
==== Front
Sci Rep
Sci Rep
Scientific Reports
2045-2322
Nature Publishing Group UK London

39300214
71445
10.1038/s41598-024-71445-9
Article
Seasonal and geographic viability of high altitude balloon navigation
Brown David dbrown3@college.harvard.edu

12
Linz Marianna 13
Leidich Jared 2
1 https://ror.org/03vek6s52 grid.38142.3c 0000 0004 1936 754X School of Engineering and Applied Sciences, Harvard University, 29 Oxford Street, Cambridge, MA 02138 USA
2 grid.524860.d Urban Sky, 4800 Race St., Denver, CO 80216 USA
3 https://ror.org/03vek6s52 grid.38142.3c 0000 0004 1936 754X Department of Earth and Planetary Sciences, Harvard University, 20 Oxford Steet, Cambridge, MA 02138 USA
19 9 2024
19 9 2024
2024
14 218613 5 2024
28 8 2024
© The Author(s) 2024
2024
https://creativecommons.org/licenses/by-nc-nd/4.0/ Open Access This article is licensed under a Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License, which permits any non-commercial use, sharing, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if you modified the licensed material. You do not have permission under this licence to share adapted material derived from this article or parts of it. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by-nc-nd/4.0/.
Control of the geographic location of high-altitude balloons has been desired for decades due to the cost and simplicity of the systems. These balloon systems rely on variations in wind direction with altitude so that when a balloon changes height, it also changes the direction of its horizontal motion. An altitude control system can thus also control the horizontal position by transitioning to a wind layer with favorable winds. The system’s ability to navigate successfully thus relies on the existence of certain wind conditions. In this paper, we explore how the ability of a balloon to station-keep varies based on the geographic location and season. We used spatially and temporally variant ERA5 wind data with a tree-search-based algorithm to traverse potential trajectories, and we selected the altitude transitions that maximize time within 50 km of a target. The simulation’s outputs show large variations in success across both latitude and season. Midlatitudes are particularly challenging for station-keeping, while lower latitudes are more favorable. Summer is typically more favorable than winter. This demonstrates that for all balloon systems, the ability to station-keep is highly variant and not universally possible.

Keywords

Station-keeping
Stratosphere
High-altitude balloon system
Tree search
Altitude control system
Navigation
Subject terms

Aerospace engineering
Atmospheric dynamics
issue-copyright-statement© Springer Nature Limited 2024
==== Body
pmcIntroduction

It has long been a goal of the aerospace industry to achieve persistent sensing via an aircraft for applications ranging from monitoring to communications1. If balloons are able to successfully station-keep or maintain close proximity to a target, they can achieve many desires of the remote sensing industry. By industry standards, successful station-keeping is when the system can maintain a distance of less than 50 km from its target station, as this will allow typical success in ground communications2 or persistent visual monitoring at nadir angles up to 45∘3 based on the flying altitude. While Google Loon was operating, they were able to perform station-keeping in limited locations2, and World View in 2022 was able to maintain 10 days within 80 km of a target station4. Through their attempts, it has become clear that the ability to station-keep is not purely a problem of system and control design.

Lighter than air balloons are some of the oldest flying machines in existence5, and they utilize hot air or a lighter-than-air lift gas like helium or hydrogen to be held aloft in the air through buoyancy. The medium inside an envelope is less dense than that outside the envelope, generating a buoyancy force that causes ascent until equilibrium with the gravitational force is reached. At this point, the altitude in an idealized system would not change unless the system were perturbed6, and in reality balloons can consume close to no energy to stay aloft and even to control their altitude7. This makes them in some ways similar to satellites, in that there is no fundamental limitation to their lifespan, making them a highly interesting remote sensing platform8,9. Google Loon has demonstrated balloons staying aloft for hundreds of days10 and Loon balloons demonstrated those types of duration while also performing altitude control maneuvers throughout the mission, showing that the energy gathered in flight through solar charging can exceed the amount of energy consumed to stay aloft and execute altitude control maneuvers11.

With the potential to operate a high-altitude balloon for a long duration, many remote sensing goals can be served using high-altitude balloons that are closer than satellites to the earth. A lower altitude vehicle allows for lower latency in communications, higher resolution in remote sensing, and a different control paradigm thereby removing some of the constraints of satellites. For example, satellites cannot be made to functionally loiter or stop unless they are very far away from the earth in Geostationary Orbit12 and must move very fast when closer to the earth presenting a variety of challenges like motion blur in remote sensing13 and intermittent connectivity in communications14,15. Finally, balloons can be deployed ad hoc in a way that does not yet exist for satellites16,17 and given the drastic difference in the energy required to launch a satellite to orbit and that required to bring a balloon to the stratosphere, the launch costs for balloons are drastically lower than satellites18. For similar reasons, industry leaders believe the latency in new technology adoption could be much lower for balloons when compared to satellites. A balloon payload adapting a new technology, such as a newly invented imaging sensor, could be brought to the stratosphere more quickly than a satellite can be brought to space because all aspects of the design and launch including the launch vehicle, mass of the equipment involved etc. are less intensive for balloons19.

