3.1 Basics of temperature and humidity sensors

Commonly used temperature and humidity sensors for UASs are mainly variants of the bead thermistor type and capacitive type sensors, respectively (Barbieri et al., 2019; Kral et al., 2018). These type of sensors are considered payload friendly for small UASs given their compact size and lightweight characteristics. Both of these temperature and humidity sensors work under very similar principles. Basically, the heat flux (diffusion) inside the sensor's material will lead to a thermal (water vapor concentration) equilibrium with the surrounding medium after a finite time. In fact, the differential equation that describes most of their behavior has the exact same form for both sensors which is given by Pletcher et al. (2013):

This is called the heat equation for the case of temperature and diffusion equation for the case of water vapor concentration. Farahani et al. (2014) explain that numerous parameters have an influence on the response time of a sensor, such as the geometry of the sensing element, the inherent thermal/water diffusivity of the sensing element, the thickness of the protective layers, and even the ambient temperature and humidity itself. Equation (3) encompasses most of these characteristics and factors and hence can be effectively used as a model to compensate for errors.

In particular, the selected sensors for this study are the iMet-XF bead thermistor from InterMet Systems and the HYT-271 capacitive humidity sensor from Innovative Sensor Technology (IST). This is for the sake of providing an example case with known sensor characteristics and specifications, and the overall modeling and method description does not lose its generality. Additionally, it is assumed the sensors are sufficiently aspirated (airflow of >5 m s⁻¹) as per the recommendation of the manufacturers so that self-heating effects are diminished.

3.2 Sensor dynamic modeling

In control theory, the method of finite difference is a commonly used numerical solution for differential equations and it is the main foundation of the IDMP method. Finite difference equations are powerful tools that can be used to create mathematical models to describe the behavior of physical systems. The method is an approximation of the derivative which is represented by the derivative taken over a finite interval around a given point. Additionally, assumptions must be made to reduce the complexity of the model and work within a linear regime. The dynamics of the sensor can be further studied after deriving the mathematical model. It can be used to trace back and restore the original signal that produced the sensor measurements as long as the inverse of the model exists and is stable.

3.2.1 Forward model of temperature sensor

The iMet-XF bead thermistor shape and dimensions are shown in Fig. 2, and the probe tip can be approximated to the shape of a sphere with radius R=0.4 mm. Given that heat fluxes can propagate anywhere around the sensor, the problem becomes three-dimensional in space. The spherical symmetry helps to significantly reduce the degree of complexity of the model to a one-dimensional case along the radius (Momoh et al., 2013). It was assumed that the heat propagates radially from the surface of the spherical glass in contact with the air all the way down to the core where the sensing element is located. Therefore, the problem can be seen as a heat transfer problem with spatial temperature gradients inside the bead thermistor. Accordingly, the differential equation for the bead thermistor is given by Eq. (3) in spherical coordinates:

$$\begin{array}{}\text{(4)}& {\displaystyle \frac{\partial T(r,t)}{\partial t}}=\mathit{\alpha }\left({\displaystyle \frac{{\partial }^{\mathrm{2}}T}{\partial r^{\mathrm{2}}}}+{\displaystyle \frac{\mathrm{2}}{r}}{\displaystyle \frac{\partial T}{\partial r}}\right),\mathrm{0}\le r\le R,t\ge \mathrm{0},\end{array}$$

where the internal temperature T is a function of time t and space r which is the radial distance from the center to a given point within the sphere, and α is the thermal diffusivity of the material. The boundary conditions are $T(R,t)=T_{\mathrm{air}}$ and T(0,t) is mirrored. The sphere was then divided into N layers with thickness $\mathrm{\Delta }r=R/N$. A finite difference method was applied to the spatial derivatives of Eq. (4), and the singularity at r=0 was solved by using L'Hôpital's rule. The following system of finite difference equations in space was obtained:

$$\begin{array}{}\text{(5)}& {\displaystyle }\begin{array}{rl}& {\displaystyle \frac{\partial T}{\partial t}}=\mathit{\beta }\left(-\mathrm{2}{T}_N+{\displaystyle \frac{N-\mathrm{1}}{N}}{T}_{N-\mathrm{1}}\right)+\mathit{\beta }{\displaystyle \frac{N+\mathrm{1}}{N}}{T}_{\mathrm{air}},\\ & r=R,\end{array}\text{(6)}& {\displaystyle }\begin{array}{rl}& {\displaystyle \frac{\partial T}{\partial t}}=\mathit{\beta }\left({\displaystyle \frac{i+\mathrm{1}}{i}}{T}_{i+\mathrm{1}}-\mathrm{2}{T}_i+{\displaystyle \frac{i-\mathrm{1}}{i}}{T}_{i-\mathrm{1}}\right),\\ & \mathrm{0}<i\mathrm{\Delta }r<R,i=\mathrm{2},\mathrm{\dots },N-\mathrm{1},\end{array}\text{(7)}& {\displaystyle \frac{\partial T}{\partial t}}=\mathrm{3}\mathit{\beta }\left(T_{\mathrm{2}}-T_{\mathrm{1}}\right),r=\mathrm{0},\end{array}$$

