Absolute Calibration method for FMCW Cloud Radars

This article presents a new Cloud Radar calibration methodology using solid reference reflectors mounted on masts, developed during two field experiments held in 2018 and 2019 at the SIRTA atmospheric observatory, located in Palaiseau, France, in the framework of the ACTRIS-2 research and innovation program. The experimental setup includes 10 cm and 20 cm triangular trihedral targets installed at the top of 10 m and 20 m masts, 5 respectively. The 10 cm target is mounted on a pan-tilt motor at the top of the 10 m mast to precisely align its boresight with the radar beam. Sources of calibration bias and uncertainty are identified and quantified. Specifically, this work assesses the impact of receiver compression, incomplete antenna overlap, temperature variations inside the radar, clutter and experimental setup misalignment. Setup misalignment is a source of bias previously undocumented in the literature, that can have an impact on the order of tenths of dB in calibration retrievals of W band Radars. 10 A detailed analysis enabled the design of a calibration methodology which can reach a cloud radar calibration uncertainty of 0.3 dB based on the equipment used in the experiment. Among different sources of uncertainty, the two largest terms are due to signal-to-clutter ratio and radar-to-target alignment. The analysis revealed that our 20 m mast setup with an approximate alignment approach is preferred to the 10 m mast setup with the motor-driven alignment system. The calibration uncertainty associated with signal-to-clutter ratio of the former is ten times smaller than for the latter. 15 Cloud radar calibration results are found to be repeatable when comparing results from a total of 18 independent tests. Once calibrated the cloud radar provides valid reflectivity values when sampling mid-tropospheric clouds. Thus we conclude that the method is repeatable and robust, and that the uncertainties are precisely characterized. The method can be implemented under different configurations as long as the proposed principles are respected. It could be extended to reference reflectors held by other lifting devices such as tethered balloons or unmanned aerial vehicles. 20


Introduction
Clouds remain to this day one of the major sources of uncertainty in future climate predictions (Boucher et al., 2013;Myhre 25 et al., 2013;Mülmenstädt and Feingold, 2018). This arises partly from the wide range of scales involved in cloud systems, where a knowledge of cloud micro-physics, particularly cloud-aerosol interaction, is critical to predict large scale phenomena such as cloud radiative forcing or precipitation.
To address this and other related issues, the ACTRIS Aerosols, Cloud and Trace Gases Research Infrastructure is establishing an state of the art ground based observation network (Pappalardo, 2018). Within this organization, the Centre for Cloud Remote ment is presented. Section 5 presents an analysis of the sources of uncertainty and bias involved in our calibration experiment.
Section 6 presents the final calibration results, the uncertainty budget and an analysis of the variability in the calibration bias correction, followed by the conclusions.

2 Equations used in Radar Calibration
The absolute calibration of a radar consists in determining the RCS Calibration Term C Γ and the Radar Equivalent Reflectivity Calibration Term C Z . They enable the calculation of Radar Cross Section Γ(r) (RCS) or Radar Equivalent Reflectivity Z e respectively, from the power backscattered by a punctual or distributed target towards the radar (Bringi and Chandrasekar, 2001). 65 Equation (2a) presents an expression for the RCS calibration term C Γ (T, F b ) of a FMCW radar as a function of its internal parameters. The deduction of this expression is shown in the supplementary material. G t and G r are the maximum gain of the transmitting and receiving antennas respectively, dimensionless. λ is the wavelength of the carrier wave in meters and p t is the power emitted by the radar in watts.
The gain of solid state components changes with variations in their temperature. Thus we make this dependence explicit in 70 the receiver loss budget L r (T, F b ) and in the transmitter loss budget L t (T ). The loss budget is the product of all losses divided by the gain terms at the end of the receiver or emitter chain, and has no dimensions.
Additionally, a range dependence is included in L r (T, F b ) to account for variations in the receiver IF loss for different beat frequency F b values. The beat frequency in FMCW radars is proportional to the distance between the instrument and the backscatterer element (Delanoë et al., 2016). Thus, changes in the IF loss for different beat frequencies introduce a range 75 dependent bias. For the 12.5 meter resolution mode used in this calibration excercise, F b ranges between 168 and 180 M Hz, and can be related to r (in meters) using Eq. (1).
In theory, C Γ (T, F b ) can be calculated by characterizing the gains and losses of every component inside the radar system and adding them. This can be very challenging, depending on the complexity of the radar hardware and the available radio 80 frequency analysis equipment. In addition, with this procedure it is not possible to quantify losses due to interactions between different components, specially changes in antenna alignment or radome degradation (Anagnostou et al., 2001). This motivates the implementation of an end-to-end calibration, which consists on the characterization of the complete radar system at once by using a reference reflector and Eq. (2b).
C Γ (T, F b ) = 10 log 10 L t (T )L r (T, F b )(4π) 3 G t G r λ 2 p t (2a) 85 Γ(r) = C Γ (T, F b ) + 2L at (r) + 40 log 10 (r) + P r (r) (2b) Equation (2b) links the calibration term C Γ (T, F b ) to the RCS Γ(r) of a target at a distance r. Γ(r) is expressed in dBsm units (decibels referenced to a square meter), L at (r) is the atmospheric attenuation between the object and the radar in dB, which can be calculated using a millimeter-wave attenuation model (for ex. (Liebe, 1989)), P r (r) is the power received from the target in dBm and C Γ (T, F b ) is the RCS calibration term in dB. The units in the RCS calibration term compensate the 90 radar power units, guaranteeing the retrieval of physical RCS values. The explicit temperature and range dependency of the calibration term has the function of compensating gain changes in P r (r) introduced by temperature effects and variations in the IF loss with distance.
This principle can be used in an end-to-end calibration by installing a target with a known RCS Γ 0 at a known distance r 0 and sampling the power P r (r 0 ) reflected back to calculate C Γ (T, F b ). However, some additional considerations must be made 95 to perform this retrieval.
In Eq. (2a) we state that the calibration value has a temperature and a range dependency. Experimental results indicate that the temperature dependency of C Γ (T, F b ) can be approximated by a linear relationship, as shown in Eq. (3). Here n is the temperature dependency term in dB • C −1 , T the internal radar temperature in • C and T 0 a reference temperature value in • C.
More details about the temperature correction can be found in Sect. (5.4).

