The stratospheric Goldilocks zone is critical for high-altitude balloon navigation

Abstract

High-altitude platforms offer a low-cost and high-quality alternative to satellites and piloted aircraft for applications from remote sensing to communications. Unlike satellites, balloons can change altitude on demand to navigate or remain stationary by leveraging natural wind variability at different altitudes. While wind variability is known to exist across the upper stratosphere, real-world platforms are operationally constrained to a subset of altitude ranges. Therefore, it is crucial to find which altitude bands contain the most diversity to maximize navigability while minimizing system size and complexity. Here, we introduce an intuitive, effective method to quantify diversity in a wind column and apply it to 338 distinct altitude bands, each evaluated using over 250 million wind profiles from 2020 to 2024. Our results reveal a pronounced ‘Goldilocks’ zone achieved by altitude floors below 16 km and altitude ceilings above 21 km. Operating in this range is essential for having predictable steering winds. This analysis provides a quantitative framework to optimize operating altitudes to maximize navigational potential for the design and operating paradigms of any high-altitude platform.

Introduction

High-altitude balloons (HABs) have become increasingly present in research for years as tools for remote sensing, communications, and atmospheric monitoring due to their unique advantages over traditional platforms. Balloons are situated to be relatively low cost compared to satellites and traditional fixed-wing aircraft, while offering powerful performance due to their close proximity to the ground and unique ability to move slowly or functionally loiter without requiring power to stay aloft.

HABS have evolved over more than a century from experimental vehicles to sophisticated sensing platforms that can be navigated long distances across known paths or made to remain in the same area deliberately over substantial time frames. In the mid-20th century, new materials and designs were developed to greatly enhance HAB performance. In the 1950s, plastic polyethylene balloons allowed for larger envelopes with heavier payloads¹. These zero-pressure balloons had a more natural shape and could rise beyond 30 km, but were not strong enough to maintain any internal pressure. As a zero-pressure balloon ascends, the gas expands, and if the balloon becomes full, any excess gas will be vented out to prevent pressurization and cause the balloon to stabilize at a near-constant altitude. Zero-pressure balloons will experience strong diurnal effects as the gas cools and contracts in the absence of solar heating. Without action, the balloon will lose altitude at night and descend towards the ground. For the next 50 years, the zero-pressure balloon dominated the industry, with NASA beginning the Long Duration Ballooning (LDB) effort in the 90s to fly for multiple weeks in the Antarctic summers, possible due to the constant daylight. In the early 2000s, NASA began a new Ultra Long Duration Ballooning (ULDB)² mission using a super-pressure balloon. These systems, utilizing multi-layer films with reinforcing tendons, could sustain constant volume under high internal pressures and extreme conditions³. A super-pressure naturally maintains a stable float when cooling occurs at night⁴. As the gas contracts, the internal super-pressure decreases, but not enough to reduce the volume of the balloon. Testing proved these balloons could fly for many months at a time, even through full day-night cycles (unlike the zero-pressure Antarctic summer trials). Innovation continued in refining the super-pressure system, and it remains dominant in the commercial industry today^5,6.

All balloon systems naturally drift with the wind, making control over their horizontal movement challenging. Some utilize onboard propulsion mechanisms to offer control⁷, but this comes at high size, weight, and power (SWaP) costs. Recently, their primary steering mechanism comes from selectively altering their altitude⁸ to place them in winds that are blowing in a desired direction⁹. This is particularly feasible at the boundaries of atmospheric layers where winds may blow in different directions at nearby altitudes, such as the transition from the troposphere to the tropopause and the tropopause to the stratosphere.