If high-altitude balloons are able to achieve navigational control, they have the potential to match, if not surpass the abilities of satellites in many applications. This makes the topic of control in high-altitude balloons of utmost importance; the ability of high-altitude balloons to flourish as an industry likely depends on whether balloons can be well controlled. This study explores the control-ability of high-altitude balloons which do not utilize lateral propulsion, making it realistic for energy parity and ultra-long duration operation. Having lateral propulsion to effectively steer a balloon in the stratosphere requires large energy consumption, and remains an ongoing area of study on methods for energy supply20. This study investigates control that can be achieved without a lateral propulsion system, thus eliminating this large energy resource from consideration.

The balloon system we study has vertical control that allows it to actively move up or down in the sky on command21. This mechanism is generally referred to as altitude control in balloons and is actuated by an Altitude Control System (ACS). The balance between the buoyancy and gravitational forces can be adjusted by adding or removing mass from the system, releasing lighter-than-air lift gas, or for a longer duration, using mechanisms such as a compressor to change the density of the system22. Because the balloon systems modeled in this study do not utilize any lateral propulsion, they are subject to the flow of any wind field they are within. But, as will be discussed in much more detail throughout this paper, as a balloon moves up and down in the sky it can access different parts of the atmosphere where winds are moving in different lateral directions to functionally steer laterally without having lateral propulsion.

For systems with efficient and limited lateral propulsion, such as the Zephyr23 system, the detailed results of this study are not applicable, but the general findings with respect to latitude and season are relevant. In situations where a system with no lateral control can navigate, a system with some lateral control will also be able to navigate with more success. For situations where a system with no lateral propulsion cannot navigate, a system with lateral propulsion may have more success, though these environments will nevertheless be more challenging for either system. This study provides a baseline from which the navigational autonomy of a variety of systems can be expanded.

Assuming a balloon will only move laterally in the direction of the wind, the amount of overall control potential a balloon has will depend on the vertical variability in wind direction. A balloon could theoretically keep station with two perfectly opposing wind layers (say one wind layer going from east to west, and one from west to east). This would be unstable, however, and any North/South drift would not be able to be corrected. To enable correction for the drift, three different wind layers with winds traveling in different directions must be present, and the sum of the interior angles between the wind layers must be greater than 180∘ (If the sum of the angles is less than 180, then at least one of the cardinal directions will be unachievable, preventing the control from spanning the lateral space). With these three vectors, the system could calculate the exact amount of time needed to spend controlled by any given wind vector to arrive at any arbitrary point. These diverse layers in theory do not have to exist simultaneously or constantly for station-keeping to work. With accurate forecasting predictions, having high temporal variability in the direction of the wind field may achieve the same goal of exposing the balloon to diverse winds.

There are good reasons to be optimistic about the potential success of station-keeping within the stratosphere. Stratospheric mean winds are subject to a strong seasonal cycle: large-scale westerlies in the winter hemisphere and large-scale easterlies in the summer hemisphere. Additional events, such as the Quasi-Biennial Oscillation24, affect the tropics, which are less dominated by the mean wind seasonality due to the lower seasonal variation in insolation. Stratospheric winds are also influenced by planetary waves25, the breaking of which drives deviations from radiative equilibrium and the global stratospheric circulation26. These stratospheric processes dynamics will affect the necessary wind diversity to effectively station-keep.

We first describe the simulation conducted for the study, and the corresponding results in the "Simulations for Station-Keeping" section. From there, we draw conclusions and investigate the implications of these results in the "Discussion" and "Conclusion" section. Finally, we detail the steps used to generate and collect the dataset analyzed in this study in the Methods section.

Simulations for station-keeping

To understand station-keeping viability, we simulate a balloon initiated at many locations and times. For each simulation, the balloon repeatedly planned two steps into the future, parsing potential altitude transitions. The balloon followed the trajectory that maximized its time within 50 km of the target, referred to as Time on Target. This allows the station-keeping duration for a given location and time to be calculated, and compared to the other simulated conditions. A sample trajectory is plotted in Fig. 1.Fig. 1 Sample Trajectory of a simulation from April 1, 2020, over Southern Chad (10 N, 18 E). This trajectory spans over 12 days of simulated balloon flight within 250 km of the initial region. Map Data: Google Earth (version 7.3.3, https://www.google.com/earth/) (Landsat/Copernicus, Data SIO, NOAA, U.S. Navy, NGA, GEBCO, © 2024 Google).

The study ran simulations across 426 locations over the world with land-based targets with an approximate distance of 500 km between each other. The temporal range includes one simulation for each season (corresponding to January, April, July, and October) for the years 2018 through 2022. This span of five years shows evolution through seasons and covers a range of Quasi-Biennial Oscillation conditions. It also includes Madden-Julian Oscillations27,28, and both El Nino (2018–2019) and La Nina phases (2020–2022)29. This totaled 20 different potential flight dates for a total of 9240 total different flight simulations.

