., the accuracy requirements and the expected deformations. To reach the VLBI 2010 goal of 1 mm position accuracy, Petrachenko et al. (2009) advise sub-millimeter for surface accuracy and path length stability for VGOS-specified radio telescopes. If these stability requirements cannot be fulfilled, corresponding correction functions for compensating signal path variations are needed that can be applied in the VLBI data processing. Due to the compact construction of VGOS-specified radio telescopes, i.e., the diameter of the main reflector is about 12–13 m, deformations of about 1 mm are expected. For comparison, the reported SPV of the conventional 100 m radio telescope Effelsberg is about 1 cm (cf. Artz et al. 2014). Following the \(3\sigma \)-rule of thumb in geodesy and metrology, which states that the measurement system should be at least three times better than the expected deformations (cf. Koch 2007, p. 47), the measurement method for VGOS-specified radio telescopes should be on the level of 0.3 mm. Optical methods such as close-range photogrammetry or laser tracker systems fulfill this rule of thumb (e.g., Baars 2007, p. 167f; Luhmann 2018, p. 696f) but need a stable platform or a crane. To overcome the necessity of a crane during the measurement process, we used an unmanned aircraft system (UAS) in this investigation.
2.1 Photogrammetric measurements using UAS
A UAS consists of an unmanned aerial vehicle (UAV) and a ground-based station for remote control. Usually, the UAV is operated in a semiautomatic mode using a predefined flight plan. The flight plan schedules the waypoints to be achieved, the velocities of the UAV, the trigger points for the camera, etc. Several onboard sensors such as an inertial measurement unit (IMU), a global navigation satellite system (GNSS) antenna/receiver, and a magnetometer are used for automated orientation and positioning. Typically, critical flight phases such as take off and landing as well as unforeseen deviations from the flight plan have to be operated manually.
Photogrammetric measurement campaigns were carried out two times from elevation \(0^{\circ }\) up to \(90^{\circ }\) using a step-size of \(10^{\circ }\), as well as once at \(34^{\circ }\). The latter was done since the telescope manufacturer adjusted the reflector panels at this elevation. For each elevation angle, the flight path consisted of two centered flight lines crossing the center of the paraboloid and two additional concentric spatial circles around the axis of symmetry of the main reflector. The distance between the geometrical reference point and the flight lines as well as the inner circle is about 20 m. The distance of the outer circle is about 25 m. Both circles have different radii, i.e., \(r_{\mathrm {in}} = 6.5\,\hbox {m}\) and \(r_{\mathrm {out}} = 11.5\,\hbox {m}\). Figure 2 depicts the flight schedule of the UAV for elevation angle \(30^{\circ }\), using the topocentric UTM system. To avoid critical flight phases at lower elevation angles, the altitude of waypoints was limited and, here, the outer circle was planed as shell shape. However, the reflector could be captured even in this configuration by tilting the camera gimbal, since the camera was controlled and triggered remotely by the pilot. The gimbal-mounted camera allowed to point the camera nearly to the diametrical part of the main reflector for each taken image. Thus, the sub-reflector was pictured in every image. Due to the close distance to the telescope, each image contained only a part of the main reflector surface. On average, 40 coded targets were captured per image.
Figure 3 depicts ONSA13NE that was equipped with 76 discrete black and white, non-reflecting 12-bit coded targets. The main reflector of ONSA13NE consists of 60 panels mounted in three concentric rings, whereas the outer and the middle rings have 24 panels each, the inner ring consists of 12 panels. Sixty coded targets were glued at the outer parts of the panels, i.e., one target per panel. To improve the distribution of the points, additionally 12 coded targets were glued at the inner part of the panels of the inner ring close to the receiver cone. Four coded targets were attached at the sub-reflector. Moreover, a coordinate cross, which approximately defines the global datum of the resulting point sets by six coded targets, was mounted close to the sub-reflector. As shown in Fig. 2, the origin of the resulting global frame is close to the sub-reflector, the Z-axis points approximately in the direction of the symmetry axis of the main reflector, the X-axis is approximately parallel to the elevation axis, and the Y-axis is perpendicular to X and Z, respectively.