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The range dependence of C Γ (T, F b ) is treated independently by defining a IF loss correction function f IF (F b ), in dB. This function is introduced to compensate for relative loss variations at different IF frequencies. The IF loss correction function is studied in Sect. 5.5.
From the afforementioned observations, we divide C Γ (T, F b ) in three components, shown in Eq.
(3). This separation consists of a constant calibration coefficient C 0 Γ , in dB, and the two correction functions n(T − T 0 ) and f IF (F b ).
As f IF (F b ) corrects for relative variations in receiver loss with distance, we define f IF (F 0 ) = 0 at the IF frequency value F 0 , associated to the reflector position r 0 (linked by Eq. (1)). Using this and Eqs. (2b) and (3), we get Eqs. (4a) and (4b).
Equation (4a) shows how the calibration term C Γ (T, F 0 ) at position r 0 is related to the calibration coefficient C 0 Γ and the temperature correction n(T − T 0 ). Meanwhile, Eq. (4b) indicates how experimental P r (r 0 ) measurements can be associated with a C Γ (T, F 0 ) value, using in-situ information to calculate 2L at (r 0 ). Then, using Eq. (4a), we can compute C 0 Γ by substracting the temperature correction function n(T − T 0 ). This temperature correction is derived independently in Sect. (5.4). Knowing C 0 Γ and the temperature correction, C Γ (T, F b ) is calcualted by adding the IF correction function, independently retrieved in 115 Sect. 5.5.
Once C Γ (T, F b ) is known, we can calculate the radar Equivalent Reflectivity calibration term C Z (T, F b ), in dB, with Eq.
(5a) (Yau and Rogers, 1996). This relationship assumes the radar has two identical parallel antennas with a Gaussianly shaped main lobe. θ is the antenna beamwidth in radians, mδr is the radar distance resolution in meters and |K| = |( r − 1)/( r + 2)| is the dielectric factor. This factor is related to the relative complex permittivity r of the scattering particles, and can be 120 calculated, for example, using the results of Meissner and Wentz (2004).
C Z (T, F b ) enables the calculation of the Radar Equivalent Reflectivity Z e in dBZ units of a distributed target located at a distance r, by using the relationship of Eq. (5b).

Experimental setup
Two calibration campaigns, that lasted one month each, were performed in May-June of 2018 and March-April of 2019 at the SIRTA observatory, located in Palaiseau, France (Haeffelin et al., 2005). The observatory has a 500 meter long grass field in an area free of buildings, trees or other sources of clutter, well suited to install our calibration setup, shown in Fig. 1.
The instrument used for the calibration experiments is a BASTA-Mini. BASTA-Mini is a 95 GHz FMCW radar with 130 scanning capabilities and two parallel Cassegrain antennas (Delanoë et al., 2016). The antennas are separated by 35 cm, and have a Fraunhofer far field distance of ≈ 50 m with a Gaussianly shaped main lobe (verified experimentally in Sect. 5.2).
Transmitted power is fixed to 500 mW , and is under constant monitoring using a diode with an uncertainty of ≈ 0.4 dB. The diode enable the monitoring of L t (T ) variations, yet our experiments have shown that T is a better indicator to capture the variability of C Γ (T, F b ). This is likely because internal temperature changes affect both L r (T, F b ) and L t (T ) simultaneously, 135 and therefore the information provided by the diode is not sufficient to capture the behavior of the whole system. The results of the temperature dependency study for our radar is shown in Sect. 5.4.
This radar also includes hardware to enable the tuning of the carrier wave frequency within a range of ≈ 1 GHz, centered at 95 GHz. During the experiments we fixed the BASTA-Mini base frequency at 95.64 GHz to avoid any interference with the other two W band radars operating in parallel at the same site.

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Our reference targets are two Triangular Trihedral Reflectors (also known as Corner Reflectors) composed by three orthogonal triangular conducting plates. Trihedral targets have a large RCS for their size and a low angular variability of RCS around their boresight (Atlas, 2002;Doerry and Brock, 2009;Chandrasekar et al., 2015). One reflector has a size parameter of 10 cm, with a maximum theoretical RCS at our radar operation frequency of 16.30 dBsm. The other is 20 cm with a maximum theoretical RCS of 28.34 dBsm (Brooker, 2006). These targets were mounted on top of masts B and C in Fig. 1 respectively.

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Only mast C was used in the 2018 campaign, while both were used in 2019.
To align the system first we aim the radar towards the approximate position of the target. Second, we aim the target by slowly changing pan-tilt angles in the motor on mast B, or axially rotating the tube of mast C to maximize the power P r (r 0 ) measured at the radar. Third, radar aiming is tuned around target position until the maximum reflected power is found. Finally, we repeat the second step, after which we have the system ready to sample P r (r 0 ).

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It must be mentioned that this procedure does not guarantee a perfect alignment. In fact, it is impossible to have every element perfectly adjusted because of limits in the positioner resolution or uncertainties introduced when installing each element.
Sections 4 and 5.6 explain how we deal with these limitations.

Methodology
This section describes the procedure followed when performing calibration experiments using the setup described in Sect. 3.

