1. Introduction

There is a growing need for wind observations in the atmospheric boundary layer (ABL) for numerical weather forecasts, air pollution, and for site investigations of wind farms. Hot-air balloon tracks can provide interesting wind information of the lower atmosphere. Hot-air balloon flights usually take place during atmospheric conditions without strong convection and turbulence. These conditions can be found just after dawn and just before dusk and are called the transition periods. The evening transition has been extensively investigated, for instance by Lothon et al. (2014), but the decaying turbulence in a stabilizing ABL is still not fully understood and more experimental data are necessary. There is a long tradition in launching upper-air balloon soundings, but these observations are infrequent and sample a vertical profile of the atmosphere and while ascending the balloon remains relatively short in the ABL. There are several other methods to measure ABL winds, such as wind profilers, sodars, and wind lidars. These observations are collected at fixed locations. Wind observations in the ABL can be also obtained from moving platforms. Laakso et al. (2007) and Petäjä et al. (2012) used a hot-air balloon to study aerosols in the ABL. Also, Doerenbecher et al. (2016) showed that balloons are useful for sampling the atmosphere and that horizontal winds can be derived from the data of drifting low-atmosphere balloons. They used constant level superpressure balloons, which could remain in the troposphere for several days.

In this paper, the focus is on a typical recreational hot-air balloon which can remain a couple of hours airborne. The principle of flying is buoyancy. Ambient air is burned with propane and the hot exhaust gas has a lower density than the surrounding air which gives buoyancy. Hot-air balloons are also called Montgolfière’s after the French Montgolfière brothers who flew the world’s first hot-air balloon at the end of the eighteenth century. Hot-air balloon flights are a leisure activity, but will be possibly applied to explore other planets. Fathpour et al. (2014) studied the feasibility of the reconnaissance of Titan, Mars, and Venus with hot-air balloons. A Montgolfière does not have to be inflated with a light gas such as helium and is not vulnerable to leaks, because leaking air can be quickly replaced and reheated. However, the balloon flight time is restricted to an external heat source.

The Royal Netherlands Meteorological Institute (KNMI) provides special forecasts for hot-air balloon operators which are useful for guidance and trajectory planning. The start location is usually selected in a way that the balloon is not going to land in undesirable areas like cities, motorways or lakes and the balloonist tries to avoid damage to crops and livestock, if a landing in a rural area is foreseen. Of course also the weather conditions should be favorable with regard to cloud base, visibility, and wind (gusts).

De Bruijn et al. (2016) revealed that trajectories from recreational hot-air balloon flights provide interesting wind information in the ABL. They showed interesting meteorological phenomena like wind jumps near fronts and low-level jets which were not captured by the standard observation network of KNMI.

This paper describes a scientific field experiment in the surroundings of the Cabauw meteorological tower (213 m), and we investigate if smartphones can be applied to track hot-air balloons and if we can derive wind data from it. The idea is that many balloon pilots or passengers will start using a special smartphone app that exchanges location data. This will result in wind measurements of the lower part of the atmosphere. The attractiveness of a smartphone app is that it can sample the position and transfer the data using the telecommunication network. A dedicated app is developed which controls this functionality. It is clear that this observation method does not require big investments but will rely on the collaboration of people on board of a hot-air balloon.

The main goal of the hot-air balloon flight experiment is to assess the accuracy of the smartphone app and the validation of the obtained wind data. To achieve this, measurements have been collected from an accurate geodetic Global Navigation Satellite System (GNSS) receiver and two smartphones. A second goal is to study the airflow relative to the balloon by measuring data from a sensitive sonic anemometer which is mounted in a rigid frame underneath the gondola. Canut et al. (2016) conducted an experiment with a 3D sonic anemometer underneath a tethered balloon, and their goal was to study the turbulence in the ABL. In our experiment, the sonic anemometer was mounted on a moving platform and the aim was to study the relative wind and the response of the balloon on the changing wind. Our hot-air balloon flight took place in the vicinity of the village of Cabauw and closely passed along the KNMI observatory which comprises also a meteorological instrumented tower.