where $\mathit{\beta }=\mathit{\alpha }/\mathrm{\Delta }{r}^{\mathrm{2}}$. A more detailed derivation of these equations can be found in Momoh et al. (2013). As a result, the system of equations is a linear time invariant (LTI) system that can be transformed to the state–space representation of the following form:

$$\begin{array}{}\text{(8)}& {\displaystyle \frac{\partial x}{\partial t}}=\mathbf{A}x+\mathbf{B}u,\text{(9)}& {\displaystyle }y=\mathit{C}x+\mathbf{D}u,\end{array}$$

where x values are the state variables, each one representing the temperature at each layer, u is the surface temperature input signal, and y is the sensor's output signal (sensor reading or measurement at the core). The matrices A and B can be easily obtained by inspection from the finite difference equations.

Figure 2Closeup of the iMet-XF bead thermistor with dimensions and composition. Image provided by International Met (InterMet) Systems.

The goal is to determine the actual temperature of the medium based on temperature readings at the core. By computing the inverse model it is possible to trace back the temperature at each layer from the core to the surface of the sensor in a stable manner. In order to do so, the C vector must be a weighted average skewed towards the core. The D matrix was chosen in a way so that the DC gain (K) of the system is equal to one: $K=\mathbf{D}-\mathit{C}{\mathbf{A}}^{-\mathrm{1}}\mathbf{B}=\mathrm{1}$.

3.2.2 Forward model of humidity sensor

In the case of the HYT-271 capacitive humidity sensor, the dynamics are mainly produced by the diffusion of water vapor concentration from the surface in contact with the air into the polymer. Figure 3 shows the sensing element configuration and the boundary conditions around it. In contrast to the bead thermistor, the capacitive sensor can be treated as a one-dimensional problem since the water vapor only exists just above the sensing element surface, and it propagates along the normal to the surface.

Figure 3Closeup of the IST HYT-271 capacitive sensor with dimensions (left) and sensing element configuration (right). Left image taken from datasheet.

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Horizontal concentration gradients were considered to be negligible given that the thickness of the polymer is small enough to prevent horizontal propagation (Wildmann et al., 2014b). As a result, the differential equation for the capacitive humidity sensor is given by Eq. (3) in one-dimensional Cartesian coordinates:

$$\begin{array}{}\text{(10)}& {\displaystyle \frac{\partial c(x,t)}{\partial t}}=D{\displaystyle \frac{{\partial }^{\mathrm{2}}c}{\partial x^{\mathrm{2}}}},\mathrm{0}\le x\le L,t\ge \mathrm{0},\end{array}$$

where c is the water vapor concentration in parts per million volume (ppmv), D is the diffusivity coefficient of the water vapor in the polymer, and L is the thickness of the polymer. The boundary conditions were defined as $c(L,t)=c_{\mathrm{air}}$, while c(0,t) is mirrored. The polymer film was then divided into N layers with thickness $\mathrm{\Delta }x=L/N$. Following a similar reasoning and steps as applied with the bead thermistor model, the system of finite difference equations is as follows:

$$\begin{array}{}\text{(11)}& {\displaystyle \frac{\partial c}{\partial t}}=\mathit{\lambda }\left(-\mathrm{2}{c}_N+c_{N-\mathrm{1}}\right)+\mathit{\lambda }{c}_{\mathrm{air}},x=L,\text{(12)}& {\displaystyle }\begin{array}{rl}& {\displaystyle \frac{\partial c}{\partial t}}=\mathit{\lambda }\left(c_{i+\mathrm{1}}-\mathrm{2}{c}_i+c_{i-\mathrm{1}}\right),\\ & \mathrm{0}<i\mathrm{\Delta }x<L,i=\mathrm{2},\mathrm{\dots },N-\mathrm{1},\end{array}\text{(13)}& {\displaystyle \frac{\partial c}{\partial t}}=\mathit{\lambda }\left(c_{\mathrm{2}}-c_{\mathrm{1}}\right),x=\mathrm{0},\end{array}$$