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The methodology has the objective of quantifying and correcting when possible all sources of uncertainty to enable a reliable estimation of the calibration terms C Γ (T, F b ) and C Z (T, F b ).
A challenge we found when using targets mounted on masts to estimate C Γ (T, F b ) is that the value of the target RCS Γ 0 may vary depending on how components are aligned. Our studies have shown that for the feasible alignment accuracy we can get when installing our setup, this effect is in the order of tenths of dB, and therefore not negligible. Additionally, we concluded 160 that if we leave this uncertainty source uncorrected, we would introduce a bias in the calibration result (see Sect. 5.6).
The flow chart of Fig. 2 illustrates the calibration procedure. To quantify the bias introduced by alignment uncertainty we decided to divide each calibration expreriment in N iterations. Each iteration consists on a system realignment, followed by sampling of the target signal P r (r 0 ) for at least one hour. Then, we select the data from the contiguous hour with the lowest variability as the iteration result.

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The period chosen to perform the sampling is important, because it will have an incidence on how stable is the calibration value. To minimize uncertainty it is recommended to perform calibration iterations when the atmosphere is clear, there is no rain and wind speed is under 1 ms −1 . However, these requirements may change depending on how robust is each setup to atmospheric conditions. FMCW radars have a discrete distance resolution. Consequently, power measurements vs distance are resolved in finite 170 discrete points, usually named gates. Because of this resolution limitation, power received from a point target is spread between the gates closer to its position (Doviak and Zrnić, 2006). This phenomena is known as spectral leakage. To reduce leakage BASTA-Mini uses a Hann time window (Richardson, 1978;Delanoë et al., 2016).
To correctly asses the total reflected power we set the radar resolution to 12.5 meters (chirp bandwidth of 12 MHz), and its integration time to 0.5 seconds. This resolution is high enough to accurately identify the reference reflector signal while 175 avoiding the introduction of additional clutter from the trees located behind the mast (see Fig. (3)).
To calculate P r (r 0 ) we add five gates: the target's gate plus two before and two after the target's position. Adding more contiguous gates increase the power value by less than 0.01 dB, thus we conclude that these five gates concentrate almost all the power reflected back from the target.
Then P r (r 0 ) is corrected considering compression effects and antenna overlap losses (Sects. 5.1 and 5.2). For each corrected 180 P r (r 0 ) sample we proceed to calculate a single C 0 Γ value with Eq. (4a) and the temperature correction function. This single sample is defined as C 0 Γs to differentiate it from the final calibration coefficient C 0 Γ of Eq. (3). Atmospheric attenuation L at (r 0 ) is calculated using in-situ atmospheric observations and the model published by Liebe (1989).
The target effective RCS Γ 0 is calculated using a theoretical RCS model, considering the beam incidence angle on the target.
Echo chamber measurements have shown that real targets RCS can be deviated from the theoretical value depending on the 185 manufacturing precision. Our corner reflectors have an angular manufacturing precision better than 0.1 • , therefore real RCS uncertainty with respect to the model can be roughly estimated to be approximately 2 dB (Garthwaite et al., 2015). Once an experimental characterization of the target becomes available, it can be used to correct any calibration bias and to reduce uncertainty by rectifying the value of Γ 0 used in the calculations.
We performed one calibration experiment with 6 iterations during the 2018 campaign using the 20 m mast. In the 2019 190 campaign we did two experiments: one with 10 iterations using the 10 m mast and another with 2 iterations on the 20 m mast ( Fig. 1).
The retrieval of the temperature dependency coefficient n and the reference temperature T 0 is done simultaneously with the calibration coefficient experiment, by extending the sampling period beyond one hour when using the 20 m mast. This is done to capture the temperature effect in the variability of C 0 Γs , by capturing a larger part of the temperature daily cycle.

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The results of this experiment can be seen in Sect. 5.4. Likewise, the retrieval of the IF correction function f IF (F b ) is an independent experiment based on sampling noise with the radar to get the IF amplification curve of the receiver. The details of this experiment are in Sect. 5.5.
From each iteration we get a distribution of resulting C 0 Γs values with a small spread introduced by second order effects. The average value of each iteration i is named C 0 Γi , and its corresponding standard deviation is named σ i . With this information we 200 proceed to calculate the bias corrected calibration coefficient C 0 Γ , by using Eq. (6).Λ is the bias correction term. The method used to calculateΛ relies on simulating the probability distribution of Γ 0 for a given set of uncertainties in the setup parameters.
More detail can be found in Section 5.6 and Section S3 of the supplementary material.
Equations (7a) and (7b) show the uncertainties δC Γ and δC Z associated with the estimation of C Γ (T, respectively. σ T is the uncertainty term associated with the temperature correction function n(T − T 0 ).
σ IF is the uncertainty term associated with the IF loss correction function f IF (F b ).
The term σ 2 i comes from the averaging operation in the estimation of C 0 Γi (Eq. 6). Since the C 0 Γi terms are corrected using the temperature correction function, the uncertainty of the later must be propagated as well, hence the term σ 2 T /N appears.

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σ Λ the uncertainty of the bias correction calculation. It is calculated from the standard deviation σ i . This procedure is explained in Section S3 of the supplementary material.
σ SCR is the uncertainty introduced by clutter. Clutter is the presence of unwanted echoes which affect our reading of P r (r 0 ), coming from reflections on other objects in the environment. The method to quantify the uncertainty σ SCR uses a parameter named Signal to Clutter Ratio (SCR), explained in detail in Sect. 5.3. 215 σ Γ0 is the uncertainty of the reference target RCS. In this work we use a theoretical model to calculate the target effective RCS, which has an uncertainty of approximately 2 dB based on the manufacturing characteristics. The inclusion of an experimental characterization of the target RCS can improve the estimation of C 0 Γ and δC Γ by reducing this uncertainty term. σ K is the uncertainty in the estimation of the backscattering particles dielectric factor. Because our objective is to calculate the calibration term of the radar, we reference this value to |K| = 0.86, corresponding to pure water at 5 • C and neglect the δ K 220 uncertainty term. However, the value of K and its uncertainty σ K must be considered when performing radar retrievals (e.g. Sassen (1987); Liebe et al. (1989); Gaussiat et al. (2003)).
σ A is the uncertainty introduced in the estimation of θ and from parallax errors and deviations from a Gaussian beam shape (Sekelsky and Clothiaux, 2002). For this work we make the assumption of parallel antennas with a Gaussian beam shape, thus we neglect this term. This problem is discussed more in depth in Section 5.2.