At first, we describe in section 2 the instrumentation and the focus is on the smartphones and the sonic anemometer. In section 3, we present results from the hot-air balloon test flight which took place in the surroundings of Cabauw. In section 4, we study the noise characteristics and temporal drift of the GNSS sensors of the smartphones. Subsequently, we determine how the error in position (longitude, latitude) propagates into the derived wind components (u, υ). In section 5, we validate the balloon derived wind with tower observations at Cabauw. In the following section 6, we will discuss in more detail the measurements from the sonic anemometer and the geodetic GNSS receiver. We will derive a mathematical explanation for the fact that the observed relative flow is not equal to zero. Finally in section 7, we will make an attempt to find an empirical value for the length scale in the response function as proposed by De Bruijn et al. (2016). This paper ends with a discussion (section 8), followed by conclusions and recommendations (section 9).

2. Instrumentation

Here we describe the instruments that we have used during the field experiment and we limit ourselves to describe the instrumentation on board of the moving platform. For a description of the meteorological tower measurements, we refer to Van Ulden and Wieringa (1996). Further, we have used off-the-shelf smartphones with proprietary technology and took their performance capabilities and limitations as a constraint for our experiment. Our focus was to calculate wind information from hot-air balloon tracks. Consecutive GNSS positions of the moving hot-air balloon and the corresponding time difference were ingested to our algorithm which delivered wind information. Our aim was not to conduct a profound investigation of the geodetic aspects of the GNSS devices.

c. Sonic anemometer

In the experiment, we have measured the relative wind using the IRGASON 3D Sonic anemometer (Campbell Scientific, Logan, Utah, United States), hereafter labeled as SONIC. This device is designed specifically for eddy-covariance flux measurements. The SONIC measures simultaneously carbon dioxide, water vapor density, barometric pressure, three-dimensional wind speed, and sonic air temperature and its sampling rate is 50 Hz. Acoustic pulses are sent and received between transducers. Wind speed is obtained by measuring the changes in travel time of the pulses. The offset error of the measurement is ±0.008 m s⁻¹ for the u, υ components, and ±0.002 m s⁻¹ for the w component.

A special frame was constructed which could be easily attached to the gondola. The SONIC was mounted in this frame (Fig. 1) after the balloon was airborne. Subsequently, the SONIC was deployed under the gondola by lowering a pivotal aluminum beam (Fig. 2). The SONIC was leveled and pointed parallel to the long side of the gondola, where the Nexus 5X smartphone was fitted. The vertical distance between the Nexus 5X smartphone and the SONIC below is 2.48 m. The vertical distance between the SONIC and the envelope of the hot-air balloon is 5.80 m. The horizontal distances between the Nexus 5X smartphone and the Trimble R7 Patch antenna and the SONIC are less than 2.0 m and given the horizontal accuracy of the smartphones, they can be neglected.

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Rigid frame for mounting the sonic anemometer underneath the gondola.

Citation: Journal of Atmospheric and Oceanic Technology 37, 2; 10.1175/JTECH-D-19-0043.1

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Sonic anemometer in operation below the hot-air balloon during the flight, one of the authors, Bert G. Heusinkveld, is checking the alignment of the sonic anemometer. This picture was taken from the gondola with a camera, which was mounted on a long telescopic golf ball retriever stick.

Citation: Journal of Atmospheric and Oceanic Technology 37, 2; 10.1175/JTECH-D-19-0043.1

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The SONIC delivers wind data in a local coordinate frame and these data have to be transformed to the coordinate system of the GNSS receiver, which is the World Geodetic System 1984 (WGS 84) Geoid surface plane. To transform data we have used data from the Nexus 5X smartphone which was properly fitted on the rim of the gondola. The smartphone Nexus 5X had sensors which measured the position and attitude like accelerators, gyros and magnetic field components. From these sensors azimuth, roll, and pitch were obtained and these data were applied to rotate the SONIC data to the appropriate coordinate system. The Nexus 5X smartphone was equipped with the Bosch BMI160 Inertial Unit which delivered gyroscope and accelerometer readings with an accuracy of respectively 0.0011 rad s⁻¹ and 0.0025 m s⁻¹ s⁻¹ and the Bosch BMM150 magnetometer which delivered 3D magnetic fields readings with an accuracy of 0.0°. The small sensors were designed for mobile applications like indoor navigation, which required high accurate real-time sensor data, and were considered as sufficiently accurate for our application. It should be noted that the Android operating system itself calculated the orientation, pitch, and roll. We used those variables to assess the attitude of the gondola and to transform the SONIC data measurements in the WGS 84 coordinate system.

