Pointing directions in the sky can be described by two spherical coordinates, where ϕ∈0°,360° denotes the azimuth angle and θ∈0°,90° is the elevation angle. We follow the common convention in radar science that 0° elevation corresponds to pointing horizontally. Scanning radars usually have a two-axis scanner mechanism, where one axis corresponds to the movement in azimuth direction and another axis on top of the first moves in elevation direction. We denote the position of those axes by γ and ω, respectively. Usually, it is assumed that a movement in γ directly corresponds to a movement in ϕ and similarly for ω and θ. However, this is only the case for a perfect scanning mechanism. In reality, axis positions are not perfectly equal to pointing directions, due to multiple system imperfections: axis encoders can have offsets, the axes are not perfectly orthogonal, or the whole system is tilted with respect to the celestial hemisphere. Therefore, a strict separation between axis positions (γ,ω) and celestial pointing (ϕ,θ) is necessary. We describe the mapping between axis positions and celestial pointing of the radar by a forward model MP: 1MP:(γ,ω)↦(ϕ,θ). Finding MP is the main goal of the methods described here. MP depends on multiple parameters P, which describe the imperfections of the scanner alignment and mechanics. There are seven static parameters: 2P=α,β,δ,ϵ,γ0,ω0,χ. The individual static scanner parameters are illustrated in Fig.  and defined as follows:

α: pedestal tilt towards West or East

We denote a perfectly accurate scanner as MI, with I=(0,0,0,0,0,0,0). For such a perfect scanner, axis positions and celestial pointing are identical (at least, as long as ω ≤ 90°): γ=ϕ and ω=θ.

In addition to the static scanner parameters, there are also dynamic parameters. Dynamic means that their effect on the beam pointing depends on the velocity of the axes (γ˙,ω˙). We consider the following dynamic parameters:

bγ: backlash in the azimuth axis. For example, this can be caused by play in the gears of the azimuth axis. As a consequence, this creates a hysteresis in the system, where the beam is always lagging behind the position indicated by the scanner motors. The backlash only depends on the direction of movement. The effect of the backlash on the actual position γ̃ of the azimuth axis can be modeled as γ̃=γ+bγsign(γ˙)

If the Sun were a perfect point source and the radar emitted an infinitely narrow beam, step 1 would not be necessary to get a pair of referenced scanner coordinates. In that idealized case, we could simply scan across the sky until the beam intersects the point-like Sun, record the corresponding scanner coordinates, and directly assign them to the absolute solar position.

In reality, the Sun appears as an extended, moving disk and the radar beam exhibits a finite width. Consequently, the measured signal represents a broadened response, and we must determine the apparent center of the Sun within the data. Similar to , we will do so by fitting a simulated response to all data points. In addition, we will also obtain information about the radar beam shape, scanner backlash and signal time offset. To achieve this, we perform multiple measurements in the region where the Sun is expected to be located. This procedure yields a set of N samples, each containing the measurement time, scanner coordinates and velocities, and recorded signal: 4Sample i:ti,γi,ωi,γ˙i,ω˙i,Si.

According to the measurement time ti, we compute the apparent solar position (ϕs,i,θs,i), accounting for the radar position and altitude.

If the scanner operated without inaccuracies, the corresponding celestial beam position at time ti would be given by the ideal mapping 5MIγi,ωi=ϕb,i,θb,i. However, with a single Sun scan it is not feasible to determine the full set of inaccuracy parameters P of a general scanner model MP, since the observations cover only a small portion of the sky. We therefore restrict the model to two parameters (Δγ,Δω), which capture the local mispointing of the system in the scanned region. In this approximation, the effective beam direction is 6MIγi+Δγ,ωi+Δω=ϕb,i,θb,i. To estimate (Δγ,Δω), we simulate the expected signal strength Q(ϕb,θb,ϕs,θs) as a function of beam and solar position, as well as several other parameters listed in Table . The optimal parameters are then obtained by minimizing the root-mean-squared deviation (RMSD) between measured and simulated signals: 7min⁡(Q Parameters)1N∑i=1N(Si-Qi)2. With Δγ and Δω available, we can finally calculate a reference pair of scanner coordinates γr, ωr with a reference position in the sky ϕr, θr: 8MIγr+Δγ,ωr+Δω=ϕr,θr. γr and ωr can be chosen arbitrarily from all samples (γi,ωi).