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Since both σ K and σ A are neglected, we get δC Γ ≈ δC Z .

Sources of uncertainty and bias in Absolute Calibration with corner reflectors
In this section we identify and quantify the uncertainty and bias introduced by several terms in Eq. (2b). Following the rec-230 ommendations in the work of Chandrasekar et al. (2015), we study the impact of receiver saturation, signal to clutter ratio, antenna lobe shape and antenna overlap. Additionally, we consider the impact of temperature fluctuations inside the radar box, loss changes with distance due to uneven amplification at the receiver IF and the effects of imperfect alignment of the reference target.

Receiver compression 235
It is advisable to design calibration experiments which avoid the appearence of compression effects. If this is not possible, compression must be considered in the data treatment so that the retrieved calibration remains valid in the receiver's linear regime, where it usually operates during cloud sampling (Scolnik, 2000).
For studying how these effects could affect our calibration, we retreived the radar's receiver power transfer curve. Receiver characterization was done by removing the radar's antennas and connecting the emitter's end to the receiver's input, with two 240 attenuators in between. The first was a 40 dB fixed attenuator, while the second was a tunable attenuator covering the range between 50 and 1 dB of losses. The adjustable attenuator enabled the retrieval of the power transfer curve by varying the attenuation and sampling the power at the receiver's end (digital processing included). Our retrieved power transfer curve is shown in Fig. 4 (a).
Compression effects must be considered in calibration, or a bias will be introduced. In consequence, we include compression 245 correction in every sample of reflected power, which consists on projecting their value to the ideal linear response using the power transfer curve.
For example, the power received from the 20 cm target on the 20 m mast returned was 4.1 dBm in average, before corrections. The power transfer curve shows that at this power values we have a loss caused by compression of ≈ 0.3 dB. After correcting each power sample by compression with the power transfer curve, we obtain a corrected power average value of 4.5 250 dBm. Meanwhile, for the 10 cm target on the 10 m mast the average power value before corrections is 3.2 dBm. As this value is lower than what is obtained the 20 m mast, the associated compression effect is also smaller, of ≈ 0.2 dB. After aplying this correction to each power sample we end with a new corrected power average of 3.4 dBm.

Antenna Properties
Manufacturer specifications indicate that antenna beamwidth should be of 0.8 • . However, data from an experimental charac-255 terization done by the same manufacturer in an anechoic chamber indicate that antenna beam shape is better approximated by a Gaussian with a Half Power Beam Width (HPBW) of θ ≈ 0.88 o . The total gain difference between the experimental curve and the Gaussian approximation of ≈ 0.0003 dB in the HPBW region. Therefore, we conclude that the contribution to uncertainty introduced by assuming a Gaussian beam shape is negligible. The Antenna beam shape and Gaussian curve are shown in Fig.   4 (b).

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Another source of bias introduced by the antennas is the parallax error. Antenna parallax errors introduce a range dependent bias, determined by the antenna beamwidth and the relative angles of deviation between the antennas boresight. This bias is usually larger in the first few hundred meters closest to the radar. For example, for a deviation of half the antenna beamwidth, losses would be on the order of 10 dB and would vary significantly over the first hundreds of meters, decreasing with distance to about 1 dB at a approximately 4 kilometers (Sekelsky and Clothiaux, 2002).

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To study this effect we took advantage of our experimental setup and the scanning capabilities of the radar, to check if the radar antennas were properly aligned. This was done by using the target on the 20 m mast. Results are shown in Fig. 4

(b). After
analyzing the results we observed that the aiming uncertainty is in the same order of magnitude of the antennas beamwidth.
Since the correction of the parallax error requires a very precise measurement of antenna alignment, we conclude that it is not possible to correct for antenna deviations directly with this information.

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However, the relativelly small difference of 0.5 dB in the estimation of C 0 Γ during the calibration experiments of 2019, obtained using two masts in the most sensitive distance range (placed at 196 and 376.5 meters of distance respectively), indicate that antennas are unlikely to have a deviation comparable to their beamwidth (calibration results in Sect. 6).
Therefore, for the present version of this calibration methodology we assume that both antennas are parallel and that they have a Gaussian beam lobe. Once a reliable method for antenna pattern retrieval is developed for W band radars, it can be 275 directly incorporated into the calibration term by adding an additional correction function f A (r) to Eq. (3). The uncertainty in this alignment estimation can also be included in the uncertainty budget, with the term σ A of Eq. (7b).
Even if antennas are parallel, it is necessary to include a correction for the loss L o (r) caused by incomplete antenna overlap.
The correction, shown in Eq. (8), accounts for the loss of power that would be received from a point target compared to a monostatic system (Sekelsky and Clothiaux, 2002). This loss occurs because a point target cannot be in the center of two 280 non-concentric parallel antenna beams.
Equation (8) assumes that the radar has two identical, parallel antennas with Gaussian beam lobes. Their main axis is separated by a distance d, and the point target is located at a distance r, facing the geometrical center of the radar, where the gain is maximum. For the BASTA-Mini d = 35 cm. This introduces a loss of 0.08 dB for the target at r 0 = 196 meters of 285 distance, and of 0.02 dB for the target at r 0 = 376.5 meters.

Signal to Clutter Ratio
The power sampled from our reference reflector is an addition of the power from the target (signal) and unwanted reflections on other elements in the environment, such as the ground or the mast (clutter). We observed that this clutter dominates above the radar noise, and thus becomes the main source of interference in our calibration signal.