The balloon is floating in a moving medium and when there is a difference in the balloon speed and the surrounding air, it will interact with it. The airflow will be deflected around the balloon which leads to a change of the relative wind speed. Batchelor (1956) describes the flow around a rigged sphere by the velocity potential:

$\varphi =U\left(r\hspace{0.17em}\cos \theta +{\displaystyle \frac{R^3}{2r^2}}\hspace{0.17em}\cos \theta \right),$(1)

where R is the radius of the sphere, r is the distance from the middle point, and θ is the angle with the horizontal velocity vector. The distances are given in meters. The flow u(r, θ) can be derived from ∇ϕ in cylinder coordinates:

$u\left(r,\theta \right)=\left[U\hspace{0.17em}\cos \theta \left(1-{\displaystyle \frac{R^3}{r^3}}\right),-\left(1+{\displaystyle \frac{R^3}{2r^3}}U\right)\hspace{0.17em}\sin \theta ,0\right].$(2)

The hot-air balloon has a volume of 6000 m³, which leads to R = 11.27 m. If we take θ = π/2 and r = 1.51R, which corresponds with the position of the SONIC in the cylinder coordinate frame, we arrive at the following expression:

$u\left(r,\theta \right)=\left(0,1.22U,0\right).$(3)

The horizontal flow will increase with a factor of 1.22 at the relative location of the SONIC. There are uncertainties in this derivation and two remarks should be made. First, we have assumed that the balloon’s shape is a sphere, but a pear shape would be more realistic. Second, we have also neglected the presence of the suspended gondola.

3. Hot-air balloon flight during 17 May 2016

Here we describe the field experiment with a hot-air balloon equipped with instruments in the vicinity of the Cabauw meteorological tower, which is situated in the western part of The Netherlands. The area is characterized by fields, meadows, and scattered villages. Southwesterly winds are predominant in this area and during the flight the meteorological conditions are typical for the time of the year. The test flight took place in the evening of 17 May 2016 from 1845 to 1930 UTC (local time = UTC + 2) and there were moderate westerly winds of 6 m s⁻¹ at 200 m height. The flight was during the evening transition and no significant wind gusts occurred, which was essential for a safe takeoff.

In Fig. 3 the flight of the hot-air balloon (call-sign PH-GSP) is depicted, which lasted 45 min and the traveled distance was 15 km. The average altitude was 200 m above surface level and the ceiling altitude was 700 m. The crew consisted of 7 persons, inclusive the pilot and the total weight of the gondola was approximately 700 kg. The volume of the balloon’s envelope was approximately 6000 m³.

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Hot-air balloon flight from Bergambacht to Uitweg [(top) ground track and (bottom) transect] during 17 May 2016. In the transect the altitude and velocity components of the hot-air balloon as measured from four GNSS devices are shown as a function of time.

Citation: Journal of Atmospheric and Oceanic Technology 37, 2; 10.1175/JTECH-D-19-0043.1

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We took off upwind of the Cabauw site and the aim was to pass over the observatory as closely as possible. The start location was chosen using the pilot’s expertise and the guidance of the KNMI duty forecasters. At 1900 UTC the pilot changed altitude to adjust the course of the balloon more toward the Cabauw observatory. In the ABL there is a so-called Ekman layer. Due the variation of the stress, the wind vector changes with height (see Holton 1979). Based on this principle, the wind usually turns clockwise with height (wind veers with height). This mechanism offers the pilot of a hot-air balloon the possibility to steer the aircraft to some extent. During our test flight, the meteorological conditions were not neutral. It became clear that the wind even backed with height. The balloonist was aware of this and descended slightly to find a more favorable wind which would bring the aircraft closer to the tower. An explanation for this nonneutral wind profile is that a small sea-breeze front has just passed the area, which caused some baroclinicity. Also the influence of the Lek river might have had impact on the wind regime in the ABL. During the flight we followed the meandering river Lek in upstream direction and we crossed the river five times (see Fig. 3).