The study was run using a set of termination conditions (discussed more in the “Methods” section) that are important for interpreting the results. Most importantly the study terminated after either 50 altitude maneuvers or two months of flight. This termination parameter was necessary to make it possible to run the study efficiently and avoid situations where simulations span multiple seasons complicating the analysis of seasonal dependency. This paradigm does not generally correspond to a real limitation of high-altitude balloons, as there is no theoretical limitation to the duration a balloon can stay afloat nor the transitions it can make. As such the study represents a relative magnitude of station-keeping ability, but does not suggest the duration of station-keeping presented would be the expected duration of station-keeping for any one balloon system.Fig. 2 Average Time on Target by latitude across all launch dates. Number of sites per latitude listed below.

Fig. 3 Average Time on Target in days by latitude region for Launch Dates of (a) January 1st, (b) April 1st, (c) July 1st, (d) October 1st (years 2018–2022) with corresponding cross-latitude averages.

Discussion

We find significant variations of Time on Target with both latitude and season. Investigating only latitudinal variation, station-keeping conditions appear to follow a centralized distribution as seen in Fig. 2, performing best in near-equatorial regions, averaging over 7 days over the target station. Due to the Quasi-Biennial Oscillation, equatorial latitudes have access to both easterly and westerly winds within the balloon operating range. This, along with low north-south wind velocities, allows the system to oscillate back and forth over the target. Beyond the equator itself, the tropical and subtropical regions more generally also maintained strong performance, averaging over 3 days on station. The winds in these regions are relatively weak, capable of sustaining longer flights over shorter distances. These weaker mean winds are easier to overcome to provide wind diversity. Performance decreases further in midlatitudes and is poor at the high latitudes, where balloons averaged just over 12 h on station, showing little successful control. This is the demonstration of dominant and homogeneous mean stratospheric winds. There are minimal improvements in performance in the polar regions. Winds within the poles tend to point towards the poles with moderate speeds, thus at small scales are able to remain within the region.

Examining the seasonality of time on target distributions in Fig. 3, there is symmetry between January and July as well as between April and October. The symmetry follows a reflection along the longitudinal axis, indicating that each hemisphere is quite similar in the equivalent season. Boreal summer and austral summer maintain similar moderate performance distributions for their respective hemispheres. Northern latitudes during boreal winter and southern latitudes during austral winter both have considerably poor performance. The existence of these symmetries suggests that station-keeping viability is driven by the overall stratospheric dynamics, and much less by more local and regional effects.

When looking at exclusively seasonal variations, we see that there are large global differences. This is seen as equinox-based launches performed considerably higher than their solstice counterparts. This demonstrates the expected effects of the strong stratospheric seasonal cycle. Figure 4 shows the mean winds shift from westerly winds throughout the depth of the stratosphere in winter to westerlies in the lower stratosphere with easterlies above in the summer hemisphere.Fig. 4 Zonal mean zonal wind from the MERRA-230 reanalysis climatology averaged from 1981 to 2010. Contours are spaced every 5 m/s, and the thick black line shows the zero contour. The colorbar is saturated for the July panel, where maximum wind speeds are about 80 m/s in the southern hemisphere polar vortex.

If the mean wind speeds are lower, as during the equinoxes when the transition of wind direction between the hemispheres is taking place, deviations can allow for diverse wind headings, supporting station-keeping performance. The uniform westerly winds in the winter hemisphere allow for little opportunity to station-keep. The effects of this cycling nature leads to performance variability of several days within the tropics, and even higher variations in station keeping viability by season in the mid and upper latitudes.

Conclusion

This study shows that for equatorial regions subject to ideal conditions, year round station-keeping could be supported. As the target region moves away from the equator the quality of station-keeping decreases and becomes only possible during specific seasons. This study clearly shows that the expectation or goal of universal station-keeping is unobtainable without significant lateral propulsion. In other words, hovering a balloon over any location on earth at any part of the year for extended time-periods is not possible. The results of the study are entirely due to the wind fields and, as such, apply to all balloon systems, regardless of design and features. The study also shows the predictability of the general performance expectations. The variable success is governed by the large-scale seasonal shifts and can be predicted and anticipated. Operations that intend to perform navigation with high-altitude balloons should expect this variability, and structure around it.

While station-keeping is thought to be one of the most important priorities for the continued development of high-altitude ballooning31,32, it is important to note that station-keeping is not the only way to create continuous coverage. As has been demonstrated with satellite networks like Starlink15, and the Planet Dove network33, a network of multiple vehicles that are all moving around the planet, or in circles locally, can functionally create constant coverage even when the diameter of the circle the vehicles are cycling in (the orbital diameter in the case of a satellite) is larger than the coverage area.

This study looks purely at the variations in the viability of station-keeping and navigation of high-altitude balloons due to the season and geographic location. From here, viable regions can be identified and studied further to understand implementation strategies. Specifically, the analytical performance of individual balloon systems in these regions can be calculated. This study does not investigate how to account for forecasted uncertainty in decision making, although a very rough estimate of how different trajectories look if they are run with a forecast instead of omniscient knowledge of the wind fields is presented in the methods. We show the potential for substantial differences in position due to the use of forecast data, motivating much more work to investigate control algorithms and models to identify these control sequences.