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To quantify the impact of clutter we use the Signal to Clutter Ratio (SCR) parameter. It is calculated as the ratio of total power received from the target to power received from clutter under the same configuration, but with the reference reflector removed. SCR enables the uncertainty σ SCR introduced by clutter in the sampled P r (r 0 ) values to be computed (Chandrasekar et al., 2015).
Clutter power is sampled and corrected following the same methodology used for reflector P r (r 0 ) retrievals, but in an 295 scanning pattern mode to capture clutter around the mast area. Figure 5 shows our results from scanning around the 10 and 20 m masts with targets removed.
We observe that the 10 m mast is more reflective than the 20 m one. This may be caused by its smaller height (more ground clutter) and its larger geometrical cross-section. We can also see that the signal in the 10 m is stronger where absorbing material is not present (below ≈ 1.5 • of elevation). In both cases we did not detect any signal from the nearby trees close to the target's 300 position.
To calculate SCR we compare the average power received from each target during the calibration experiments with the maximum clutter power observed in a region of 0.125 o around the target's coordinates, vertically and horizontally. The value is taken from the radar's positioner resolution.
The average power received from the 10 cm target on the 10 m mast is 3.4 dBm. This provides an SCR value of 19.4 dB, 305 which implies a σ SCR uncertainty value of ≈ 0.93 dB. From the 20 cm target on the 20 m mast, the average received power is 4.5 dBm. Its SCR equals 40.1 dB, which is translated as an uncertainty contribution of σ SCR ≈ 0.09 dB. From the results we see that even if target alignment is better with the 10 m mast, calibration results may not get less uncertain because the motor used for target alignment acts as a big source of clutter.

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BASTA-Mini has a regulation system to control temperature fluctuations inside the radar box. However, since the radar is based on solid state components, even small temperature fluctuations may impact the performance of the transmitter and receiver, and therefore affect the calibration stability. To account for this effect we introduced a temperature dependency in the calibration term, shown in Eq. (3).
During the experiments we verified the need of this correction by observing that the retrieved calibration term C Γ (T, F 0 ) 315 has a consistent change depending on the time of the day, and that this change is strongly correlated to the temperature inside the radar. (c) presents the timeseries of (a), but in a daily cycle perspective. Here we plot hourly means of the deviation of C Γ (T, F 0 ) with respect to the total average, with its hourly standard deviation as errorbars. We also superimposed atmospheric attenuation and the radar amplifier temperature to show that the first has a much smaller impact in calibration variability compared to the second. The reference T 0 value is chosen because it is approximately the average internal temperature when considering all the experiments.

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To maximize the range of temperatures covered we choose to not limit the sampling period to one hour. This decision has the drawback of increasing the noise of the dataset due to the inclusion of some data taken under suboptimal conditions, for example with wind speed velocities above 1 meter per second or with the presence of drizzle. Yet, this step is necessary to enable the retrieval of the temperature correction function for the widest range of temperatures possible.
To retrieve the temperature dependency we perform a linear regression over the results from all the experiments done in 2018 335 and 2019, shown in Fig 7. The regression shows that the variability in the calibration term has an almost linear relationship with internal radar temperature, in the dB scale, and it is the same for both campaigns. This analysis allows us to estimate the value n = 0.093 dB • C −1 for the temperature correction function of Eq. (3). To estimate the uncertainty of the temperature correction function we calculate the RMSE between the linear regression model and the whole dataset, for each degree of deviation in temperature. The RMSE value for the complete dataset is of 0.13 dB, while its value per degree ranges between 0.07 and 0.23 dB for a deviation of 0 and +3 • C respectively. These results enable us to conclude that the temperature correction function uncertainty σ T is ≤ 0.23 dB.

IF loss correction function
FMCW radars rely on estimating the beat frequency of the received signal to estimate the distance of an object. This signal may suffer uneven amplification depending on its frequency, because of a frequency dependent gain function in the amplifiers 345 of the IF chain of the radar. Since there is a direct relationship between the IF frequency F b and the target distance r, this dependency on the beat frequency introduces a gain variability with respect to the target distance r. As introduced in Sect. 2, this distance dependency is compensated in the calibration term with a IF correction function f IF (F b ) The power P r (r) measured by the radar receiver when no active signal is input corresponds to the system noise temperature N s (F b ) plus the environmental noise power density N 0 , reduced by the radar total receiver loss log 10 (L r (T, F b )), as indicated 350 in Eq. (9a) (Pozar, 2009).
The standard way to retrieve each of this terms is to perform a two point calibration. This requires the use of two noise sources at significantly different and well known temperatures. Usually, the temperatures of the loads are environmental temperature 355 (298 K) and liquid nitrogen (77 K) (Rodríguez Olivos, 2015).
However, this approach was not practical with the equipment available on site, thus a different approach is used. BASTAmini radar has a IF bandwith of 12 M Hz, while the receiver Low Noise Amplifier (LNA) has a bandwidth of 35 GHz. Since the operational bandwidth is much narrower than the receiver full bandwidth, it is reasonable to assume that system noise is constant in the IF bandwidth (168 to 180 M Hz).

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This enables the approximation of Eq. (9b). Here, we assume that system and environmental noise are constant in the IF frequency range, and that their addition can be expressed as a constant noise power N c . To estimate the uncertainty introduced by this approximation, we observe that the gain and noise figure of the radar LNA have variations smaller than 0.1 dB in the operational frequency range. This indicates that uncertainty coming from assuming a constant system and environmental noise power for the IF bandwidth should be in the order of 0.1 dB or lower.