During the launch the gondola was out of equilibrium relative to the balloon, underwent a shock which resulted in an oscillation. This is clearly recognized in the data of the Trimble R7 geodetic GNSS receiver where in the beginning wiggles are observed. These wiggles are also recognized later during the flight. A Fourier decomposition of the derived speeds from the Trimble R7 data is made (Fig. 4). There is a peak at 0.1 Hz and we have found a similar spectrum from the acceleration sensor of the Nexus 5X smartphone (not shown) and it is clear that the system balloon with gondola experienced a small pendulum motion during the flight. We have processed all the measurements to eliminate this oscillation. We realize that this oscillation might have caused turbulence which would have had impact on the measurements. We have therefore checked the amplitude of the oscillation, which appeared to be small (0.4 m) and could be further neglected.

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Velocity spectrum derived from data of the Trimble R7 geodetic GNSS receiver with a peak at 0.1 Hz.

Citation: Journal of Atmospheric and Oceanic Technology 37, 2; 10.1175/JTECH-D-19-0043.1

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4. Assessment of the noise in location and speed

As stated previously in section 2b, we consider the measurements from a Trimble R7 geodetic GNSS receiver as the reference. This instrument has also position errors, but they are an order of magnitude smaller than the other GNSS devices. In our setup the Trimble GNSS Receiver does not deliver speed components, so the speed is calculated, based on position data only. The Nexus 5 and 5X smartphones have different sampling rates, namely 1 and 0.1 s, respectively. The Trimble R7 GNSS receiver data are available every 1 s and for a fair comparison all the data are resampled to 1 s.

a. Quantification of the position errors in the x–y–z plane

In Fig. 5 the time series of the position errors are shown. Dx, Dy, and Dz are the deviations in meters from the Trimble R7 geodetic GNSS receiver which is the reference. The errors in the horizontal plane are smaller than the altitude errors. The position errors are not evenly distributed around the time axis which is an indication of correlated noise. This becomes also manifest when we study the errors in a x–y diagram as depicted in Fig. 6. The biases are not centered around the origin. The statistics in terms of bias and standard deviation are summarized in Table 1.

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Time series of the 3D position errors of the smartphones Nexus 5, Nexus 5X, and the Garmin 60Csx relative to the Trimble R7 GNSS receiver. Note that the sampling rate is 1 s.

Citation: Journal of Atmospheric and Oceanic Technology 37, 2; 10.1175/JTECH-D-19-0043.1

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Horizontal position errors of the smartphones Nexus 5 (green), Nexus 5x (red), and Garmin 60Csx (blue) relative to the Trimble R7 GNSS receiver, for the same period as shown in Fig. 5.

Citation: Journal of Atmospheric and Oceanic Technology 37, 2; 10.1175/JTECH-D-19-0043.1

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Table 1.

Statistics of the position noise: Trimble R7 GNSS receiver vs the smartphones Nexus 5, Nexus 5X, and the Garmin 60Csx.

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Now we study in more detail the vertical position noise in Fig. 5. In the lower panel, we clearly see that the Garmin 60Csx outperforms the other devices, which is also obvious from the scores in Table 1. The Garmin 60Csx is a portable navigation system of the pilot, which is on board for the aircraft’s safety. The Garmin 60Csx is equipped with a pressure sensor and the algorithm applies this data for the calculation of the 3D position. It should be noted that the altitude information does not play a role in the calculation of the horizontal speed.

The largest deviations for the Nexus smartphones occur in the interval 1913–1928 UTC, the period of ascent to 700 m and the subsequent descent. The lack of a pressure sensor data could have caused these spurious data.

b. Characterization of the horizontal position and velocity noise

Now we investigate the horizontal position noise in more detail and we study how this error propagates in the derived wind components. Now we elaborate on how an error in position (x, y) is propagated in an error of the wind (u, υ). The balloon speed components are derived from the displacement of the balloon and are discretized as follows:

$u_i={\displaystyle \frac{x_{i+1}-x_{i-1}}{2\mathrm{\Delta }t}},$(4)