In regions where navigation and persistence are possible, balloons can be launched periodically to form a network of coverage for providing services from communication to surveillance to wildfire detection. Computational algorithms can be created to identify reachable regions and corresponding trajectories for desired navigation, and used to achieve a wide range of remote sensing goals.

In regions where navigation and persistence are not possible, flight safety becomes a crucial consideration. The flight operator must acknowledge the limitations to steering and the implications that will have on the trajectory of the balloon. Using predictions to characterize where the balloon will float will be essential to understanding what regions the balloon will fly over. For safe and effective balloon flight and termination, it must not be brought down over highly populated or inaccessible regions. Without navigation, these future scenarios may not be avoidable, and thus the balloon should either not be flown, or should be terminated before entering those unsafe environments.

Methods

Wind data

For the study, we utilized the ERA5 reanalysis dataset34, a comprehensive and global atmospheric product produced by the European Centre for Medium-Range Weather Forecasts (ECMWF). The ERA5 reanalysis dataset utilizes numerical models and data assimilation to reinterpret historical weather observations with enhanced resolution. ERA5 contains historical data on a multitude of meteorological variables, the ones utilized for this study were wind speed and direction35.

As having diverse winds is essential to the success of station-keeping, the study uses the full resolution global data to best model the development and dynamics of stratospheric conditions. Figure 5 shows a sample slice of this wind field at a given time and longitude. The ERA5 dataset is available on a horizontal 0.25 × 0.25 latitude-longitude grid. The vertical resolution is 137 hybrid-sigma pressure levels. This gives a maximum range of 1000–0.01 hPa or sea level to 80 km. For the study, only levels 28–64 were utilized, giving data for altitudes between about 15 km and 32 km, which span the entire range of potential flight altitudes for typical stratospheric balloons. There may be altitude layers between those modeled by the reanalysis data, which may flow in unique directions. This will only increase the ability for navigation in a field balloon setting.

The ERA5 dataset’s high spatial resolution of 0.25∘ equates to a grid size of approximately 28 km at the equator. Such high spatial resolution enables a detailed representation of the atmospheric conditions, capturing local wind patterns that lower-resolution datasets might miss. For lagrangian trajectories, limited improvement in the tropics was identified beyond 0.5 × 0.536, but due to the nature of the cost function maximizing time within 50 km, having spatial resolution smaller than the region is more valuable than reducing the computational cost.

The temporal resolution of the ERA5 data is also particularly advantageous for the study. The output is updated hourly, which allows the simulation to account for the changing wind patterns over time in a high level of detail. Such granularity is crucial for the tree-search-based algorithm, which requires frequent updates of atmospheric conditions to make accurate predictions and decisions.

Due to the resolution of the ERA5 dataset, the wind data at any given spatial-temporal location was approximated to the nearest point in the dataset. This data was then used to calculate the balloon’s predicted trajectory under various control actions. By using the ERA5 data in this way, the simulation can account for the complex, dynamic, and four-dimensional nature of the wind patterns in the stratosphere. This allows the station-keeping performance of the balloon to be evaluated under a wide range of realistic atmospheric conditions.Fig. 5 Wind Direction for Wind Columns at various latitudes at Longitude 315 in July 2021. Arrow magnitude is normalized, and the color and direction correspond to the direction of the wind layer at each altitude. This show little variability in the Southern Hemisphere during Austral winter, and wind diversity in the equatorial region and the mid northern latitude region.

Dynamic model

For operating the balloon simulation, the dynamic model has been linearized, while maintaining valuable characteristics of balloon-based flight. The simplification deals with two assumptions that were made regarding the dynamic model to reduce computational complexity and maintain generic balloon system assumptions.

Firstly, it is assumed that the balloon’s horizontal speed always matches the horizontal wind speed at its current location given by the wind model. The generic balloon system is one without powered steering, thus they are primarily wind-driven and have no alternative propulsion systems for horizontal movement. Winds affect the balloon’s speed is through a drag force. The drag force FD is given byFD=12ρatmv2CDA

and is proportional to the relative speed of the wind v, the atmospheric density ρatm, the system’s surface area A and drag coefficient CD. This force will cause the balloon system’s lateral velocity to match that of the wind field over time. The characteristic time it will take for the balloon to reach this equilibrium is dependent on system-specific conditions (surface area and drag coefficient). With the intention of the study being the viability of a generic system, avoiding a model system that relies on the introduction of these arbitrary constants is ideal. When not accounting for forces or acceleration, the physics model is able to remain linear, and thus the simulation integration step can be significantly increased to reduce computational run time.