where $\mathit{\lambda }=D/\mathrm{\Delta }x$. A more detailed derivation of the above equations can be found in Wildmann et al. (2014b). Again, the resulting system of equations is an LTI system that can be transformed to the state–space representation. The matrices A and B can be determined by inspection from the finite difference equations. The output y of the system is equal to the average of all the water vapor concentrations in each layer. This is because the capacitance is measured across the entire polymer film and not just one particular spot as compared to the bead thermistor. Therefore, the parameter C is a length N row vector with elements equal to $\mathrm{1}/N$, and it maps the state variables to the output resulting in the sensor measurement.

The HYT-271 humidity sensor comes hard-coded to output relative humidity values. The model only works with water vapor concentration in parts per million volume (ppmv) units. Therefore, the relative humidity input must be converted to water vapor concentration and then back to relative humidity after applying the model. The following equations for the conversion were taken from McRae (1980), who states that the error involved in using these equations over a temperature range of −50 and 50 ^∘C is less than 0.5 %:

$$\begin{array}{}\text{(14)}& {\displaystyle }c={\mathrm{10}}^{\mathrm{4}}H{\displaystyle \frac{P_{\mathrm{s}}}{P_{\mathrm{a}}}},\text{(15)}& {\displaystyle }\begin{array}{rl}{P}_{\mathrm{s}}& =P_{\mathrm{a}}\exp \left(\mathrm{13.3185}t-\mathrm{1.9760}{t}^{\mathrm{2}}-\mathrm{0.6445}{t}^{\mathrm{3}}\right.\\ & \left.-\mathrm{0.1299}{t}^{\mathrm{4}}\right),\end{array}\text{(16)}& {\displaystyle }t=\mathrm{1}-{\displaystyle \frac{\mathrm{373.15}}{T_{\mathrm{a}}}},\end{array}$$

where H is relative humidity in percentage, P_s is saturation vapor pressure in millibars, P_a=1013.25 mb is the standard atmospheric pressure, and T_a is the ambient temperature in Kelvin.

6.1 General procedures and limitations

In order to detail and formulate the IDMP restoration technique, it will be beneficial to start by summarizing a basic general procedure, part of which is taken from Wildmann et al. (2014b):

1. Identify the sensor noise floor from the actual measurements by inspecting its power spectral density (PSD) and suppress the noise using a zero-phase low-pass filter.

2. Apply the inverse sensor model to the filtered measurements after adjusting the model parameters based on the weather conditions and sensor characterization.

3. Re-apply the filter from step 1 to the restored signal to filter out any amplified noise.

4. Quantify the improvement by inspecting the time series and spectral response of the signal before and after applying the inverse sensor model.

5. Validate the correction by comparing the structure function with the theoretical $\mathrm{2}/\mathrm{3}$ slope for locally isotropic turbulence in the ISR.

Although the general procedure seems to be robust in the sense that it includes a validation step, the validation step only works for CBL weather conditions as explained in Sect. 2. Furthermore, step 5 is feasible only if the atmosphere is assumed to be horizontally homogeneous and locally isotropic without any rapid atmospheric evolution. Additionally, the flight patterns are limited to steady hover in windy conditions and horizontal transects assuming Taylor's hypothesis of frozen field holds true. For the case of flights across FTIs, only steps 1 to 3 are feasible, and the results can be trusted based on prior testings and calibrations.

6.2 System description and tuning

The signal conditioning of the raw sensor measurements is necessary to extract a more accurate representation of the atmospheric parameters. The challenge is to remove the unwanted distortion and contamination caused by slow sensor dynamics and sensor noise, respectively. If the considerations from Sect. 5 are correctly implemented, then the effects of the UAS and its immediate surroundings on the sensor measurements are considered to be small and can be ignored. The type of low-pass filter chosen for noise removal was a non-causal zero-phase finite impulse response (FIR) digital filter. This kind of filter has the advantage that it does not introduce phase delay to the output signal with respect to the input signal, which is only possible in an offline processing.