To transform the resulting point sets of the bundle adjustment (see Sect. 2.2) into a metric system (cf. Luhmann 2018, Ch. 7.1.5.2), six scale-bars with coded targets were attached. The scale-bars are made of carbon fiber with a thermal expansion coefficient of about \(\gamma _{\mathrm {c}} = 10^{-7}\,\hbox {K}^{-1}\). Three scale-bars were evenly distributed at the rim of the main reflector, covering the measurement space in the XY-plane of the image–block configuration defined by the coordinate cross. To avoid block deformations in the Z-direction, two scale-bars were mounted on the struts, and one at the sub-reflector. To avoid possible unanticipated time-dependent effects and to decorrelate the campaigns observed by equal elevation angles, no direct repetition was carried out, cf. Table 1. The photogrammetric camera, the Sigma DP3, was carried by the hexacopter HP-TS960 (HEXAPILOTS) and appears in the left upper corner of Fig. 3.
2.2 Data preparation
Consumer cameras such as the Sigma DP3 Merrill are not geometrically stabilized. Turning off the camera after a flight (to exchange the battery of the UAV) may change the interior orientation of the camera. As a result, all images that were acquired in an image–block configuration were taken without turning off the camera, which means they were taken in one flight for each elevation angle. Additionally, the interior orientation of the camera model, including the principal distance, the coordinates of the principal point, the radial–symmetric lens distortion and the decentering distortion, were calibrated in situ, during the bundle adjustment for each single campaign (e.g., Förstner and Wrobel 2016, Ch. 15.5; Luhmann 2018, Ch. 4.4.2).
Each photogrammetric bundle was processed separately as a free-network adjustment. The formal error of the coordinate components of the coded target positions in the global frame was about \(10\,\upmu \hbox {m}\) w.r.t. the datum. However, the formal error does not take into account the limited accuracy of the GNSS sensor, the unpredictable slight wind gusts, or different illumination environments, which all affect the resulting image–block configuration. Moreover, the in situ-derived calibration parameters as well as the telescope temperature may slightly change during the flight campaigns. Based on extensive additional experience with the measurement system, more reliable uncertainties for the coordinate components of the coded target positions were estimated to be between 80 and \(120\,\upmu \hbox {m}\) w.r.t. the datum. These single-point uncertainties are slightly larger than for high-precision photogrammetric cameras (e.g., Subrahmanyan 2005; Baars 2007, p. 167f; Luhmann 2018, p. 696f) but fulfill the requirements for detecting the expected deformations.
The telescope monument, i.e., the base and the mechanical structure of a radio telescope, is made of concrete and metal parts which are affected by temperature changes (e.g., Haas et al. 1999; Nothnagel 2009). In contrast to the thick structural elements of the monument, where the time between a change in the air temperature and the corresponding expansion is delayed by several hours, the construction parts of the main reflector are thinner. Therefore, the time delay between a temperature change and the corresponding expansion has been assumed to be less than 30 min.
The variation in the campaign temperature is \(<2\,\hbox {K}\) but yields an expansion sensitivity of \(20\,\upmu \hbox {m}\hbox { m}^{-1}\) if \(\gamma _{\mathrm {s}} = 10^{-5}\hbox {K}^{-1}\) is used as the expansion coefficient of steel. In accord with Artz et al. (2014), the estimated point sets of each campaign are scaled uniformly to the reference temperature^{1}\(T_0 = 9\,^{\circ }\hbox {C}\), even if a small delay may exist. In total, 21 point sets were prepared for evaluating possible changes in the ray path of ONSA13NE.
3 Ring-focus paraboloid
The main reflector of most of the conventional radio telescopes is designed as a rotationally symmetric paraboloid (RSP). Depending on the radio telescope type, either the sub-reflector or the receiver, which is located close to the unique focal point of the RSP, will shade the main reflector and result in a field of decreased intensity (cf. Cutler 1947). Depending on the dimension of the main reflector, the shading effect may become significant. The new generation of the so-called VGOS-specified radio telescopes is characterized by drive systems that allow high rotational velocities of about \(12^{\circ }\,\hbox {s}^{-1}\) and \(6^{\circ }\,\hbox {s}^{-1}\) in azimuth and elevation, respectively, and also by a small diameter of about 12–13 m for the main reflector (e.g., Petrachenko et al. 2009). Due to the small diameter of the main reflector, the lower signal strength becomes significant if a RSP is used. For this reason, many of the VGOS-specified radio telescopes are designed as rotationally symmetric ring-focus paraboloids (RSRFP).