Secondly is the implemented an idealized Altitude Control System (ACS) that guarantees the balloon will be in one of three states at any given time: Float, Ascent, or Descent. In the Float state, the vertical velocity of the balloon is always zero (vz=0), causing the system to maintain a constant altitude. This represents a situation where the ACS is ensuring a constant balance between the buoyancy and gravitational forces. In the Ascent and Descent states, the balloon’s vertical velocity is assumed to be constant at vz=±0.5 m/s for ascent or descent, respectively. These speeds were designed as the effect of a small deviation of equilibrium by the ACS, causing a slight acceleration in a given direction. A more accurate model would have an ACS to regulate the effective vertical force on the system via a ballast-based system. Then, an understanding of the buoyancy and volume would be required. This would involve a thermal model of the balloon film and gas, introducing numerous system-specific configurations, and once again limiting the maximum integration time.

These assumptions simplify the model to allow the simulation to run efficiently, while still remaining a realistic representation of the dynamic nature of the balloon’s movement, and maintaining the characteristic qualities required for station-keeping a balloon. The intention of the study is to identify relative performance shifts in different geographic and temporal environments. Added fidelity to the balloon model would risk losing generality, and would not affect relative performances. The linearized model equations are a simple time evolution of the position following the velocity field, and our published code shows their implementation.

Tree-search optimization algorithm

To determine when to transition and to which layer, an algorithm was designed to return an optimized transition decision to maximize the time with 50 km of the target. The system runs this algorithm when certain necessary criteria are met, and for the viability study, the algorithm used is a tree-search algorithm37. Figure 6 displays sampled trajectories of both successful and unsuccessful station-keeping simulations under this algorithm.

The tree search algorithm is only engaged when the system needs to explore alternative routes, determined by certain criterion, known as the investigation criterion. This criterion is that the balloon is moving away from its target station and has exceeded a distance of 40 km. This combination of conditions ensures that the algorithm works proactively to prevent substantial drifts while avoiding unnecessary actions when the balloon is near its target or moving toward it.

A tree-search algorithm is a strategy that explores potential solutions or states within a tree-structured diagram. Every decision point or state on the tree is known as a node. In this context, a node signifies a given transition time and transition altitude. Each of these nodes is traversed by executing the altitude transition, and then floating until the investigation criterion is met. At this point, a new set of nodes is created and expanded from the current state. In this process, the algorithm will traverse several sequences by navigating through each unique tree path. Upon completion of this traversal, the system can evaluate the optimal sequence of altitude transitions which maximize Time on Target, to be followed by the true system.Fig. 6 (a) Example Trajectory and corresponding (b) altitude and velocity over time plot for a successful trajectory and (c) unsuccessful trajectory. Demonstrates balloon moving to altitudes that keep it within station at slow wind speeds to maximize Time on Target for a successful 48 h of station-keeping. Whereas the unsuccessful balloon is unable to keep within the necessary radius.

Node creation is done with a generator function. The function calculates the future states of the balloon if it continues to float at its current altitude until the system exceeds its maximum distance from the target, set at 250 km. Each point in time along this trajectory is assigned to the set of potential transition altitudes, which are then returned as nodes.

The tree is constrained by its depth or the number of nodes traversed by a single path. Once the algorithm arrives at a new state achieved after a simulated transition, the algorithm repeats the process, generating a new set of potential future states. It continues this iteration for a predetermined depth, creating a branching structure of possible paths that the balloon could follow. Allowing the algorithm to traverse deeper will give it better foresight for decision-making, but at the cost of exponentially increasing computational complexity. An analysis was done, weighing the improvements in decision-making versus runtime, and the decided depth assigned to the study is 2. This provides some complexity but is not unreasonable computationally. Because we are interested in geographical and seasonal differences, the minimal additional improvements in quality due to greater depth is unnecessary. This, along with all other hyperparameters are listed in Table 1. Table 1 Hyperparameters defining the simulation algorithm.

Hyperparameter	Value	
Minimum altitude	15,003 m	
Maximum altitude	32,106 m	
Coverage radius	50 km	
Max radius	250 km	
Investigation radius	40 km	
Planning depth	2	
Max transitions	50	
Max Duration	2 months	

A tree-search algorithm will allow the system to anticipate flight futures, and strategically select the most favorable path based on the wind conditions in the given future. This style of system relies on two conditions not present in a real balloon situation. Firstly, it assumes perfect information on forecasted wind data about future conditions. That allows this system to have total confidence in the future trajectory. In reality, the accuracy of forecasted information decreases dramatically the further into the future it’s predicted. Forecasting error and uncertainty will have consequential effects on the ability of a high-altitude platform to navigate effectively. Using the proposed tree search algorithm, the system may make a multi-day trajectory to arrive at a target point with a heading error of 180∘, potentially making the navigational attempt unsuccessful. The incorporation of forecast uncertainties for navigation is an area of control theory that requires further research to identify an effective navigational controller. To get a rough estimate for how much using a forecast instead of reanalysis might impact the results, we examine several of our most successful trajectories as follows. For each trajectory segment of a selected simulation, we take the calculated trajectory (x, y, z, t) and consider the ith transition time, ttransition(i). We calculate an alternative trajectory using the GFS forecast at ttransition(i) using the initial position of the calculated trajectory at that time (x(i), y(i), z(i)). This forward trajectory is run until the time ttransition(i+1), at which point the new distance from the target is calculated. The difference in the distance from the target for the forecast-based trajectory and the reanalysis-based trajectory gives a rough indication of how much additional error one might expect from using forecasted rather than reanalysis winds. For our most successful trajectories, that distance is 435 km ± 360 km averaged across 205 transitions using GFS versus the 67 km ± 49 km using reanalysis. We note, however, that the calculation is a very rough estimate since it does not incorporate any information that can be gained from the trajectory itself or any learning methods (as discussed in the Deep RL work of Google Loon2), and so we do not perform this exercise more generally. The control algorithm would obviously initiate a transition sooner than time ttransition(i+1) if the balloon were drifting so far from the target, and so this estimate does not reflect what would happen during flight. Instead it serves to highlight the importance of the development of a good control and learning algorithm, which is beyond the scope of this current study.Fig. 7 Data flow diagram for implementation of tree search optimization algorithm.