The potential for using different winds at different altitudes for navigation and station keeping has been shown experimentally by multiple organizations in multiple configurations. Notably, the Google Loon¹⁰ project logged over a million hours of flight using their autonomous altitude control systems to sail with the winds¹¹. This powerful technology has considerable potential as it allows these platforms to functionally steer and travel considerable distances without any onboard propulsion mechanisms. This paradigm offers massive potential given that the systems can stay aloft for extremely long durations, like satellites, while navigating laterally in a deliberate direction like airplanes. The energy requirements for performing altitude control maneuvers are insignificant when compared to the energy requirements of propelling an aircraft sideways. The systems are functionally powered by the sun, riding currents of air and using solar power to regulate altitude to ideal conditions. The potential longevity of flight, unique navigational ability, and close proximity to the ground (for low-latency communications¹² and high spatial resolution sensing) have driven interest in bringing these platforms into use.

To achieve this wind-based navigation, many super-pressure balloons employ compressor-driven inflatable air bladders for altitude control, necessitating the use of large, high-speed mechanical systems¹³. These systems critically increase the cost, mass, and overall complexity of the balloon platform. Moreover, their operational efficiency is constrained by the limitations of compressor impellers, which can be optimized for only a single air density. As a result, performance degrades when operating across a wide range of altitudes, where variations in atmospheric density reduce compressor efficiency. Producing enough solar power to remain energy positive in flight necessitates even larger systems with more solar panels.

Zero-pressure balloons, on the other hand, have been designed to release limited consumables during diurnal transitions, enabling multiple days of flight. This has since been adapted to additionally be used to regulate the system’s altitude. This mechanism has numerous advantages over those used by super-pressure systems, as they require less complicated operating mechanisms and offer a larger altitude range. The zero-pressure balloon has no theoretical lower limit of downward altitude control maneuver, excluding the ground. Additionally, using altitude control to lift the balloon back towards its ceiling requires only a marginal amount of ballast to offset the equilibrium of forces. Since a zero-pressure balloon does not aim to fly for months at a time, a consumable altitude control system can be utilized, allowing the system to stay simple, small, and affordable.

The primary uncertainty lies in the ability of the systems to navigate using the winds alone, with no support of onboard propulsion. Previous studies have investigated the viability of this form of navigation, and through simulation of reanalysis data¹⁴ have demonstrated that it remains viable, dependent on the seasonal and geographic conditions, with further validation using radiosonde data¹⁵. Navigable winds are more prominent near the equator, in the tropics except in summer, and in hemispheric summer in the mid latitudes. This correlates well in regions with low zonal mean winds, where deviations can occur and offer opposing wind regimes. While these works have established the general seasonal and geographic viability of wind-based navigation, the current study focuses on optimizing the specific vertical operating bands to maximize this potential, a critical step for platform design and operational efficiency.

Additionally, studies have investigated control algorithms for achieving effective wind-based navigation. Traditional approaches range from using a predetermined set of rules or models to search for optimal winds¹⁶. More recent work has used machine learning methods, typically using deep reinforcement learning, to perform station-keeping maneuvers^17,18. In these AI-based systems, the goal is to learn an optimal policy for adjusting altitude using the available wind forecasts and onboard data systems.

These studies have offered substantial advancements in understanding what drives wind-based balloon navigation, but they also offer a major assumption regarding the operating regime of the system. Existing studies of wind-based navigation conditions investigate winds ranging through the entire lower stratosphere and below between 15 and 30 kilometers. Most major ballooning systems do not operate through this entire 15-kilometer range. Many companies and research systems specialize in using super-pressure balloons, which typically have an operating range of just a few kilometers. This poses a potentially cumbersome challenge, as these traditional super-pressure balloons are inherently limited in their ability to exploit wind diversity because they cannot sample through the entire vertical column. In contrast, a zero-pressure balloon system does not have this constraint on its operating range. All balloons have an upper altitude limit dictated by their fixed mass and total envelope volume, defined when the buoyancy force lifting the system in the atmosphere at its altitude is equal to the gravitational force exerted by its mass. Zero-pressure balloons have no theoretical lower altitude bound. Where a super-pressure balloon will lose its super-pressure from going too low in most architectures, a zero-pressure balloon can continue getting smaller until it encounters the Earth. Due to their lack of an altitude floor, zero-pressure systems can potentially access a broader range of altitudes, albeit for a shorter flight duration in most cases.