3.1 Double-elliptic ring-focus paraboloid
The RSP as well as the RSRFP are two possible sub-models of Eq. (7). The RSRFP design results by introducing the parameter constraints \(a_1 = a_2\) and \(b_1 = b_2\). Moreover, the RSP design is a special type of the RSRFP and results from the further simplification \(r_i = 0\) (cf. Lösler et al. 2017, 2018b).
Due to structural deformations induced by gravity, the focal length F of the radio telescope may change when it rotates around the elevation axis (e.g., Sarti et al. 2009; Nothnagel et al. 2013; Bergstrand et al. 2019). According to Artz et al. (2014), two deformation patterns can exist. The first one describes an unpredictable deformation of the main reflector. This pattern cannot be modeled by quadric surfaces and needs a specific compensation model. The second one describes an affine deformation. Here, the resulting deformations provide surfaces that are similar to each other and can therefore be modeled geometrically. Whereas Artz et al. (2014) assume a homologous deformation, i.e., the focal lengths change while the main reflector maintains its parabolic shape, the use of Eq. (7) extends this homologous deformation pattern to a wider range of applications because Eq. (7) also allows modeling changes in the surface type (Lösler et al. 2018b, c).
Having a discrete set of observed spatial points lying on the paraboloid surface of the ring-focus antenna, the datum-invariant parameters, as well as the isometric parameters, are estimable using an errors-in-variables model. In Sect. 3.2, the sequential quadratic programming (SQP) is proposed for the data analysis.
3.2 Parameter estimation
To estimate the parameters of the implicit Eq. (7) of a double-elliptic ring-focus paraboloid, an errors-in-variables (EIV) model is needed. In numerical optimization, a well-known solver that belongs to the class of EIV models is the SQP. The SQP approach is recommended for nonlinear constrained optimization and estimates the unknown parameters \({\mathbf {u}}\) by solving sequences of quadratic sub-problems (cf. Nocedal and Wright 2006, Ch. 18).
To avoid an over-parameterization, the number of model parameters should only be expanded to the maximum size if the realized surface type significantly deviates from its ideal surface type. Usually, the surface type is restricted to the designed rotationally symmetric ring-focus paraboloid that yields the seven model parameters to be estimated \({\mathbf {x}}^{\mathrm {T}} = \left( \begin{array}{ccccccc} a&b&X_0&Y_0&Z_0&\xi _x&\xi _y \end{array} \right) \).
3.3 Sub-reflector variations
4 Signal path variation
The OTT are secondary focus Gregorian-type radio telescopes, and the fixed sub-reflector is located behind the focal point \(\mathbf {F}_{1}\) of the main reflector, see Fig. 7. The sub-reflector is designed as a rotationally symmetric elliptical torus, cf. Eq. (3), and reflects the signal toward the system focal point \(\mathbf {F}_{0}\). The sub-reflector of ONSA13NE does not provide a movable mounting to adjust the focal length during observations because in geodetic VLBI, the sub-reflector is kept fixed. As shown in Figure 7, the cross section of the rotationally symmetric elliptical torus yields two ellipses, and the ray path can be simply modeled in 2D. However, in contrast to a conventional Gregorian-type radio telescope having an axially parallel ellipse as cross section of the sub-reflector, here, the ellipses of the sub-reflector of the ring-focus telescope are tilted. Thus, the incidence angle \(\gamma \) of the ray at the system focal point \(\mathbf {F}_{0}\) becomes zero if the ray is reflected at the rim of the main reflector.