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Then, to retrieve the f IF (F b ) we turn off the radar emitter and sample the environmental noise with the radar operating in its calibration configuration (12.5 meters distance resolution and 0.5 seconds integration time). After retrieving a significant amount of noise samples we calculate the average value of the difference P r (F 0 )−P r (F b ) for each IF frequency F b , to remove the effect of the unknown noise power density. This operation is done to quantify relative gain variations around the calibrated frequency F 0 .
370 By using Eqs. (2a) and (3), we get that the difference P r (F 0 ) − P r (F b ) is equivalent to the difference between C Γ (T, F b ) and C Γ (T, F 0 ), and therefore it is equivalent to the IF correction function f IF (F b ) (Eq. (10)). The temperature effect in gain is removed because both P r (F 0 ) and P r (F b ) are sampled simultaneously, and therefore under the same temperature conditions.
For this experiment only, P r (F 0 ) corresponds to the power measured at the gate closer to the reference target position, 375 without integrating other gates. This is done because there is no significant leakage and, as results of Fig. (8) show, L r (T, F b ) changes are negligible in the five gates used for integration. Figure 8 shows the results of the IF correction function retrieval referenced to P r (F 0 ), using F 0 associated to the target distance r 0 = 376.5 meters (corresponding to the 20 m mast experiment setup). We can observe that all functions retrieved in 2019 are in close agreement, without significant variations between different dates or time of the day chosen for the plots.

Misalignment Bias
The retrieval of C Γ (T, F 0 ) using Eq. (4b) requires a precise knowledge of the reference target effective RCS Γ 0 . Each dBsm of difference between the theoretical value used in calculations and the effective target RCS will introduce a bias of the same magnitude in the estimation of the calibration coefficient C 0 Γ , and thus in C Γ (T, F 0 ). The effective reflector RCS is the actual physical value that would be measured by a perfectly calibrated radar. It is different 395 from the target intrinsic RCS which only depends on its physical properties. Effective RCS changes when the experimental setup is modified. For example, if the point target is not exactly in the beam center, antenna gain will not be maximum and therefore the effective RCS will decrease compared to the intrinsic value. Effective RCS also changes when the incidence angle of the radar beam is modified. This latter effect may increase or decrease effective RCS depending on the original situation.
A common approach in these type of experiments is to set Γ 0 to be the maximum theoretical RCS of the target, assuming 400 misalignment will cause a negligible deviation from this value. This procedure can be refined for cases where the system default configuration does not have the target boresight aligned with the radar position. In these cases, effective RCS can be calculated using equations derived from geometrical optics (more complex optical calculations may be necessary for other wavelengths or target sizes). For example, we use the equations published by Doerry and Brock (2009) when calculating the effective RCS of our Triangular Trihedral target on the 20 m mast. 405 Unfortunately, this approach does not correct the impact of alignment uncertainties. We observed that random errors in the element positioning will statistically impact the effective Γ 0 in a single direction. Thus, simply taking the average of many target sampling iterations would result in a biased estimation of the calibration.
With the objective of quantifying the impact of alignment uncertainties we developed a geometrical simulator of effective RCS. This simulator receives as input the position of each element in the setup and calculates the effective RCS considering 410 the beam incidence angle and antenna gain variations when the target is not in the center of the beam. The degrees of freedom included in the simulator are shown in Fig. 9 (a). It enables the modification of the radar aiming angles, the mast dimensions and the positioning and orientation of the target. The equations used in the simulator can be found in the article support material.
We now use the simulator to study how uncertainty in alignment can affect the value of Γ 0 . For this, we model an example experiment based on the 20 m mast setup. In this model we separate input variables between known and uncertain. Known 415 terms can be fixed or measured very precisely in the field experiment, hence they are set as fixed values. Meanwhile, uncertain terms represent the parameters that cannot be fixed or measured very precisely, and for that reason are better expressed as probability distributions (terms defined in Fig. 9 (a)).
In the uncertain variables, θ * r = 87.82 • , φ * r = 0 • and τ * = 0 • represent the nominal alignment angles, which are the values expected under an ideal field experiment where the radar aims directly to the target and the mast is perfectly vertical. To these nominal values we associate a distribution shape and the uncertainty set σ θr = 0.075 • , σ φr = 0.075 • , σ θ = 1.5 • , σ τ = 5 • . Each term, known and uncertain, is estimated from observations done during the experimental field work.
With these input parameters we sample the Γ 0 distribution that would arise after a large amount of experimental iterations. 435 Figure 9 (b) shows the results from this sampling. The black dashed line shows the effective RCS under our experimental configuration, when each element is in its nominal position. We can see that this effect cannot be neglected in our case, since its value is 0.8 dB lower than the maximum theoretical RCS.
However, this single correction does not suffice. The results of the model show that the addition of uncertainty into the process induces another bias of ≈ 0.3 dB, in average. Since this is withing the order of magnitude of our desired uncertainty 440 in the calibration, the example clearly illustrates the need of including a bias correction step in our calibration methodology.
The standard deviation σ between N experimental retrievals of C 0 Γi cannot be used directly as an estimation of uncertainty because the RCS distribution shape is not Gaussian. The uncertainty introduced by this variability is studied by sampling a large set of possible RCS distributions based on our experimental configuration, and selecting the candidates matching our observed spread σ . This set provides an estimation of the expected bias correctionΛ and of the effective RCS uncertainty σ Λ .

445
The uncertainty of the C 0 Γ estimator of Eq. (6) will correspond to the uncertainty of each C 0 Γi estimation propagated through the calculation of their average (terms σ 2 i /N 2 and σ 2 T /N of Eq. (7a)) plus the effective RCS uncertainty σ Λ . The details on how this estimator works and how the RCS distribution sampling is done are fully explained in Sect. S3 of the supplementary material.

450
In 2018 we used the 20 m mast only, performing six iterations. For 2019 we did 10 iterations using the 10 m mast and 2 iterations with the 20 m mast. The distributions of C 0 Γ obtained in each iteration and experiment is shown in Fig. 10. The radar hardware changed between 2018 and 2019 campaigns due to experiments required to retrieve the power transfer curve and perform maintenance operations. This implies that we cannot compare absolute calibration values between both campaigns. What remains valid is to compare properties such as the variability, and the results from both experiments of 2019.