This poses a critical design question: can balloons with a limited altitude control range access enough wind diversity to usefully navigate? Given that balloon types that can theoretically access very large altitude control ranges (zero-pressure balloons) typically have very limited durations, and balloon types with very long duration potential (super-pressure balloons) have limited altitude control ranges, is there enough wind variability within the achievable altitude control ranges for ultra long duration balloons and what is the range that works?

In addition to the mechanics of high altitude balloon flight, there are global and national norms and standard operating procedures for aircraft that functionally impact operating zones for balloons such as the global standard class A controlled airspace that extends to 60,000 feet (18 km) and the regular operation of passenger aircraft which extends functionally to about 43,000 feet (13 km) altitude. While there are some references to the ceiling of class A airspace at 60,000 feet in balloon regulations, they are generally not restrictive, and high-altitude balloons operate regularly in the space between the highest aircraft flights and the top of class A airspace.

In this study, we address this question of optimal operating altitudes and altitude ranges of balloon systems through a systematic analysis of wind diversity across a thorough range of altitude bands on high spatial resolution global reanalysis data. The objective is to determine the optimal vertical operating regime (the altitude floor and ceiling) to maximize the diversity offered by the available winds. By quantifying the degree of wind variability within different altitude bands, we can infer potential enhanced navigational capabilities.

We first describe the extent of the study and the corresponding results in Section 2 “Results”. Then, we investigate the implications of these results in Section 3 “Discussion” and Section 4 “Conclusion”. Finally, we detail the metrics designed and the steps conducted to generate the dataset created in this study in Section 5 “Methods”.

Results

For each of the 338 altitude bands investigated in the study, the diversity at each lateral-temporal location was computed. The study maps the wind column of a given altitude band to a single value representing the wind diversity using the developed diversity function (see Section “Wind Diversity Computation” for computation details and Section “Altitude Configuration” for altitude band definition). The global diversity for the entire studied region (10–32 km) in February is seen in Fig. 1. The study was conducted globally through weekly snapshots from 2020 to 2024 to capture all relevant upper atmospheric phenomena, including the Quasi-Biennial Oscillation¹⁹, Madden-Julian Oscillations^20,21, and both La Nina (2020–2022) and El Nino phases (2023–2024)²².

Fig. 1: Global wind diversity for February 16, 2020.

Wind diversity globally for February 16, 2020. Colors represent the diversity percentage, with 100% indicating maximum potential diversity and lower percentages indicating more uniform wind directions, where 0% is no wind diversity, or all winds at all altitudes blowing in the same direction. Due to the large 22 km range, we see strong performance nearly globally, except for the high northern latitudes, as expected during their winter.

Since it is known from previous studies that navigation is dependent on both season and latitude band¹⁴, the results are shown for specific latitude bands and seasons that are known to have strong diversity. The mean wind diversity of each altitude band configuration is studied for each season and latitude band. Contour plots are arranged in a 2-by-2 layout to visualize the altitude band diversity for each season for a given latitude zone. The plots of the most diverse seasons in each of the tropic and mid-latitude regions are shown in Fig. 2, and those of all seasons in the equatorial region are given in Fig. 3.

Fig. 2: Diversity contour plots of selected high diversity regions.

Altitude floor/ceiling plots for selected high diversity regions. A Southern Mid Latitudes (30^∘S–60^∘S) during hemispheric Summer. B Northern Tropics (10^∘N to 30^∘N) during hemispheric Spring. C Northern Mid Latitudes (30^∘N–60^∘N) during hemispheric Summer, D Southern Tropics (10^∘S–30^∘S) during hemispheric Spring. Colors represent the mean wind diversity percentage. Contours plotted for each 5% interval.

Fig. 3: Diversity contour plots of equatorial regions.

Altitude floor/ceiling plots for equatorial region (10^∘S–10^∘N) in each season (A DJF, B MAM, C JJA, D SON). Colors represent the mean wind diversity percentage. Contours plotted for each 5% interval.