5 Measurement results
Campaign-wise averaged temperature as well as estimated overall \(\hbox {RMS}(\mathbf {v}_{c})\) w.r.t. different survey elevations \(\epsilon \). During the first ten measurement campaigns, the rotation direction of the telescope was upward, and afterward downward. At \(34^{\circ }\), the telescope was measured only once (rotation direction upward). Horizontal lines indicate the subdivided campaigns. The campaign numbers are in chronological order
Campaign | \(\epsilon \) in \(^{\circ }\) | RMS in \(\upmu \hbox {m}\) | T in \(^{\circ }\hbox {C}\) |
---|---|---|---|
1 | 0 | 204 | 18.6 |
2 | 10 | 187 | 18.5 |
3 | 20 | 200 | 18.1 |
4 | 90 | 282 | 18.6 |
5 | 30 | 167 | 17.6 |
6 | 40 | 192 | 17.8 |
7 | 50 | 182 | 17.7 |
8 | 60 | 173 | 17.9 |
9 | 70 | 178 | 17.9 |
10 | 80 | 221 | 17.7 |
11 | 90 | 292 | 17.8 |
12 | 80 | 227 | 18.1 |
13 | 70 | 167 | 18.2 |
14 | 60 | 162 | 18.4 |
15 | 50 | 155 | 18.6 |
16 | 40 | 154 | 18.6 |
17 | 30 | 147 | 18.9 |
18 | 20 | 166 | 19.0 |
19 | 10 | 194 | 19.1 |
20 | 0 | 190 | 19.2 |
21 | 34 | 169 | 18.3 |
In 2017, the surface of the main reflector was observed at \(34^{\circ }\) during the panels adjustment. The resulting focal length (cf. Lösler et al. 2017) is symbolized by a black star. The difference of this focal length and the prediction derived by Eq. (30) is \(-\,0.2\,\hbox {mm}\) and confirms the latest results.
During the first ten measurement campaigns, the rotation direction of the telescope was upward, and afterward downward for the repeated measurements. By comparing the derived focal lengths of the upward and the downward configurations, a small, but insignificant hysteresis is visible. The reason might be an unrepresentative temperature compensation. For example, a change of \(2\,\hbox {K}\) yields a corresponding focal length change of about 0.1 mm. However, further investigation, including more than one repetition measurement, is needed to verify this behavior.
To verify the shape of the curve described by the function, the variation \({\Delta }R'(\epsilon )\) was derived independently by Eq. (20). The nominal position of the strut \({\mathbf {S}}\) as well as the nominal length of the strut element \(\left| \mathbf {SR}\right| \) were taken from the mechanical drawings and provide the nominal arc length Open image in new window via Eq. (19), see Fig. 8.
Amplitude \(c_1\) of the cosine function, cf. Eq. (29), derived by direct measurements of the sub-reflector variations \({\Delta }R(\epsilon )\) and evaluated by Eq. (20), which uses basic geometrical conditions \({\Delta }R'(\epsilon )\). \({\hat{\sigma }}_{c_1}\) is derived by applying the law of propagation of uncertainty
Parameter | \(c_1\) in mm | \({\hat{\sigma }}_{c_1}\) in mm |
---|---|---|
\({\Delta }R\) | 0.59 | 0.1 |
\({\Delta }R'\) | 0.65 | 0.1 |
Nominal values of the construction components of the Onsala twin telescopes
Name | Abbr. | Value in mm |
---|---|---|
Focal length of the paraboloid | \({\tilde{F}}_{\mathrm {1}}\) | 3700.00 |
Distance from apex to sub-reflector | \({\tilde{F}}_{\mathrm {0}}\) | 3611.66 |
Diameter of the main reflector | \({\tilde{D}}_{\mathrm {m}}\) | 13,200.00 |
Diameter of the cylinder | \({\tilde{D}}_{\mathrm {c}}\) | 1480.00 |
For each elevation, the geometrical ray path is sampled between the boundaries of the paraboloid, i.e., \(\frac{{\tilde{D}}_{\mathrm {c}}}{2}\) and \(\frac{{\tilde{D}}_{\mathrm {m}}}{2}\). Applying the derivations \({\Delta } F(\epsilon )\) and \({\Delta } R(\epsilon )\) yields a geometrical ray path length map of ONSA13NE. Figure 14 illustrates the resulting map for \(0^\circ \). The ray path length varies over a range of about 3 mm. It should be noted that the yellow part of the map is the integrated cylinder of the ring-focus paraboloid and, therefore, is insensitive by definition. However, rays that are reflected close to this cylinder region are affected by rather large variations compared to rays reflected at the rim. Other areas (dark blue) show almost no ray path variation. This latter results from the combination of the reflection at the main reflector and the sub-reflector, respectively, during ray tracing when \({\Delta } F(\epsilon )\) and \({\Delta } R(\epsilon )\) have opposite signs and cancel out for some ranges. In general, Fig. 14 shows clearly that the largest (outer) part of the map covering the area between about 3 and 6.5 m from the center shows very small variation in the ray path, i.e., is stable. These areas are weighted highest by the illumination function Eq. (26), whereas the more central parts showing larger ray path variations are weighted less by the illumination function.