${\upsilon }_i={\displaystyle \frac{y_{i+1}-y_{i-1}}{2\mathrm{\Delta }t}},$(5)

where x_i, y_i are respectively the longitudes and latitudes (°), Δt is the time step (s), and u_i, υ_i are the speed components (m s⁻¹). In Fig. 7 the velocity error scatterplots are given, note that Δt = 1 s. Now du and dυ are the velocity deviations in meters per second relative to the velocities from the Trimble R7 GNSS receiver data. The velocity differences have a standard deviation ranging from 0.14 to 1.01 m s⁻¹. Apparently the relative positions have a smaller error than the absolute positions. To study the characteristics of the noise of the GNSS receivers more profoundly, we average the data over an increasing interval nΔt within the time series and for each subinterval we calculate the variance. We know from the theory that if the variance decays according to the 1/n curve, the noise can be regarded as uncorrelated. From Fig. 8 it is evident that the noise is correlated over small intervals. For intervals beyond 100 s the decay of the noise is more according to the 1/n curve. We repeat the abovementioned procedure for the velocity components and from Fig. 9 it is clear that the error is significantly smaller. The noise in the speed components is correlated for small averaging periods and beyond 10 s the decay approaches the 1/n curve.

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Velocity error scatter in the smartphones Nexus 5 (green), Nexus 5X (red), and Garmin 60Csx (blue).

Citation: Journal of Atmospheric and Oceanic Technology 37, 2; 10.1175/JTECH-D-19-0043.1

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Variance of position errors as function of averaging time n, the solid and dashed lines correspond respectively with the position errors in x and y. The solid black line represents the 1/n curve, which is the reference for the white noise decay.

Citation: Journal of Atmospheric and Oceanic Technology 37, 2; 10.1175/JTECH-D-19-0043.1

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Variance of velocity errors as function of averaging time n, the solid and dashed lines correspond respectively with velocity errors in u and υ. The solid black line represents the 1/n curve, which is the reference for the white noise decay.

Citation: Journal of Atmospheric and Oceanic Technology 37, 2; 10.1175/JTECH-D-19-0043.1

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5. Validation of hot-air balloon winds with data from the meteorological tower of Cabauw

For the comparison of the hot-air balloon wind data we selected an episode during the flight where the balloon was between the levels of the anemometers of the Cabauw tower at respectively 10, 20, 40, 80, 140, and 200 m. These requirements were met during 1850 and 1910 UTC and for this period we compare the balloon wind data with the Cabauw tower wind data (see Fig. 10). Because the hot-air balloon is traveling toward the Cabauw mast, we also present the distance balloon tower in Fig. 10. In Fig. 10, we compare the balloon wind data with tower data, which are averaged to 600 s. The anemometers are calibrated and the data are corrected for flow obstruction of the mast. The hot-air balloon wind data are averaged in time to meet the same time coordinate as the mast data (600 s). Subsequently, the mast wind data are interpolated to the level of the balloon using the wind shear as observed at the mast. We compare data from the Trimble R7 geodetic GNSS receiver, the two Nexus smartphones, and the Garmin 60Csx navigator. Although the horizontal distance from the hot-air balloon to the meteorological tower varies from 10 km in the beginning to 4 km at the end of the validation interval, the hot-air balloon wind data corresponds reasonably well. Deviations observed at 1900 UTC may be related to the fact that the hot-air balloon is still 10 km away from the tower, but despite this horizontal mismatch the hot-air balloon wind data are in accordance to some extent with the observations. It should be noted that the hot-air balloon approached the observatory, but did not fly over the tower and the closest distance was approximately 1500 m. By varying the height the pilot has tried to approach the tower as near as possible. Note that the tower observations hardly vary during the depicted interval. At 1900 UTC when the hot-air balloon has passed the small city of Schoonhoven, the wind speed decreases which is realistic; the hot-air balloon feels the wind of the increased roughness of the built up area. Further, the hot-air balloon follows the river in the upstream direction, the temperature of the water which differs from surface temperature at Cabauw might have had impact on the stability and therefore also on the wind profile.

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A comparison of hot-air balloon wind data with tower measurements at Cabauw during 17 May 2016. Note that also the distance from the hot-air balloon to the Cabauw mast is shown by the dashed line.

Citation: Journal of Atmospheric and Oceanic Technology 37, 2; 10.1175/JTECH-D-19-0043.1

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6. Interpretation of the SONIC data

Here we study the SONIC data which represent the relative speed of the hot-air balloon to the surrounding air. We will study the small-scale variations in the relative speed in relation to the acceleration of the hot-air balloon and we will address some typical features. From the GNSS data accelerations of the balloon can be derived, which are representative for the forces on the balloon. The sonic data represent the relative speed between the hot-air balloon and the surrounding air and are a metric for the thrust on the hot-air balloon. In Fig. 11, the measurements of the SONIC and the GNSS derived accelerations are depicted. Recall that the SONIC data have been processed as described in section 2. Note that the time series start 15 min after the takeoff, because some time was needed to deploy the SONIC. Between 1900 and 1918 UTC the averaged SONIC data show a negative bias in both u and υ components of respectively −0.39 and −0.27 m s⁻¹. Here we try to understand this bias.