Secondly, it gives the system no time constraints, giving plenty of time to consider all potentially valuable options. This algorithm is computationally expensive and inefficient. For a live balloon, transition altitudes must be determined quickly as the balloon is constantly moving.

For the Python code implementation, a system was initialized at a given target position (latitude and longitude), time, and an altitude corresponding to the minimum lateral velocity in the vertical wind column. The balloon system is then propagated forward in time, using the Tree-Search Optimization Algorithm to determine the future altitude decisions to stay within the target 50 km radius of the station. It will continue to do this until it exceeds the set outer radius. This was set at 250 km to keep the viability of the location localized, as there was approximately 500 km of distance between stations in the study. A visualization of the algorithm’s processes is given in Fig. 7. Two additional boundary conditions were also set to prevent an infinite runtime case of a perfectly viable environment. First, the system terminates after 2 months of flight. This extends through the majority of a given season and gives a strong indication of relative performance, without extending into the next season being investigated. This end case occurred in 0.8% of simulations ran. The second condition gives a limit of altitude transitions the system can make at 50 transitions. This prevented the system from infinitely transitioning. Under the average duration of flight in the simulation (5.9 days), this puts the rate of transitions at 1 per every 3 h, which exceeds the dynamic rate of wind evolution. This end case led to the termination in 0.4% of simulations. Both of these end cases were only seen in high-success simulations, thus the understanding of the performance relative to lower-success simulations is not restricted.

Author contributions

D.B wrote the main manuscript text and prepared Figs. 1, 2, 3, 5, 6, and 7. M.L prepared Fig. 4. All authors reviewed the manuscript.

Data availibility

Output data used to replicate the results in the study and conduct further studies can be found at https://doi.org/10.7910/DVN/C4V84D38. Code used to generate the simulation results and associated graphics can be found in the repository linked at https://github.com/Davidb8/Station-Keeping-Viability-Study34 was downloaded from the Copernicus Climate Change Service (2023). The results contain modified Copernicus Climate Change Service information 2020. Neither the European Commission nor ECMWF is responsible for any use that may be made of the Copernicus information or data it contains. MERRA-2 data are publicly available at https://disc.gsfc.nasa.gov/datasets?project=MERRA-2, managed by the NASA Goddard Earth Sciences (GES) Data and Information Services Center (DISC).

Competing interests

The authors declare no competing interests

Publisher's note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
==== Refs
References