A single Sun scan provides information only about the local mispointing in one specific region of the sky. However, in other regions the mispointing may differ, depending on the specific inaccuracies of the scanner introduced in Sect. . To estimate these inaccuracies systematically, we implemented the forward model MP(γ,ω) as a kinematic chain using ikpy, a Python library for robotics and inverse kinematics . The kinematic chain consists of four links: Origin, Azimuth joint, Elevation joint, and Antenna. Each link defines a rotated coordinate frame relative to the previous one, with the rotations depending on the parameters α, δ, β, ϵ, γ̃, and ω̃. The effects of backlash bγ, time offset t0, angle offsets γ0 and ω0, as well as structural flexibility χ, are combined into the effective angles γ̃ and ω̃.

A single Sun scan is not sufficient to unambiguously determine all parameters in P. For example, when scanning the Sun in the South, no information can be inferred about a potential East-West pedestal tilt α. To resolve such ambiguities, Sun scans at multiple positions across the sky are required, ideally covering the course of a full day in the summer months. Each Sun scan yields a pair of referenced coordinates (γr,ωr) in the scanner system and the corresponding celestial reference coordinates (ϕr,θr).

For a given set of estimated parameters F, we define an objective function based on the mispointing angle between the predicted beam position and the celestial reference position: 19ΔΩMF(γr,ωr),(ϕr,θr).

The objective is defined as the RMSD of the mispointing across all referenced pairs: 20O=1N∑i=1NΔΩi2. Minimizing this objective yields the set of parameters F that best explains the observed radar pointing across all Sun scans. To carry out the minimization, we employ a two-step strategy: first, a three-point brute-force search provides reasonable starting values; second, a Nelder-Mead simplex optimization minimizes the objective .

Global optimization in a high-dimensional parameter space is challenging, and simultaneous fitting of all parameters can cause the optimizer to become trapped in local minima. We found that sequential fitting, leveraging the specific mispointing signatures of each parameter, avoids this issue. Figure  shows the mispointing caused by each static scanner parameter over the celestial hemisphere. In our approach, previously fitted parameters are held fixed as best guesses in subsequent steps: 0.

Backlash bγ and time offset t0. As described in Sect. , the dynamic parameters are assumed constant across the sky and can be estimated from a single Sun scan. We recommend averaging the results from multiple scans.

Once an estimate of the pan-tilt system parameters F is available, misalignments can either be corrected mechanically – for example, by leveling the scanner pedestal – or compensated for in software. The latter requires inverting the forward model MF, such that for a desired celestial position (ϕ,θ), the corresponding scanner axis positions (γ,ω) are determined: 21(γ,ω)=MF-1(ϕ,θ). This inversion can be performed numerically by minimizing the mispointing between the desired celestial coordinates and the beam position predicted by the forward model, i.e., by finding the scanner position (γ,ω) that minimizes 22ΔΩMF(γ,ω),(ϕ,θ). The feasibility of such inversion depends on the specific set of parameters. For example, offsets in the azimuth or elevation encoders (γ0,ω0) can be compensated straightforwardly. By contrast, an antenna tilt ϵ renders the zenith position unreachable, as will be discussed further in Sect. .

To test whether the inversion is successful, the obtained scanner position (γ,ω) is inserted back into the forward model, and the resulting pointing is evaluated against the target celestial position: 23ΔΩ(MF(γ,ω),(ϕ,θ))=ΔΩ(MF(MF-1(ϕ,θ)),(ϕ,θ))=!0°.

Sun scans were carried out in August 2025 in Munich, southern Germany, using a Mira-35 Doppler cloud radar manufactured by Metek GmbH

The instrument is a central component of the Munich ACTRIS National Facility for cloud remote sensing (https://nflabelling.actris.eu/facility/51, last access: 24 April 2026). Continuous measurements and derived products are available via the Cloudnet portal https://cloudnet.fmi.fi/ (last access: 24 April 2026).

As described in Sect. , each Sun scan provides a referenced pair of scanner and celestial coordinates (γr,ωr) and (ϕr,θr). With Sun scans distributed over the course of a full day, these data can be used to estimate the scanner inaccuracies following the procedure outlined in Sect. .

Figure a presents Sun scans performed on 11 and 12 August 2025. The figure shows both the actual beam positions (ϕb,θb) in the celestial hemisphere and the expected beam positions obtained from the scanner coordinates mapped into the sky using the identity scanner model, 24(ϕI,θI)=MI(γr,ωr). The comparison reveals large discrepancies between expected and actual beam positions, due to an uncorrected north angle γ0 = 202.7°. After adding the northangle, the residual differences are much smaller (< 1°), as shown in Fig. b. For visibility, in Fig. b, the residuals are enhanced.