Discussion

The results presented offer clear evidence that wind diversity is highly non-linear through the upper atmosphere. For a super-pressure’s given operating range, the specific altitude floor and ceiling chosen will have crucial effects on the expected amount of wind diversity and thus navigational feasibility. As would be anticipated, the diversity of wind increases as altitude control range increases, given that more altitude range offers more opportunity for wind diversity. This is particularly relevant for zero-pressure balloons that can take advantage of lowering their floor.

Firstly, when equating wind diversity with navigation and station-keeping potential, we see a validation of the existing literature where, regardless of altitude band configuration, the equator exhibits high wind diversity year round, the tropics exhibit strong diversity in all seasons except for summer, and the mid and high latitudes exhibit relatively strong wind diversity only in summer.

Across the latitude regions and seasons where strong diversity exists, we see two major topological regions. First, many of the lower diversity contours take a hockey stick shape, where we see (with varying severity) that the contour slopes increase rapidly, going from nearly horizontal to nearly vertical. This characteristic contour shape generally begins at an altitude ceiling around 21 km [70 kft] and extends downward. The sharp transition in the contours occurs at altitude floors between 15 km [50 kft] and 18 km [60 kft]. These low to moderate diversity contours are extremely dense, revealing rapid increases in diversity. The second region is a ‘Goldilocks’ zone of high diversity potential, which generally appears when altitude floors are extended below 16 km [52 kft] and altitude ceilings are raised above 21 km [70 kft]. In this region, the hockey stick shape devolves, and we begin to see more linear contour lines. Additionally, the spacing between the contours increases significantly.

These two topological regions offer key insight into optimizing navigability for both zero-pressure and super-pressure balloons. For any balloon, we see that it’s imperative to extend the floor below 18 km. The slope of the contours changes to nearly vertical around this point, resulting in a sheer inability to increase navigability when operating only above 18 km. Increasing the ceiling with such a high floor will result in little or no improvements to the wind diversity. For a zero-pressure balloon, this is particularly inefficient. Since these systems have no functional altitude floor, they can easily enter this high diversity Goldilocks zone, assuming they have a modestly high ceiling. This suggests that a balloon utilizing an altitude control system can better exploit the vertical structure of the wind field in these regions. Assuming a low floor, the contours become nearly horizontal below the Goldilocks zone. Here, where the contours are dense, there is a major opportunity to increase diversity by increasing the altitude ceiling. This extends until the high diversity zone, where the contours become much more sparse, and there are diminishing marginal returns on increasing diversity by increasing the altitude ceiling. Increasing the ceiling altitude significantly beyond 21 km yields only marginal improvements. This plateau implies that the upper stratospheric winds begin to rehomogenize, exhibiting similar or identical directional behavior. Hence, while it’s generally desirable to offer a broader range of altitudes, the critical benefit is unlocked through the incorporation of lower altitudes, as opposed to an indefinite extension of the altitude ceiling.

For the super-pressure balloon, the operating paradigm is more complex. These systems are designed to have only a fixed and limited range of operating altitudes, so selecting the right ones is crucial. Additionally, increasing the operational range is expensive compared to a zero-pressure balloon. For a super-pressure system to keep a constant ceiling and lower its floor, it needs to be able to hold more air ballast, requiring more power and equipment, and thus an even larger and more complicated balloon. Therefore, for super-pressure systems, it would be advantageous to get the most out of the range it has. This is generally achieved in the bottom right corner of the Goldilocks zone, or the crux of any of the hockey stick contours. Along a contour of constant diversity, this elbow will have the lowest ceiling and the highest floor, and thus the smallest operating range. It is in this tighter range that pivotal shifts in wind direction occur; capturing these opposing wind structures is essential, and may allow for solid maneuverability over just a few wind layers.