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Processed relative winds from the SONIC (y axis, left) and accelerations of the balloon (GNSS receiver) (y axis, right). Note that the average sonic speed components are ${\overline{u}}_{\text{sonic}}=-0.39\hspace{0.17em}\text{m}\hspace{0.17em}{\text{s}}^{\mathrm{-}1}$ and ${\overline{\upsilon }}_{\text{sonic}}=-0.27\hspace{0.17em}\text{m}\hspace{0.17em}{\text{s}}^{\mathrm{-1}}$ during an interval when the balloon’s displacement is almost horizontal. The interval for the estimate of the a-coefficient is marked with the dashed vertical lines.

Citation: Journal of Atmospheric and Oceanic Technology 37, 2; 10.1175/JTECH-D-19-0043.1

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a. Influence of wind shear

The distance between the virtual point where the wind forces impact on the balloon and the position of the SONIC is 18 m. Thus when a wind shear is present, the SONIC will measure a bias. To quantify the wind shear, we have applied the Cabauw mast data. The hot-air balloon flies between 100 and 300 m altitude during this period and we use the 140 and 200 m levels of the mast to calculate a representative wind shear. For the u- and υ-shear components we have found values of respectively −0.015 and −0.008 s⁻¹. Applying this information we find relative speed components of −0.27 and −0.14 m s⁻¹ for the u and υ components, respectively. For clarity we summarize the bias components in Table 2. If we confront these values with the averaged SONIC components we notice a certain mismatch. In an attempt to explain the residual, we take a closer look to the equations of motion of the ABL and the hot-air balloon in the ABL.

Table 2.

Overview of the bias components in the SONIC data during 1900–1918 UTC.

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c. Stationary solution for a constant level balloon

Under stationary conditions the acceleration terms on the left-hand side of Eqs. (6), (7), (9), and (10) will be zero. We will show that the balloon movement will deviate from the actual wind due to the absence of the turbulent stress divergence and we will call this deviation the inertial drift. Similar to the inertial oscillation in the stable boundary layer, which may lead to the formation of the low-level jet, the vector $\mathrm{\Delta }{\mathbf{U}}_b=\widetilde{{\mathbf{u}}_b}+\mathrm{\Delta }{\mathbf{U}}_a$ (ageostrophic balloon vector) start to turn around the geostrophic wind (Van de Wiel et al. 2010). This turning stops as soon as the thrust is in balance with the pressure and Coriolis force. We will now solve this system of equations for the stationary condition. The force balance of the balloon reads

$0=f_c\left|\mathrm{\Delta }{\mathbf{U}}_b\right|-a{\left|\widetilde{{\mathbf{u}}_b}\right|}^2$(11)

and thus $\left|\widetilde{{\mathbf{u}}_b}\right|$ can be solved

$\left|\widetilde{{\mathbf{u}}_b}\right|=\sqrt{{\displaystyle \frac{f_c}{a}}\left|\mathrm{\Delta }{\mathbf{U}}_b\right|}.$(12)