455
In the results we can notice a difference in C 0 Γi spread when comparing the 10 and 20 m masts. The 6 iterations of 2018 ( Fig. 10 (A)) have an spread of σ = 0.33 dB, while the spread of the 10 iterations of 2019 is 0.11 dB (Fig. 10 (B)). This happens because the 10 m mast has a motor on top which enables a much finer adjustment of the target position, improving the repeateability of the experiments.
There is also a small difference in the spread of the curves. The dynamic, specially wind speed variability. The introduced variability was not fully compensated by our corrections and thus bimodal distributions remained. However, individual spread is still small, within ≈ 0.1 dB, so we decided to accept these samples for calibration purposes.
To study the dependency of the bias correction on the amount of iterations we calculate the bias correction termΛ and its uncertainty σ Λ of experiments (A) and (B) with different amounts of repetitions. The order of the iterations used in each row 470 match the sequential order indicated in Fig. 10. The results are shown in Table 1. For both cases we have the best estimate when we use all the samples available for each experiment, and thus we use this bias correction and uncertainty when computing the calibration coefficient.
For experiment (C) we followed a different approach. Because we only have two samples, the calculated σ = 0.2 dB is very likely to be underestimated. Consequently, and because the experimental procedure was identical to what was done in 475 2018, we assume our parameters σ ,Λ and σ Λ to be equal to the best estimation of experiment (A). This is possible because in our methodology we assume that the bias probability distribution of a given system is unique, even if it is unknown, and what is done by performing many iterations is to successively restrict the possible sets of uncertainties that can generate results consistent with the observations. This latter hypothesis is consistent with the decrease in uncertainty for the bias correction when increasing the amount of iterations.
480 Table 2  and C Z (T, F b ), alongside their uncertainty. Since the term σ Γ0 is much larger than all other uncertainty sources, we calculate a partial calibration uncertainty including all but this term, to simplify the comparison of uncertainty contributions between different experimental setups. This term is then added for the calculation of the final result. C Z (T, F b ) is calculated for the 485 range resolution δr = 12.5 m, which is the same mode used for target sampling. T is the radar amplifier temperature in o C and is the IF loss correction function.
-(A) 20 m mast -2018: -(B) 10 m mast -2019: f IF (F b ) = 7.60 · 10 −6 F 6 b − 7.97 · 10 −3 F 5 b + 3.48F 4 b − 8.10 · 10 2 F 3 b + 1.06 · 10 5 F 2 b − 7.40 · 10 6 F b + 2.15 · 10 8 [dB] calibration campaigns (for ex. 3 iterations means we used iterations 1, 2 and 3 of the experiment). We include the average and spread σ between the retrieved C 0 Γi for each case. This variability σ is introduced in the bias estimation procedure to determine the bias correctionΛ and its uncertainty σΛ. -(C) 20 m mast -2019: These results enable the analysis of relative uncertainty contributions from different sources, however the total calibration uncertainty may be underestimated. As indicated in Sects. 4 and 5, some bias terms remain unknown. Specifically, target physical RCS must be measured in an echo chamber to improve the misalignment bias estimation. In addition, the method to 505 characterize antenna alignment must be improved to determine if there is a need for an additional distance correction function (Sect. 5.2). The uncertainty of these retrievals will impact the total uncertainty value, however, it is possible to quantify this effect through the terms σ Γ0 and σ A of Eq. (7b).
To finalize, we perform a test of the calibration results by measuring a altostratus cloud in both campaigns (Fig. 11). The sampling was done with the 25 m resolution, and thus 6 dB had to be substracted from the C Z (T, F b ) calibration calculated 510 for the 12.5 m resolution. In this correction, 3 dB come from the change in the distance resolution term δr (Eq. (5a)), and the other 3 dB are substracted to compensate the additional digital gain coming from doubling the amount of points in the chirp fourier transform (Delanoë et al., 2016). A Signal to Noise Ratio threshold of 8 dB is used to remove noise samples. We observe that for both campaigns the reflectivity measured in altostratus cloud is within −30 -0 dBZ, which are typical values reported in literature (Uttal and Kropfli, 2001).

Conclusions
This study presents a cloud radar calibration method that is based on cloud radar power signal backscattered from a reference reflector. We study the validity of the method and variability of the results by performing measurements in two experimental setups and analyzing the associated results. In the first experimental setup we use a scanning BASTA-Mini W-band cloud radar, that aims towards a 20-cm triangular trihedral target installed at the top of a 20-m mast, located 376.5 m from the radar. For 520 the second experimental setup, we use the same radar, aimed towards a 10-cm triangular trihedral target mounted on a pan-tilt motor at the top of a 10-m mast. The mast is located 196 m from the radar.
The first consideration in the design of the experimental setup is the need to avoid excessive compression or saturation in the radar receiver. This must be checked before any calibration attempt by comparing measurements of radar backscattered power with the radar receiver power transfer curve. In both our setups we find losses due to compression on the order of 0.2 ∼ 0.3 525 dB. There is a compensating effect between target RCS and radar-to-target distance (Eq. 2b). Since the compression effect is small, we correct it using our receiver power transfer curve. However, in cases where the radar is operating close to saturation, or when compression effects are larger than the calibration uncertainty goal, it is advisable to compensate by reducing target size or by positioning the target farther away from the radar.
Secondly, the reflector must be positioned far enough from the radar to be outside the antennas near-field distance and 530 to ensure that the received power has low antenna-overlap losses. The BASTA-Mini cloud radar has a Fraunhofer near-field distance of 50 m. The estimated maximum overlap loss is less than 0.1 dB for the closest (10-m) mast setup. Thus we conclude that the target positioning is far enough for both setups.
Thirdly, the experimental setup should strive to reduce clutter in the radar measurements. This can be achieved by operating in an open field that is several hundred meters in length and free of trees or other signal-inducing obstacles. It is also advisable 535 to perform radar measurements under clear conditions, without fog or rain, with wind speed below 1 ms −1 and low turbulence.
Next, the proposed calibration method requires performing several iterations in the same setup configuration. In each iteration the setup is first realigned, followed by approximately one hour of sampling of the reference reflector backscattered power. The sampled power is then corrected for compression effects, incomplete antenna overlap, variations in radar gain due to temperature and atmospheric attenuation, before being used to estimate a RCS calibration term value. Once all iterations are completed, the final RCS and Equivalent Reflectivity calibration terms can be computed with their respective uncertainties.
Iterations are necessary because they enable the quantification of bias introduced by inevitable system misalignment. Our experiments indicate that, for our setup, at least 5 iterations are necessary to reach convergence in the calculation of bias and uncertainty associated with misalignment. We find a bias correction of ≈ 0.4±0.3 dB for the 20-m mast, and of ≈ 0.2±0.1 dB for the 10-m mast. This difference can be explained by the more precise alignment attainable with the pan-tilt motor installed 545 on the 10 m mast.
Calibration is also impacted by changes in the gain of radar components associated with internal temperature variations.
For the radar used in our experiment, these changes reach up to ±0.6 dB. Our experiments enabled us to retrieve a correction function for the temperature dependence and to reduce the temperature uncertainty contribution to σ T = 0.23 dB. This result indicates that lower calibration uncertainties can be achieved by studying temperature effects, especially for solid state radars.