The only major exclusion to the typical contour shaping occurs in the equatorial region. Through each of the months, there is a ripple effect on the contour lines, where the Goldilocks zone still is prevalent, but additionally, the wind diversity at high altitude bands (in the 21 km to 27 km range) remains moderately strong. This effect is likely induced by both the considerably higher tropopause at the equator, 17 km vs 12 km in the mid latitudes, as well as by the Quasi-Biennial Oscillation¹⁹. The Quasi-Biennial Oscillation (QBO) is present in the equatorial region and extends into the tropical regions, and is a downward propagation of opposing easterly and westerly winds every two to three years. This results in strong shears that correlate highly with wind diversity localized around the current location of the shear. Tracking the current state of the QBO may offer opportunities for additional favorable operating altitudes for steering. Although due to these winds being sets of directly opposing winds, their actual diversity is only moderate.

These results demonstrate inherent trade-offs in operating altitude, offering clear implications for design configurations for all balloon systems. For a zero-pressure balloon system, one first needs to select an operating floor, for example, 13.1 km, to remain entirely above commercial airspace. From here, the marginal benefit curve, defined as the additional diversity gained per kilometer increase in the altitude ceiling to exploit more wind layers, can be found and is plotted in Fig. 4. The importance of entering the Goldilocks zone is seen clearly as the marginal benefit shoots up as the ceiling increases through 21 km. As can be inferred by the spacing of the contour lines, there are diminishing marginal returns of increasing the altitude ceiling, and it’s validated as the marginal benefit begins to decline rapidly beyond 21 km. By 25 km, each additional kilometer of altitude range is only increasing diversity by 2%, and by 30 km, that decreases to just 1%. Increasing the altitude ceiling is directly achieved by increasing the size of the envelope, and thus the volume of the balloon. This then promotes a metric of the marginal cost of increasing the operating ceiling by that given amount. Cost can come in many structures, and doesn’t have to be financial. For balloons, common costs include those of increased capital expenditures (CapEx), operating expenditures (OpEx), and regulatory compliance. For CapEx, increasing the balloon volume increases both the surface area (driving up the cost of the envelope) and the required lift gas. Bigger balloons require larger and better-trained teams to deploy and launch, increasing OpEx. Larger systems can enter new classes of balloons, exposing one to new regulatory constraints. One must first find a common metric of value to evaluate the marginal benefit and marginal costs, and from there design a system where these are equalized and utility is maximized.

Fig. 4: Zero-pressure diversity curves.

A Diversity as a function of altitude ceiling for a zero-pressure balloon with an altitude floor of 13.1 km. B Marginal Diversity gain as a function of altitude ceiling.

Super-pressure balloons have a more nuanced set of design trade-offs, as these are restricted to a specific band or range based on system design. As said above, their altitude control systems typically utilize large, power-hungry pumps and compressors to operate an air ballast system, with efficiency varying based on the density of the atmosphere in that band. This results in a more sophisticated equation to understand the effect that increasing the altitude range has on the system’s cost and complexity. It’s optimal to achieve as much navigability with as little range as required, which is often achieved along the knee of the contours. To investigate this, for each operating range r, the diversity of the best band, defined where z_ceiling − z_floor = r, is recorded. This allows for the construction of a new marginal benefit curve, visualized in Fig. 5 (the optimal bands for each range are also listed in Table 1). Here, we see a similarly shaped exponential decay curve. The marginal benefit curve decays by $\frac{1}{2}$ after a range of 4.09 km and achieves a diversity score of 43%, and decays by $\frac{1}{e}$ after a range of 5.96 km and achieves a diversity score of 48 %. Again, the specific equilibrium optimization depends on the specifics of the system and its operations, but this offers a foundational framework and reasonable expectations. Google Loon often operated in the 5 km range between 15 km and 20 km, offering 41.8% diversity on average²³. World View predominantly images in the 6.1 km range between 50 and 70 kft, reaching 47.7%²⁴. Aerostar’s primary thunderhead system operates in the 6.1 km range between 55 kft and 75 kft, reaching 48.0%²⁵. For a 6 km range, the optimal diversity of 48% is achieved between 15.9 km to 21.9 km [52–72 kft]. We see further validation that the optimal diversity at low ranges is centered around the 18 km region, and quickly expands to enter the Goldilocks zone of 16 km to 21 km.