An adaptive contamination removal algorithm was not developed for the IDMP method. Instead, the cut-off frequency of the low-pass filter was manually tuned on a case by case basis. The PSD of the raw sensor measurement was used to assess the dynamics of the sensor, identify the noise floor, and define the cut-off frequency of the low-pass filter. Moreover, the logic used to identify distortion and contamination in the measurements taken in CBL conditions is as follows:

• If the slope of the PSD is greater than $-\mathrm{5}/\mathrm{3}$ at high wavenumbers, then the region is considered to be contaminated by sensor noise. This is because of the natural downward trend of the turbulence power spectrum that eventually encounters the noise floor of the sensor. Therefore, it must be removed using the low-pass filter before going through the restoration phase to prevent the noise from getting amplified.

• Otherwise, the measurement is considered to be attenuated and distorted by the slow sensor dynamics which cannot keep up with the turbulence dynamics. Consequently, the bad trend is corrected in the restoration phase, and the new PSD approximates the desired $-\mathrm{5}/\mathrm{3}$ slope line.

The next step is to run the conditioned signal through the core of the IDMP method, the inverse sensor model itself. The transfer function H(z) of the sensor model was computed by taking the z transform of the state–space system of the sensor model. Figure 5 shows an example of the frequency response of the continuous and discrete transfer function H of a capacitive humidity sensor with a response time of about 4 s. The continuous system is the approximation to the analog behavior of the sensor and can be used to simulate the “real” sensing element. The discrete system is a “sampled” version of the continuous system with sampling rate equal to that of the real sensor or the analog-to-digital converter (ADC) to be precise. The truncation effect that comes with sampling a system is what produces the mismatch in the frequency response at high frequencies, which is seen in Fig. 5. The only way to improve the frequency response of the discrete system is by increasing the sampling rate of the ADC to better capture the dynamics of the sensing element.

Figure 5Bode diagram of the sensor model. Magnitude and phase are shown in panels (a) and (b), respectively.

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The inverse system of H(z) exists, and it is stable if and only if H(z) is minimum phase, meaning that all the poles and zeros are within the unit circle of the z plane. Subsequently, the transfer function of the inverse sensor model is obtained by simply taking the inverse of $H\left(z\right)=G^{-\mathrm{1}}\left(z\right)$. Even though the process of finding the inverse sounds simple and straightforward, special attention must be given to the poles of G(z). The resulting poles of G(z) might end up being too close to the unit circle of the z plane which could cause instability and oscillations with high-frequency input signals. The stability parameter for the presented models is defined by Pletcher et al. (2013) as $\mathit{\varphi }=\frac{\mathit{\gamma }\mathrm{\Delta }t}{\mathrm{\Delta }{x}^{\mathrm{2}}}<\mathrm{0.5}$, where γ is an intrinsic parameter of the sensor, such as the thermal and diffusivity coefficients. The equation shows how fast the sampling rate must be in order to precisely capture the dynamics of the sensor within a small spatial interval. Given that the sampling rate of the sensors is fixed at Δt=0.1 s, then the only way to adjust the poles is by varying Δx. The model and hence its inverse G(z) are stable with large values of Δx at the expense of reducing the order of the model and, consequently, the overall accuracy and resolution of the method. Moreover, notice that the degree of correction made on the sensor measurements is limited by the sensor's sampling rate. If the same sensors are enabled to sample at higher rates, then the IDMP will become more effective.

After tuning the parameters and creating a stable system, G(z) was then fed with the filtered sensor measurements to produce the corrected sensor measurement. Finally, the same low-pass filter from the beginning was applied to the corrected signal to remove any amplified noise that survived the process.

7.1 Weather signal generation and sensor simulation

As described in Sect. 2, CBL weather signals from a horizontal transect tend to have a particular PSD with a $-\mathrm{5}/\mathrm{3}$ slope in the logarithmic scale. Moreover, assuming horizontal CBL weather signals are produced by a wide-sense stationary (WSS) random process with Gaussian probability density function, then it is possible to generate the artificial weather signals by modifying the PSD of a Gaussian white noise signal to look like Kolmogorov's energy cascade power spectrum. This was achieved by taking the Fourier transform of a white noise signal and dividing the magnitude by its frequency to the power of $\mathrm{5}/\mathrm{6}$ while keeping the phase unchanged. Figure 6 shows an example of an artificially generated CBL weather signal.

Figure 6Spatial temperature signal in CBL conditions after converting the time-series data given a constant wind speed of 10 m s⁻¹.

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For the case of FTI conditions, the focus is mainly on the evaluation of the sensor response in strong gradients with less importance on features with small amplitude and high frequency. Therefore, the FTI weather signals were constructed in a piece-wise manner using straight lines. The horizontal transect of an FTI was modeled using a ramp function, whereas the vertical transect follows a dry adiabatic lapse rate model with a temperature inversion in the middle. Both weather signals were low-pass filtered to smooth out the corners so that it looks more realistic. The gradient (or inversion) strengths, length, and altitude scales of the weather signals were adjusted using the simulation results shown by Houston and Keeler (2018) as a reference. Figure 7 shows examples of generated FTI weather signals.