550
Another necessary consideration is the inclusion of gain variations with distance, introduced by frequency dependent losses in the IF of the radar receiver. We found calibration variations with distance up to 0.9 dB for the 2019 campaign. Therefore, characterizing the IF loss is a necessary step to validate the calibration results for all ranges. For future work we envisage to develop a technological solution to allow target orientation without introducing additional clutter. Another interesting prospect is to improve the accuracy of the radar positioner, to enable direct retrieval of antenna 570 pattern directly with the radar, following the method proposed by Garthwaite et al. (2015). This retrieval would improve bias correction arising from parallax errors. We also plan to perform an echo chamber characterization of our reference targets, to remove any possible bias caused by manufacturing imprecision, to reduce its contribution to total uncertainty and to improve the estimation of our misalignment bias correction.
Further, there is ongoing research on calibration and antenna pattern characterization methods based on reference targets 575 held by Unmanned Aerial Vehicles (UAVs) (Duthoit et al., 2017;Yin et al., 2019). Since the underlying principle is the same, most considerations written here should be directly applicable in these new experiments. Here the UAV takes the role of the mast, holding the reflector (usually a sphere), and therefore it is important to characterize the UAV RCS and verify that it does not interfere with the experiment. The main difference would be in the procedure necessary to estimate bias, because the reference target (usually a sphere) will be always moving due to wind. Here an adaptation of the effective RCS simulator would 580 be necessary to account for the target type and different alignment protocol.
Author contributions.
All authors contributed to the planning of the campaigns and the design of the calibration experiments.
Author Julien Delanoë was responsible of radar installation and operation.
Authors Jean-Charles Dupont and Felipe Toledo worked in the preparation, development and operation of the necessary 585 infrastructure for the experiments.
Authors Julien Delanoë and Felipe Toledo retrieved the Power Transfer Curve of the Radar Receiver.
Data analysis and the establishment of the calibration methodology presented in the paper was done by Felipe Toledo.
Authors Martial Haeffelin and Felipe Toledo worked in defining the paper structure and content.
Authors Felipe Toledo, Susana Jorquera and Cristophe Le Gac worked in developing the method to retrieve the IF correction 590 function, and in its calculation.
Author Christophe Le Gac contributed with technical information about the radar.
All authors reviewed the paper.

Symbols
Symbol Description Units  Summary of a complete calibration process. Each calibration requires repetition of system realignment and sampling steps, called iterations. During each iteration we continously sample the power reflected from the reference target position for one hour (power corrections in Sect. 5.1). The retrieval of N iterations enable the 700 estimation of the system bias due to misalignments in the setup (Sect. 5.6). Temperature dependency is retrieved in an independent experiment (Sect. 5.4). Uncertainty introduced by clutter signals at the target location is also included in the total uncertainty budget (  Scanning BASTA-Mini radar located in a reinforced platform 5 m above the ground. (B) 10 mast with a 10 cm triangular trihedral target mounted on a pan-tilt motor with an angular resolution and repeateability better than 0.1 • . This mast has microwave absorbing material wrapped to it to reduce its RCS (clutter). The 10 m mast was only installed in the 2019 calibration campaign. (C) 20 m mast with a 20 cm triangular trihedral target. The target aiming is fixed relative to the mast. This mast is used in both 2018 and 2019 calibration campaigns. Angular separation between the masts is enough to sample both targets without mutual interference. Figure 2. Summary of a complete calibration process. Each calibration requires repetition of system realignment and sampling steps, called iterations. During each iteration we continously sample the power reflected from the reference target position for one hour (power corrections in Sect. 5.1). The retrieval of N iterations enable the estimation of the system bias due to misalignments in the setup (Sect. 5.6). Temperature dependency is retrieved in an independent experiment (Sect. 5.4). Uncertainty introduced by clutter signals at the target location is also included in the total uncertainty budget (Sect. 5.3).  Input power is relative to the minimum attenuation value of the curve characterization experiment. All our signal retrievals from the target are slightly under the 5 [dBm] line, thus the correction required due to compression effects is small (< 0.3 dB). (b) Normalized antenna pattern of the BASTA-Mini antennas. We can observe that the Gaussian fit with a beamwidth of θ = 0.88 • is very close to the antenna gain curve measured at the manufacturer's laboratories. This figure also shows the results from mast scans around the target to compare with the theoretical curves. To enable the comparison with the laboratory antenna pattern we assume that the gain of both antennas is identical. Then, the received power in dBm is normalized with respect to the maximum measured value and divided by 2, to represent the gain of a single antenna.       . Lower reflectivities are easier to capture at lower altitudes because of lower distance and attenuation losses (Eq. (5b)). In the altostratus reflectivity histograms (c) and (d) we observe that for both campaigns measurements are within the ranges reported in literature.