1. Cathey HM Smith M Stephens R Design and testing of the uldb vehicle Adv. Space Res. 2002 30 5 1215 1220 10.1016/S0273-1177(02)00534-3
Cathey, H. M., Smith, M. & Stephens, R. Design and testing of the uldb vehicle. Adv. Space Res. 30(5), 1215–1220. 10.1016/S0273-1177(02)00534-3 (2002).
2. Bellemare MG Candido S Castro PS Gong J Machado MC Moitra S Ponda SS Wang Z Autonomous navigation of stratospheric balloons using reinforcement learning Nature 2020 588 7836 77 82 10.1038/s41586-020-2939-8 33268863
Bellemare, M. G. et al. Autonomous navigation of stratospheric balloons using reinforcement learning. Nature 588(7836), 77–82. 10.1038/s41586-020-2939-8 (2020).33268863
3. Yi H Chen X Wang D Du S Guo N Ma Y Satellite imaging direction angles estimation method based on rational polynomial coefficients Int. Arch. Photogramm. Remote. Sens. Spat. Inf. Sci. 2020 43 527 532 10.5194/isprs-archives-XLIII-B2-2020-527-2020
Yi, H. et al. Satellite imaging direction angles estimation method based on rational polynomial coefficients. Int. Arch. Photogramm. Remote. Sens. Spat. Inf. Sci. 43, 527–532 (2020).
4. World View Enterprises. https://www.worldview.space/. Accessed: 2024-01-27 (2023).
5. Tsien, T.-H. Chemistry and chemical technology, part 1: Paper and printing. In: Needham, J. (ed.) Science and Civilisation in China vol. 5, p. 128 (Cambridge University Press, 1985).
6. Jones, W. V. Evolution of scientific ballooning and its impact on astrophysics research. Adv. Space Res. 53(10), 1405–1414 (2014).10.1016/j.asr.2013.12.028.
7. Mohapatra, M.C. Buoyancy based balloon altitude control. PhD thesis, WORCESTER POLYTECHNIC INSTITUTE. Buoyancy of a Balloon (2022).
8. Cathey, H.M. & Pierce, D. Development of the nasa ultra-long duration balloon (2007).
9. Cathey, H. M. Development of the nasa long duration balloon vehicle. Adv. Space Res. 26(9), 1345–1348. 10.1016/S0273-1177(00)00058-2 (2000).
10. Serrano P Gramaglia M Mancini F Chiaraviglio L Bianchi G Balloons in the sky: Unveiling the characteristics and trade-offs of the google loon service IEEE Trans. Mob. Comput. 2023 22 6 3165 3178 10.1109/TMC.2021.3135976
Serrano, P., Gramaglia, M., Mancini, F., Chiaraviglio, L. & Bianchi, G. Balloons in the sky: Unveiling the characteristics and trade-offs of the google loon service. IEEE Trans. Mob. Comput. 22(6), 3165–3178. 10.1109/TMC.2021.3135976 (2023).
11. Katikala, S. Google project loon. InSight: Rivier Acad. J. 10(2), 1–6 (2014).
12. Perek L The scientific and technical aspects of the geostationary orbit Acta Astronaut. 1988 17 6 589 598 10.1016/0094-5765(88)90202-0
Perek, L. The scientific and technical aspects of the geostationary orbit. Acta Astronaut. 17(6), 589–598 (1988).
13. Anger, J., Franchis, C. & Facciolo, G. Assessing the sharpness of satellite images: Study of the planetscope constellation. In IGARSS 2019-2019 IEEE International Geoscience and Remote Sensing Symposium, pp. 389–392 (IEEE, 2019).
14. Elbert, B.R. Introduction to Satellite Communication (Artech house, 2008).
15. Ma, S., Chou, Y.C., Zhao, H., Chen, L., Ma, X. & Liu, J. Network characteristics of leo satellite constellations: A starlink-based measurement from end users. In IEEE INFOCOM 2023 - IEEE Conference on Computer Communications, pp. 1–10. 10.1109/INFOCOM53939.2023.10228912 (2023).
16. Friedl-Vallon, F., Dannenberg, K., Raizonville, P. & Vargas, A. Stratospheric balloons: low-cost platforms for science and technology development. In International Conference on Space OpticsICSO, 11180, 2714–2721 (2019).
17. Crisp N Smith K Hollingsworth P Small satellite launch to leo: A review of current and future launch systems Trans. Jpn Soc. Aeronaut. Sp. Sci. Aerosp. Technol. Japan 2014 12 ists29 39 47
Crisp, N., Smith, K. & Hollingsworth, P. Small satellite launch to leo: A review of current and future launch systems. Trans. Jpn Soc. Aeronaut. Sp. Sci. Aerosp. Technol. Japan 12(ists29), 39–47 (2014).
18. Weigel AL Hastings DE Evaluating the cost and risk impacts of launch choices J. Spacecr. Rocket. 2004 41 1 103 110 10.2514/1.9270
Weigel, A. L. & Hastings, D. E. Evaluating the cost and risk impacts of launch choices. J. Spacecr. Rocket. 41(1), 103–110 (2004).
19. Smith, I. The nasa balloon program: looking to the future. Adv. Space Res. 33, 1588–1593. 10.1016/j.asr.2003.07.052 (2002).
20. Wynsberghe E Turak A Station-keeping of a high-altitude balloon with electric propulsion and wireless power transmission: A concept study Acta Astronaut. 2016 128 616 627 10.1016/j.actaastro.2016.08.017
Wynsberghe, E. & Turak, A. Station-keeping of a high-altitude balloon with electric propulsion and wireless power transmission: A concept study. Acta Astronaut. 128, 616–627 (2016).
21. Aaron KM Heun MK Nock KT A method for balloon trajectory control Adv. Space Res. 2002 30 5 1227 1232 10.1016/S0273-1177(02)00526-4