Fig. 5: Super-pressure diversity curves.

A Diversity as a function of altitude range for a super-pressure balloon. B Marginal Diversity gain, representing the derivative of the best diversity curve A with respect to range, as a function of altitude range. An exponential function is fit to the data, with an R² = 0.9961. The point where marginal diversity has decreased by a factor of $\frac{1}{2}$ and $\frac{1}{e}$ is also marked on the plot.

Table 1 Optimal altitude bands and diversity scores for each range

Conclusion

Our study demonstrates that wind diversity within the upper atmosphere is highly sensitive to the chosen altitude band, and thus, the feasibility for high-altitude balloon navigation also depends heavily on a balloon’s achievable altitude range. In regions with suitably strong wind diversity, there are two major curves produced and investigated. First, there exists a tight set of “hockey stick" shaped contours for low to moderate diversity bands. Under this regime, there is a critical transition where the contours go from horizontal to vertical, indicating there is a key region of shifting winds that is essential to achieving diversity. With altitude floors that are too high or ceilings that are too low, the opportunity to navigate quickly vanishes as the winds in these extremes are more homogeneous, but improvement can clearly be achieved by lowering the floor or raising the ceiling.

Once this key region of the atmosphere is included, one enters the ‘Goldilocks’ region of high diversity. The contours begin to lose their characteristic shape and become much more spaced out as specific additional altitudes offer only marginal improvement. This zone begins once the balloon’s operating floor drops below 16 km and its ceiling rises above 21 km.

Existing outside of this region, such as flying exclusively above 16 km or exclusively below 21 km, is not recommended. There will be systematically limited wind diversity, both limiting the feasibility and value unlocked in balloon navigation and persistence, as well as posing an increased flight safety risk. When unable to steer, the balloon may be forced to fly into inconvenient or contested areas. Continuing to increase the altitude ceiling yields diminishing returns due to the homogenization of upper stratospheric winds. Similarly, decreasing the altitude floor will begin to have a similar effect as one descends into the troposphere. This conveniently means that lowering the altitude range such that it would become an airspace safety concern is not an issue, as there is little benefit in flying below 13 km.

As HABs become more prevalent, establishing appropriate altitude range regulations in regulatory frameworks is essential. Organizations such as the Federal Aviation Administration (FAA) and the International Civil Aviation Organization (ICAO) must carefully consider lower altitude boundaries in particular. Adoption of thresholds such as the top of Class A airspace at 60,000 ft could severely restrict HAB operations, preventing balloons from effectively utilizing diverse wind conditions for optimal navigation. Notably, the most advantageous altitude layers identified in the Goldilocks wind bands crossover class A airspace while being virtually entirely above the airspace commonly used by piloted heavier-than-air aircraft, with a substantial buffer between them.

These results have major implications for choosing between zero-pressure and super-pressure balloon systems. With no functional altitude floor, a zero-pressure balloon can operate in larger altitude regions, offering more wind diversity than is functionally possible by existing super-pressure systems, thus offering a clear navigational advantage. Assuming an altitude floor of 13.1 km to keep the balloon above typical commercial aviation, we see that average diversity already exceeds 50% with a 21 km ceiling (more than any existing super-pressure system achieves), and reaches 60% by 23.5 km and 70% by 30 km. Although we also quickly reach diminishing marginal returns of increasing the altitude ceiling. This offers a clear framework for designing zero-pressure systems, where one can directly conduct a cost-benefit analysis of the economic, technical, and logistical trade-offs of increasing the altitude ceiling. For a super-pressure balloon, we investigate how diversity increases with altitude range (assuming optimal selection of the altitude band). These systems have design-specific altitude floors and ceilings and require substantial engineering effort to support larger ranges. The analysis reveals an exponentially decaying marginal benefit curve that can similarly inform design decisions. We see that the curve had pronounced decay by the 6 km range of current systems, and thus, while they perform lower than zero-pressures, super-pressures with this altitude range are achieving sufficient diversity.