In Fig. 16, the illustrated uncertainties are derived by Monte Carlo simulation (e.g., JCGM102 2011) starting at the beginning of the data preparing, see Sect. 2.2, and comprising the full analysis process. The SPV approach depends on several input quantities, i.e., the derived point set of the bundle adjustment, the campaign temperature, and the elevation angle of the telescope. The uncertainties of the point set, as well as the uncertainty of the elevation angle, are assumed to be normally distributed. The campaign temperature given in Table 1 is assumed to follow a uniform distribution, cf. Fig. 6.
Besides the input quantities that are related to the metrology part affecting the geometrical deformation pattern in Eq. (21), the uncertainties of \({\Delta }L\) also depend on the weighting coefficients. The weighting coefficient \(\alpha _R\) in Eq. (23) depends on the selected function h as well as \(I_n\). Assuming a uniform distribution of \(\alpha _R\) within the boundaries given by \(\alpha _R(h) = 0.65\) and \(\alpha _R(h') = 0.62\), the uncertainty derived by Eq. (2) reads \(\sigma _{\alpha _R} = 0.01\). The illumination function \(I_n\) is known and, thus, no further uncertainty must be specified.
By applying 100, 000 samples, the uncertainties are evaluated by a Monte Carlo simulation from \(0^{\circ }\) to \(90^{\circ }\) using a step-size of \(1^{\circ }\). The resulting error band regarding only the metrology part is plotted in gray, and the error band, extended by the assumed uncertainties of the weighted coefficients, is colored in red.
6 Conclusion
GGOS aims for 1 mm position accuracy on the global scales for the ITRF. Gravitational deformations of VLBI radio telescopes yield systematic errors and bias the estimated vertical position of the radio telescopes in the geodetic VLBI analysis and, therefore, potentially affect the ITRF scale. For the upcoming ITRF2020, the VLBI community has been encouraged to include gravitational deformation models for VLBI radio telescopes to compensate for the undisputed systematic errors caused by elevation-dependent reflector deformations for as many VLBI radio telescopes as possible. For that reason, mathematical models as well as metrology methods have to validate, to capture, and to model the expected structural deformations.
In this investigation, for the first time, a UAS was used to study the gravitational deformation behavior of a VLBI radio telescope. To our best knowledge, it is also the first time that an observation-based model for the gravitational deformation of a VGOS-specified radio telescope was derived. This model can be easily implemented into VLBI data analysis software packages and used in future VGOS data processing. Several coded targets were mounted at ONSA13NE. The surface of the main reflector and the position of the sub-reflector were measured by photogrammetric methods from elevation \(0^{\circ }\) to \({90}^{\circ }\) using a step-size of \(10^{\circ }\). To increase the reliability, each position was measured twice. In total, 21 measurement campaigns were carried out in August 2018. In contrast to measurement approaches with laser scanners, each target position is highly redundantly observed during a single measurement campaign. This allows for evaluating the quality of the observed points using established statistical methods during the bundle adjustment right at the beginning of the analysis process. The single-point uncertainties of the bundle adjustment are about \(100\,\upmu \hbox {m}\) w.r.t. the datum.