To calculate the ageostrophic wind vector |ΔU_b| = |U_b − U_g| the geostrophic wind is required. The geostrophic wind is derived from pressure sensors in a 75 km range from Cabauw (Bosveld et al. 2014). The geostrophic wind at Cabauw is depicted in Fig. 12 with the black line. We estimate |U_b| by taking the balloon wind averaged between 1900 and 1918 UTC. This is allowed, because we expect that the inertial drift will be small. If we calculate the averaged ageostrophic wind between 1900 and 1918 UTC, we arrive at a value of |ΔU_b|=2.24 m s⁻¹ and substituting this in Eq. (12), we find $\left|\widetilde{{\mathbf{u}}_b}\right|$ = 0.186 m s⁻¹. This is defined in the wind vector system and if we convert this to the GNSS coordinate system we obtain $\widetilde{u_b}$ = −0.175 and $\widetilde{{\upsilon }_b}$ = 0.064 m s⁻¹. We have obtained a similar result by solving the differential Eqs. (9) and (10) numerically with stationarity as a constraint. The obtained values together with the shear correction do not explain fully the systematic wind bias as observed with the SONIC (see Table 2). The norm of the inertial drift corresponds satisfactorily with the residual, but there is some mismatch in the direction. One reason may be that the geostrophic wind is derived for the surface level, but we need to have it at the balloon level. Thus, the presence of a thermal wind will have an influence on our estimation of the inertial drift. A second reason is that the meteorological situation is rather complex, because the atmosphere is baroclinic and the flow is nonstationary. In general, under barotropic conditions the geostrophic wind is larger and veered to the ABL wind. Apparently this is not the case here (see Fig. 12), and if we study the mast observations in more detail we notice that from 1845 until 1917 UTC the wind backs with height; after 1917 UTC the levels below 80 m veer with height and the upper levels slightly back with height. Due to a horizontal gradient in temperature, causing a baroclinic effect, the wind profile is obviously affected. What becomes clear is that the flow is nonstationary in contrast with the assumption under which we derived the inertial drift. This is also manifest if we study the wind profiles in Fig. 13. In the time span of the balloon flight the wind changes considerably, which is related to the passage of a front.

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Tower wind measurements at fixed levels at Cabauw and the balloon wind speed based on GNSS data. Also, the geostrophic wind at surface level (black line) is depicted.

Citation: Journal of Atmospheric and Oceanic Technology 37, 2; 10.1175/JTECH-D-19-0043.1

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Tower measurements of the wind at Cabauw during 1840–1940 UTC 17 May 2016 in a 10-min interval. Note the nonstationary conditions.

Citation: Journal of Atmospheric and Oceanic Technology 37, 2; 10.1175/JTECH-D-19-0043.1

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Another relevant issue is the response time toward the inertial drift. The time scale τ associated with the adjustment of the balloon can be found by observing that initially the acceleration is given by the Coriolis term of Eq. (11). The time τ required to reach the equilibrium speed of Eq. (12) with this acceleration is then given by

$\tau ={\displaystyle \frac{1}{\sqrt{a{f}_c\left|\mathrm{\Delta }{\mathbf{U}}_b\right|}}}.$(13)

From the solution we can deduce that it takes about 600 s before an equilibrium is established. This means that the balloon is adapting during the major part of the analyzed period and therefore added to the effects of nonstationary synoptic conditions, we have a further uncertainty in the validation of the inertial drift.

7. Empirical assessment of the inverse length scale (a-coefficient)

8. Discussion

In this paper, we have found that wind observations can be obtained from hot-air balloon flights by tracking them with smartphones. Using the telecom network, position data can be transferred to a server, and from this wind information can be obtained. The GNSS chip in the smartphone is less accurate than the geodetic GNSS used in this study. The accuracy of the position and other features in smartphones will always be a moving target, so our findings depend very much on the used smartphones. Improvement of smartphones is expected in the combined processing of GNSS and cellular signals. Currently the API software delivers the 3D coordinates of the position. If pseudo ranges become available, a more sophisticated mathematical model with Kalman filters can be applied in which more sensor data can be integrated. The presence of the pseudo ranges is foreseen in a coming version of the android operating system (Banville and van Diggelen 2016). Vertical positions are less accurate than horizontal positions. The errors in altitude do not have immediate impact on the calculated wind, because only the horizontal displacement is needed. Of course, spurious data can be removed by a time filter, and a beneficial effect is expected when more sensors like pressure and temperature are used to calculate the vertical displacement.

The second objective of this paper is the interpretation of the SONIC data in combination with the accurate geodetic GNSS receiver measurements. Unexpected behavior of the balloon’s acceleration and the relative speed between the balloon and the air turned up, which could not be described by the simple dynamic model introduced by De Bruijn et al. (2016). In this paper we have made an effort to improve the dynamic equation of the hot-air balloon. We realize that the reality might be even more complicated. In our newly developed model we neglected for instance baroclinic effects. Further, the pressure term, which was cast in terms of the geostrophic wind [Eqs. (6) and (7)] is usually attributed to the large-scale component related to the synoptic pressure distribution. It is likely that this term will also contain smaller scale components which may vary in space and time. This mesoscale variation will exert the same influence on both the balloon and the surrounding air. They will therefore be characterized by variations in the balloon speed but without a change in the wind speed difference between the balloon and the surrounding air.