Figure 7(a) Temperature change across a frontal inversion (cold front). (b) Vertical dry adiabatic lapse rate temperature profile with a strong thermal inversion.

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To simulate the sensor measurement process, the sensor model was divided into three parts: the analog sensing element, the ADC discrete sampling, and sensor noise generation. The sensing element was simulated using the forward models shown in Sect. 3.2 with a much smaller sampling period Δt=0.01 s. This is because analog signals can not be generated in a computer, and, therefore, the best approximation is to increase the resolution of the discrete model. The actual weather signal was run through the high-resolution sensor model to add the effects of sensor dynamics. This signal then goes through the ADC which down-samples the signal to the actual sampling period of the sensor Δt=0.1 s. The down-sampling process may add aliasing which makes it more realistic. Finally, the down-sampled signal gets its characteristic noise floor by adding additive white Gaussian noise (AWGN) to it. The noise amplitude from each sensor was taken from previous steady-state calibrations done in a controlled environment by the Center for Autonomous Sensing and Sampling (CASS) of the University of Oklahoma.

7.2 Validation of the IDMP method in simulated CBL conditions

The following measurement validation method for CBL conditions exploits the ISR of turbulent fluctuation theory, described in Sect. 2, by using the PSD and structure function calculations. In real-world CBL conditions, the data are collected by conducting horizontal transects or stationary flights in windy conditions at a constant altitude with a multi-rotor UAS. In order to keep this article short and because the correction results of temperature and humidity are very similar, results will be shown in an alternated fashion between temperature and humidity.

Following the procedures of the IDMP from start to end, the PSDs of the signals are computed and compared in Fig. 8. Notice that the cut-off frequency of the low-pass filter was selected near the constant and flat power level of the measured signal, and the sensor noise was effectively removed as a result. The effect of the slow sensor dynamics is noticeable as a downward trend with respect to the $-\mathrm{5}/\mathrm{3}$ slope line. The IDMP successfully restores the power levels of the measured signal at high frequencies. Next, Fig. 9 shows a comparison of the time-series signals, and the RMSE of the measured and corrected signals were computed with respect to the actual weather signal using Eq. (17). The time-series plot clearly shows an improvement in the time response of the sensor which is confirmed by the lower RMSE value. Finally, Fig. 10 shows results from the two-point spatial correlation calculation, namely, the structure function. Assuming locally isotropic turbulence conditions, the deviations of the computed structure function from the theoretical $\mathrm{2}/\mathrm{3}$ slope in the ISR region are indications of the effects of sensor dynamics and sensor noise on the measurement. Moreover, the results can be used to slightly tune the IDMP until getting a best-possible agreement with the theory. Additionally, all the presented results show that the IDMP method is effectively restoring the signal without any signs of instability and oscillations.

Figure 8Power spectral density of the simulated weather signal in CBL conditions and processed signals using the sensor models. Results from before and after applying the IDMP are shown.

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Figure 9Time series of the simulated weather signal in CBL conditions and processed signals using the sensor models. Results from before and after applying the IDMP are shown.

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Figure 10Structure function of the simulated weather signal in CBL conditions and processed signals using the sensor models. Results from before and after applying the IDMP are illustrated.

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7.3 Simulation of flights across strong gradients in FTI conditions

Similar to the CBL simulation, flights across FTIs also exhibit the expected lag in the measurement as a consequence of the sensor time response. Additionally, the lag within this condition becomes more apparent and sensitive to the relative wind speed with respect to the UAS. Houston and Keeler (2018) explain that the number of errors is fewer when flying at low speeds; however, the observation might not be representative because the weather phenomena might have evolved faster than the observation period. Therefore, it is of significant importance to study the performance of the IDMP method with different speeds across the thermodynamic boundary. For the case of frontal inversions, results from the simulated humidity sensor will be shown since it has a large time response, and the correction is more noticeable compared to the faster temperature sensor, whereas for the thermal inversion, results from the simulation of the temperature sensor will be shown.