Like many of the VGOS-specified radio telescopes, ONSA13NE makes use of the improved main reflector design and is manufactured as a so-called ring-focus paraboloid. A ring-focus paraboloid results from the combination of two quadric surfaces and, therefore, cannot be modeled as a common paraboloid. The mathematical parameterization of a double-elliptic ring-focus paraboloid was given in detail in this investigation. Similarities of the presented unified model to the simplified rotationally symmetric paraboloid, which has been used in general for modeling the main reflector of conventional VLBI radio telescopes, were mentioned.
Based on the resulting coordinates of the bundle adjustment, the surface parameters of ONSA13NE were estimated using the SQP approach. The focal length varies by about 2 mm. However, the focal length variation itself is a descriptive parameter and does not represent the variation of the optical ray path because the physical position of the sub-reflector also varies. To reconstruct the position variation of the sub-reflector, two approaches were considered. Whereas the first one results from a geometrical modeling of the deformation of the main reflector, the second one is based on direct measurements. Both approaches yield sub-reflector variations of about \(\pm \,0.3\,\hbox {mm}\), which partly counteract the focal length variations. For ONSA13NE, the variations of the focal length and the sub-reflector position can both be expressed by cosine functions.
The model of the main reflector and the model of the sub-reflector are combinations of quadric surfaces, i.e., a ring-focus paraboloid and an elliptic torus, respectively. Considering the geometric properties of both reflectors and the derived deformation pattern, the length of the geometric ray path was derived. As shown in Fig. 14, the length of the geometric ray path depends on the distance from the optical axis and varies over the reflector by about 3 mm at \(0^{\circ }\).
The Gaussian illumination function, which is used by ONSA13NE, was introduced to weight the ray path depending on the incidence angle at the aperture. The resulting amplitude of the modeled signal path variations is about \(-\,0.5\,\hbox {mm}\). For the modeled signal path variations, uncertainties of about \(200\,\upmu \hbox {m}\, (2\sigma )\) were derived by applying a Monte Carlo simulation to the entire analysis process. For VGOS-specified radio telescopes, the requested RMS for modeled path length variations is \(300\,\upmu \hbox {m}\) (Petrachenko et al. 2009). The measurement and analysis concept presented in this investigation fulfill these requirements.
Future work will focus on metrological methods for deriving the shift of the vertex to obtain the SPV independent of the finite element method. Moreover, investigations are needed to verify the possible hysteresis of the focal length variations. Furthermore, the Onsala Space Observatory is in the possibility of having the SPV variation model externally validated by comparing the baseline derived by high-precision terrestrial observations and the baseline obtained by the VLBI data analysis (e.g., Carter et al. 1980). For this purpose, terrestrial measurement campaigns are planed to derive the local baseline vectors between the hosted space geodetic techniques at Onsala.
The OTT are just one type of several different VGOS-type telescopes that have been designed and are currently deployed at various international observatories. To our best knowledge, there are at least three other designs of VGOS-telescopes. We are not aware that any other investigations corresponding to the one presented here have been performed at any of the other VGOS stations. However, a first assumption is that other VGOS-designs also fulfill the VGOS-specifications and will have a similar deformation behavior as the Onsala twin telescopes. Nevertheless, it is strongly advised to derive at least type-specific SPV models for each VGOS telescope type. Regarding the small magnitude of the detected SPV and the small uncertainty of the derived SPV model for the OTT, there are high expectations that VGOS will be able to live up to the challenging GGOS requirements.
In general, the use of a UAS provides a promising and practicable surveying method for free-standing radio telescopes because neither does additional heavy equipment have to be mounted on the radio telescope structure nor is a crane required during the measurement process. This method thus appears very applicable for both VGOS-type and conventional radio telescopes.
Footnotes
Notes
Acknowledgements
We thank Lars Wennerbäck and Christer Hermansson from the mechanical workshop at the Onsala Space Observatory for their support mounting the coded targets and the scale-bars at the radio telescope. Moreover, we thank Jonas Flygare for providing the illumination function of ONSA13NE. We would also like to show our gratitude to Eberhard Sust (MT Mechatronics GmbH) for supporting this research project by providing results of their finite element analysis. This research project is part of the JRP 18SIB01 “Large-scale dimensional measurements for geodesy” (GeoMetre) and has received funding from the EMPIR programme co-financed by the Participating States and from the European Union’s Horizon 2020 research and innovation programme.
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