In the ideal case the SONIC should be placed on the level, where the balloon feels the maximum drag, preferably in an undisturbed flow, but for practical reasons this was impossible. In our evening transition case the balloon is immersed in a stabilizing atmosphere and turbulent eddies are becoming smaller. Large eddies will have a similar wind effect along the total balloon–gondola system, on the contrary the smaller eddies will introduce more vertical variability in the wind. This causes uncertainty in the interpretation of the measurements of the SONIC in terms of the difference between balloon and airspeed. In summary, the effective wind speed that drags the balloon–gondola system will be different from the wind underneath the gondola at the level of the SONIC. Further the ABL is stabilizing and it is well known that the drag coefficients for laminar and turbulent flows have different values. In the transition zone the drag coefficient can suddenly drop to very small values (Munson et al. 1990) and this process causes another uncertainty in our experiment. For the calculation of the response coefficient (a) we have assumed that the balloon is a sphere. However, in reality the balloon is not a sphere and is moreover slightly deformed during the flight. Also the volume is not constant. The pilot can rotate the balloon or change altitude by letting escape air via (lateral) vents and warm air is released to reduce buoyancy. Therefore the shape might change during the flight and this may have impact on the response coefficient.

9. Conclusions and recommendations

Smartphones are not very accurate in absolute positions, compared to geodetic GNSS receivers. The standard deviations are (σ_x = 5 m, σ_y = 5 m, σ_z = 12 m), but the relative positions have a better accuracy. Therefore, horizontal speeds, which are based on relative positions and a time step of 1 s, have standard deviations of σ_u = 0.8 m s⁻¹, σ_υ = 0.6 m s⁻¹. We have found that the instrumental noise is correlated and averaging in time reduces the standard deviation in the speed error less quicker than that would have been the case for uncorrelated noise. However, averaging wind speeds over longer time periods will inherently lead to smaller errors. For time scales beyond 600 s, which is a typical time step of an NWP model, the σ_u and σ_υ become smaller than 0.03 m s⁻¹.

The vertical position of the current smartphones is inaccurate, compared to the horizontal position. In future the vertical coordinate might be improved if sensors like pressure and accelerations are integrated in the algorithm that delivers the position. In this way the bias and standard deviation can be reduced.

We have conducted a field experiment with a hot-air balloon equipped with sensors on special instruments and smartphones. We have collected an unique dataset of a SONIC to measure the relative speed of the balloon to the air and a geodetic GNSS receiver from which the accelerations of the balloon could be derived. This allowed us to study the dynamics of a hot-air balloon. By performing the flight in the proximity of the Cabauw meteorological tower we were able to give further interpretation of the atmospheric conditions during the flight. During most part of the flight a relative flow of less than 1 m s⁻¹ was recorded with the SONIC and it was peculiar that the crew hardly experienced this flow. Only during the ascending/descending excursion at the end of the flight when a relative wind speed of about 1 m s⁻¹ was observed, a slight wind sensation was felt. We may conclude that the SONIC is a more sensitive device than the receptors on the human face. The response coefficient a as estimated from theoretical considerations in De Bruijn et al. (2016), was found to be in reasonable agreement with the empirical derived value from the current experiment.

On the basis of accurate measurements, we have seen that a hot-air balloon does not precisely follow the wind in the ABL, but reveals a slight deviation, the so-called inertial drift. For a quantitative test of the inertial drift it is important to have a balloon flight under conditions with a stationary geostrophic wind. The authors would like to underline that the meteorological conditions were more complex than foreseen. Especially the passing of a baroclinic disturbance made it difficult to validate the theory of the drifting balloon. A possible next experiment should be planned preferably under more synoptic stationary conditions.

Not every phenomenon in the data of the geodetic GNSS receiver and SONIC is completely understood. Using accurate GNSS and SONIC devices on hot-air balloons might be a way to detect the footprint of mesoscale variations in pressure.

It is recommended to make the developed apps available in the repositories Play Store and App Store, so that the crowd sourcing can be initiated. In this way ABL wind information during a leisure activity can be collected in an economic manner.