Frontal inversions, such as cold fronts, are usually sampled using a fixed-wing UAS. The typical ground speed of a fixed-wing UAS in flight is around 25 m s⁻¹, while the wind can reach speeds of 20 m s⁻¹. Assuming that the fixed-wing UAS is able to fly in very windy conditions, then the relative wind across the fixed-wing UAS is 45 m s⁻¹ when flying into headwinds. It has to be mentioned that such an airspeed is extremely high for a typical fixed-wing UAS. However, testing the IDMP outside the operating envelope of the UAS is a good way to show its robustness. Figure 11a shows the comparison between the actual, measured, and corrected humidity in a frontal inversion. Notice how far the measured signal settles with respect to the air mass boundary; this agrees with the results seen in Houston and Keeler (2018). The corrected signal shows a much better transient response, but some oscillations are present when the weather signal is constant. This is because of some remaining sensor noise leaking into the IDMP where it gets slightly amplified. Additionally, the IDMP can not fully recover the shape of the actual weather signal because of missing parts in the frequency content due to noise filtering, low sampling rate, and poor capturing of the weather dynamics.

Figure 11Comparison of weather signals in simulated (a) frontal inversion conditions with relative wind speed of 45 m s⁻¹ and (b) thermal inversion conditions with climbing rate set to 5 m s⁻¹. Results from before and after applying the IDMP are illustrated.

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Vertical profiles are typically carried out using a multi-rotor UAS, and it is common to encounter thermal inversions aloft. The vertical speed of the multi-rotor UAS was assumed to be equal to the radiosonde's climbing speed of about 5 m s⁻¹, much slower compared with the horizontal speed of a fixed-wing UAS. Figure 11b illustrates a comparison between signals in the simulated thermal inversion environment. It can be seen that the IDMP corrects for the constant offset produced by the sensor dynamics not being able to keep up with the temperature change rate. Once the multi-rotor UAS encounters the inversion, the result is almost identical to the frontal inversion case.

7.4 Case study using real data

The presented simulation results show the feasibility of the framework and the IDMP technique on measurements taken with a UAS. However, several assumptions were made to produce the models and simulations which may not hold true for real observations in the field. Therefore, to begin exploring the mitigation of slow sensor dynamics and sensor noise for realistic UAS flights, the IDMP was applied on real data collected using the CopterSonde UAS (Segales et al., 2020) from the University of Oklahoma (OU). Two flights were picked from the extensive database made in the past years throughout several field campaigns. Although these two flights are not available in an open online repository, a similar dataset can be found in Greene et al. (2020) and described in Pillar-Little et al. (2021) which was collected with the same UAS in this study.

These flights were conducted in CBL and FTI weather conditions, respectively, at the Kessler Atmospheric and Ecological Field Station (KAEFS) in Purcell, Oklahoma, USA, located 30 km southwest of the OU Norman campus. The Certificate of Authorization (COA) with number 2020-CSA-6030-COA, issued by the Federal Aviation Administration (FAA), allowed us to fly the CopterSonde above 122 m (400 ft) with a flight ceiling of 1524 m (5000 ft). It has to be mentioned that the sensors were not properly characterized as described in the presented literature; instead the sensor models were tuned by trial and error and best guessing the physical parameters of the sensing elements. Although the sensor characterization was poor, the implementation still showed the potential benefits of using the IDMP.

In the CBL conditions, the CopterSonde was flown stationary at a constant altitude of 10 m for about 15 min with a mean wind speed of 10.2 m s⁻¹. Figure 12a shows a close-up of a portion of the measured and corrected relative humidity time series, while Fig. 12b illustrates the degree of correction made by the IDMP. This is noticeable by observing the amount of deviation in the structure function.

Figure 12(a) Comparison of the measured against corrected relative humidity signal. (b) Structure function comparison.

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In the FTI conditions, the CopterSonde was flown shortly after a cold front moved through KAEFS, leaving a shallow cold pool behind. The climbing rate was set to 3.5 m s⁻¹, whereas the descent rate was set to 5 m s⁻¹. The CopterSonde was sent to 1300 m above ground level, collecting temperature and humidity data in the ascent and descent legs. The flight took about 10 min from take-off to landing. Figure 13a shows a comparison between the measured and corrected vertical profiles of relative humidity. The correction is not very noticeable due to the large spatial scale and the slow vertical speed of the UAS. However, Fig. 13b is a zoomed-in plot of the lower altitude region where the small correction is visible. Assuming that no other environmental factors influenced the measurements, this may confirm that the separation shown by the arrow is in fact an atmospheric evolution and not a result of sensor lag.

Figure 13(a) Relative humidity vertical profile measured against corrected values. (b) Close-up of the vertical profile in the lower altitude region.

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