Over the course of the day, the discrepancies exhibit a characteristic pattern. The largest deviations occur in the southwest, where the beam consistently points lower than expected. Since this effect is present in both forward and reverse scans, it indicates a pedestal tilt of approximately 0.2° towards the southwest.

We apply the sequential fitting procedure introduced in Sect.  to the data in Fig. a. Table  lists the optimal scanner parameters F1 obtained from this procedure. As expected, a positive α and negative δ confirm the southwest tilt. Compared to the pedestal tilt, the other inaccuracy parameters are about an order of magnitude smaller.

These comparably small axis offsets, gimbal tilt and antenna tilt are to be expected, since the scanner had undergone a factory calibration prior to the measurements shown in this study. The factory procedure is based on a manual Sun scan analysis. For the experiment presented here, the pedestal was intentionally misaligned to test the fitting method.

Figure c shows the expected scanner positions when the optimal parameters F1 are taken into account. The numbers next to the points in Fig. c indicate which samples are used for which step of the sequential fit, as presented in Sect. . Relative to Fig. b, the discrepancies are reduced by a factor of seven, yielding an average residual of only 0.02°. The remaining deviations are likely caused by effects not explicitly represented in the scanner model, such as position-dependent offsets, nonlinear elastic deformations, or differential thermal expansion of the steel support structure during the day.

With precise knowledge of the pedestal misalignment, we can proceed to correct it mechanically. Our radar is mounted on four adjustable screws arranged in a 977 mm by 1328 mm rectangle. Each full turn of a screw raises the corresponding corner by 2 mm, enabling fine adjustments. Figure d presents the results from a second series of Sun scans after adjusting the feet to compensate for the pedestal tilt. The remaining discrepancies are now dominated by other scanner inaccuracies, such as motor offsets, gimbal tilt β, and antenna tilt ϵ. As a consistency check, we re-estimated the optimal scanner parameters F2 from these new scans. The results, shown in Table  (second column), confirm that the pedestal is now aligned to within 0.01°, while the other parameters remain similar to their previous values. Taking MF2 as the basis for the expected pointing in Fig. e, the remaining discrepancies almost vanish and we get a similar agreement between expectation and measurement as in Fig. c.

Instead of correcting the pointing errors mechanically, the ability to invert the scanner model MP allows for an automatic correction in software. For example, if the radar is required to point to (ϕ=0°,θ=30°) in the sky, the corresponding scanner coordinates according to the fitted parameter set F1 are: 25MF1-1(ϕ=0°,θ=30°)=γ=157.30°,ω=29.91°,γ=337.38°,ω=150.10°.

The two solutions correspond to the forward and backward scanner configurations. Both solutions account for the complete set of inaccuracies represented in the scanner model. For cloud radar applications, the zenith position is typically the most critical. In this case, the inversion yields: 26MF1-1(ϕ=0°,θ=90°)=γ=171.06°,ω=89.85°,γ=47.47°,ω=90.15°.

At zenith, the prescribed azimuth ϕ = 0° is irrelevant for the pointing accuracy. The scanner azimuth positions γ listed in Eq. () were chosen to optimize the final pointing.

Figure a and b illustrate the corrections required to be added to the scanner coordinates γ and ω, respectively, to achieve perfect pointing across the full celestial hemisphere. Figure a is dominated by the system's north angle of 157.3°. Near zenith, the azimuth corrections increase, reflecting the reduced effectiveness of azimuth corrections at high elevation angles. Figure b is dominated by the southwest tilt of the pedestal: in the southwest, the elevation axis must be increased by about 0.2°, while in the northeast it must be reduced by roughly 0.1°.

It should be noted that inverse kinematics does not always allow for complete correction. Certain scanner configurations restrict accessibility to specific sky regions. For illustration, consider the extreme case where the dish is tilted by ϵ = 90°, effectively aligning the beam with the elevation axis. In this configuration, the elevation motor no longer affects the beam pointing, making it obviously impossible to reach the zenith position. Figure a shows the hypothetical case of an antenna tilt of ϵ = 10°. Here, the maximum achievable beam elevation is 80°, leaving the zenith position unreachable. Above this limit, the residual mispointing after inverse kinematics correction increases continuously, reaching up to 10° at zenith.

A straightforward solution for Doppler cloud radars is to deliberately shift the unreachable polar region away from zenith by tilting the pedestal. Figure b illustrates this effect for a virtual scanner tilted by 6° towards both the west and north. The unreachable region is displaced accordingly, and zenith can again be reached with nearly perfect accuracy. In fact, the same result would be achieved for a sufficiently large pedestal tilt into any direction. This leads to the paradoxical effect that with the ability to perform inverse kinematics corrections, a perfectly leveled pedestal represents the least favorable configuration, as it maximizes the risk of an unreachable zenith.