Aaron, K. M., Heun, M. K. & Nock, K. T. A method for balloon trajectory control. Adv. Space Res. 30(5), 1227–1232. 10.1016/S0273-1177(02)00526-4 (2002).
22. Borges, R.A., Battistini, S., Cappelletti, C. & Honda, Y.M. Altitude control of a remote-sensing balloon platform. Aerosp. Sci. Technol. 110, 106500. 10.1016/j.ast.2021.106500 (2021).
23. Rapinett A Zephyr: A high altitude long endurance unmanned air vehicle 2009 Doctor Department of Physics, University of Surrey
Rapinett, A. Zephyr: A high altitude long endurance unmanned air vehicle (Department of Physics, University of Surrey, Doctor, 2009).
24. Baldwin MP Gray LJ Dunkerton TJ Hamilton K Haynes PH Randel WJ Holton JR Alexander MJ Hirota I Horinouchi T Jones DBA Kinnersley JS Marquardt C Sato K Takahashi M The quasi-biennial oscillation Rev. Geophys. 2001 39 2 179 229 10.1029/1999RG000073
Baldwin, M. P. et al. The quasi-biennial oscillation. Rev. Geophys. 39(2), 179–229. 10.1029/1999RG000073 (2001) https://agupubs.onlinelibrary.wiley.com/doi/pdf/10.1029/1999RG000073.
25. Matsuno, T. Lagrangian motion of air parcels in the stratosphere in the presence of planetary waves. Pure Appl. Geophys. 118, 189–216 (1980).
26. Plumb, R.A. Stratospheric transport. J. Meteorol. Soc. Japan. Ser. II 80(4B), 793–809 (2002).
27. Zhang, C. Madden-julian oscillation. Rev. Geophys. 43(2). 10.1029/2004RG000158 (2005).
28. Alexander MJ Grimsdell AW Stephan CC Hoffmann L Mjo-related intraseasonal variation in the stratosphere: Gravity waves and zonal winds J. Geophys. Res. Atmos. 2018 123 2 775 788 10.1002/2017JD027620
Alexander, M. J., Grimsdell, A. W., Stephan, C. C. & Hoffmann, L. Mjo-related intraseasonal variation in the stratosphere: Gravity waves and zonal winds. J. Geophys. Res. Atmos. 123(2), 775–788. 10.1002/2017JD027620 (2018) https://agupubs.onlinelibrary.wiley.com/doi/pdf/10.1002/2017JD027620.
29. Domeisen DIV Garfinkel CI Butler AH The teleconnection of el nio southern oscillation to the stratosphere Rev. Geophys. 2019 57 1 5 47 10.1029/2018RG000596
Domeisen, D. I. V., Garfinkel, C. I. & Butler, A. H. The teleconnection of el nio southern oscillation to the stratosphere. Rev. Geophys. 57(1), 5–47. 10.1029/2018RG000596 (2019) https://agupubs.onlinelibrary.wiley.com/doi/pdf/10.1029/2018RG000596.
30. Global Modeling and Assimilation Office (GMAO): MERRA-2 tavgC_3d_ltm_Np: 3d, Long Term Mean 3-Dimensional Meteorological Fields V1. Goddard Earth Sciences Data and Information Services Center (GES DISC), Greenbelt, MD, USA. Accessed: 01/23/2021 (2020).
31. Fesen, R. & Brown, Y. A method for establishing a long duration, stratospheric platform for astronomical research. Exp. Astron. 39, 475–493. 10.1007/s10686-015-9459-9 (2015).
32. Kayhan, Ö. Station keeping of wind driven stratospheric balloon via propulsion unit. Mhendislik Bilimleri ve Tasarm Dergisi 8(1), 252–261. 10.21923/jesd.397265 (2020).
33. Safyan, M. Planets dove satellite constellation. In: Handbook of Small Satellites: Technology, Design, Manufacture, Applications, Economics and Regulation, pp. 1057–1073 (Springer, 2020).
34. Hersbach, H., Bell, B., Berrisford, P., Hirahara, S., Hornyi, A., MuozSabater, J., Nicolas, J., Peubey, C., Radu, R., Schepers, D., Simmons, A., Soci, C., Abdalla, S., Abellan, X., Balsamo, G., Bechtold, P., Biavati, G., Bidlot, J., Bonavita, M., De Chiara, G., Dahlgren, P., Dee, D., Diamantakis, M., Dragani, R., Flemming, J., Forbes, R., Fuentes, M., Geer, A., Haimberger, L., Healy, S., Hogan, R.J., Hlm, E., Janiskov, M., Keeley, S., Laloyaux, P., Lopez, P., Lupu, C., Radnoti, G., Rosnay, P., Rozum, I., Vamborg, F., Villaume, S., & Thpaut, J.-N. Complete ERA5 from 1940: Fifth generation of ECMWF atmospheric reanalyses of the global climate. Copernicus Climate Change Service (C3S) Data Store (CDS). Accessed on 01-09-2023. 10.24381/cds.143582cf (2017).
35. Hersbach H Bell B Berrisford P Hirahara S Hornyi A MuozSabater J Nicolas J Peubey C Radu R Schepers D Simmons A Soci C Abdalla S Abellan X Balsamo G Bechtold P Biavati G Bidlot J Bonavita M Thpaut J-N The era5 global reanalysis Q. J. R. Meteorol. Soc. 2020 10.1002/qj.3803
Hersbach, H. et al. The era5 global reanalysis. Q. J. R. Meteorol. Soc.[SPACE]10.1002/qj.3803 (2020).
36. Bourguet, S. & Linz, M. The impact of improved spatial and temporal resolution of reanalysis data on lagrangian studies of the tropical tropopause layer. Atmos. Chem. Phys. 22, 13325–13339. 10.5194/acp-22-13325-2022 (2022).
37. Munos, R., et al. From bandits to monte-carlo tree search: The optimistic principle applied to optimization and planning. Found. Trends Mach. Learn. 7(1), 1–129 (2014).
38. Brown, D. Replication Data For: Seasonal and Geographic Viability of Stratospheric Balloon Station-Keeping. 10.7910/DVN/C4V84D.