As described in Sect.  and Fig. , the Sun scan method requires multiple Sun scans at different positions in the sky. However, the accessible positions may be restricted by a limited solar path, for example at high latitudes or during winter months, or by radar-specific constraints such as surrounding obstacles or mechanical limitations of the scanner.

To assess the impact of such restrictions on the applicability of the method, Fig.  shows the mispointing caused by a deviation of 0.1° in each scanner parameter across all sky positions. In general, a parameter can only be inferred from Sun scans if measurements are available from sky regions where the pointing is sensitive to that parameter. The markers in each panel illustrate the Sun positions for Munich and a high-latitude site (Neumayer III station, 70°40^′ S, 008°16^′ W). For the polar site, the Sun traverses at least one sensitive region for all parameters except the gimbal tilt β. Hence, even in polar regions with only low solar elevation angles available, we expect that all parameters except β can be estimated.

For campaigns and mobile radars, the pedestal tilts α and δ are likely the most relevant parameters. As described in Sect. , their estimation requires only samples close to the horizon in the West and East, or North and South, respectively. However, during polar winter conditions, when the Sun remains confined to the North or South, α cannot be reliably estimated because no sensitivity exists near the horizon in the North or South (Fig. a).

These considerations apply analogously to local obstacles or mechanical constraints. In the latter case, reverse scans may not be feasible. Consequently, certain parameter pairs, such as γ0 and ϵ, cannot be determined independently (see also Sect. ). In such situations, we recommend fixing one parameter and excluding it from the fit. For example, the gimbal tilt β and the antenna tilt ϵ may be assumed small and prescribed according to manufacturer specifications.

Figure  already provides a measure of the relative uncertainty of the Sun scan parameters by showing the variability of repeated scans throughout the day. Beyond this variability, sources of absolute bias must also be considered.

A potential source for uncertainty is the accuracy of the solar position algorithm. We compared four different algorithms. Skyfield , Sunpy and PySolar are free and open source Python modules. SolCalc is the solar position calculator from the US National Oceanic and Atmospheric Administration. We found general agreement in the solar azimuth and elevation of better than 0.02° between the libraries. The main cause for this difference is whether refraction is considered and how a correction is implemented. For the calculations in this manuscript and SunscanPy, we rely on the geometrical (i.e. not refraction corrected) position provided by Skyfield. We then apply the microwave refraction formula proposed by . Their coefficients were fitted to simulations of the Starlink positional astronomy library , which uses the refractivity model from . The Starlink implementation is generally valid for microwave radiation in the GHz range, with a reported accuracy better than 1^′′. It does not include an explicit wavelength dependency. According to , dispersion of the refractive index would be most relevant near oxygen and water vapor resonance lines, which are deliberately outside of the frequency bands used by weather and cloud radars. Figure a shows the correction to the geometric solar elevation as a function of Sun elevation. The effect is largest near the horizon, but already at elevations above 10°, the correction remains below 0.1°. The dominant source of uncertainty in this parameterization is atmospheric humidity. Figure b illustrates how the apparent solar elevation changes when relative humidity increases from 0 % to 50 %, and from 50 % to 85 %. Above 10°, the associated effect is less than 0.01°. For the calculations in the previous sections, we assumed a constant humidity of 50 %. While future refinements could incorporate vertical humidity profiles from radiosonde measurements, this level of detail is not necessary to achieve an absolute calibration accuracy well below 0.1°. This conclusion is confirmed by sensitivity tests: repeating our calculations with 85 % humidity altered the derived scanner parameters by in general less than 10^′′.

In order to estimate the Sun position accurately, precise timing is important. As described in Sect. , we use different azimuth velocities to estimate the relative time offset between signal and scanner encoder readings. However, this method cannot correct for an absolute time offset being present in signal processor and scanner encoder alike. Absolute time offsets will cause an apparent mispointing, which is always directed along the trajectory of the Sun. Depending on the latitude and season, it may therefore be possible to derive an absolute time offset from the characteristic misalignment pattern of multiple Sun scans. Nevertheless, for practical applications, we recommend simply synchronizing the time of the radar before a Sun scan calibration. Using for example GPS time or the standard network time protocol (NTP) synchronization available in modern operating systems, an accuracy of 100 ms or better should be easily achieved . From sensitivity tests, we found time shifts of 1 s to cause in general less than 10^′′ shift in the derived scanner parameters.