These conclusions offer a quantitative foundation for optimizing the altitude band in HAB design and can provide insight for rulemakers on what is important to consider when imposing operational limitations. When designing systems with zero-pressure or super-pressure, it’s imperative to consider the effect the balloon design will have on navigability. Zero-pressure balloons can achieve considerably higher wind diversity than their super-pressure balloon counterparts, due to the ease of increasing their range. Super-pressure balloons can achieve good diversity, but due to their limited range, must be expertly placed to exploit variable winds. By optimizing these balloon design regimes, trajectory planning algorithms will be better able to arrive at effective navigational solutions. Steerability has huge implications for mission success and flight safety, and the insights learned through this work can inform both the design and operation strategies of high-altitude balloon systems, guiding further development for more effective and efficient navigational strategies.

Methods

ERA5 reanalysis data

We utilized ERA5 global reanalysis data²⁶ from 2020 to 2024, taking advantage of the full quarter-degree spatial resolution wind fields. This allows for the most accurate diversity averages by maximizing the number of collected samples. To capture the evolution of the field’s behavior temporally, analysis was conducted in weekly snapshots for each year, allowing for analysis to be broken down by season. The ERA5 model level data was used to provide the highest vertical resolution. Specifically, the study was conducted at levels 28 through 80, offering 52 discrete wind vectors between the altitudes of 10 km and 32 km, which well extends beyond typical operating altitudes of HABs²⁷.

Altitude configuration

To investigate the wind diversity as a function of balloon operating range, we define a set of altitude band configurations, defined by the ERA5 model levels. We generated configurations using a range of z_mean values from 28 to 80 in steps of 2, and for each z_mean, we considered all possible z_width values (from 1 to 26) that satisfied the constraint:

$$28\le {z}_{{{{\rm{mean}}}}}-{z}_{{{{\rm{width}}}}}\quad \,{\mbox{and}}\,\quad {z}_{{{{\rm{mean}}}}}+{z}_{{{{\rm{width}}}}}\le 80.$$

This procedure yielded 338 valid altitude configurations.

Wind diversity computation

Wind diversity is computed by a newly defined metric. For a given altitude band, all wind vectors are normalized to unit length. These vectors are then used to form the vertices of a polygon. The area of this polygon is then computed. The area is then normalized by dividing by π, giving a diversity score D between 0 and 1 (0–100%), offering an interpretable percentage diversity. This allows for the intrinsic nonlinearity of wind diversity, where the score steeply increases as the number of unique cardinal directions achievable increases, and then begins to rapidly level off as new wind directions are similar in angle to existing ones. This non-linearity is implicit in the area inscribed by a set of vectors. Each additional vector extends the straight line distance between the two vectors surrounding it to instead point to the outer edge of the circle. As there are more wind vectors and the angular distance between the surrounding vectors decreases, the additional area gained by the added vector decreases. There is one edge case that forms where the column can have moderate diversity while having a low area. This occurs when the vectors are all approximately opposed across 2 directions (i.e., all east or west). The area computed will be quite low, but the diversity would be moderate, as persistence would be possible specifically. This situation is remedied by applying a minimum score of 0.5 (50%) to all columns where the origin is an interior point of the convex hull of the vector endpoints. This adjustment addresses niche scenarios, such as winds being tightly clustered in only two opposing directions, where the raw polygon area would be very small, but a moderate, usable diversity for station-keeping or bi-directional travel still exists. A score of 50% was chosen to represent this baseline maneuverability. Sample wind columns and their associated diversity score are given in Fig. 6. The wind diversity is calculated for each spatio-temporal sample independently, based on the vertical wind profile at that specific location and time. This identifies diversity in the wind column at a given point in time. However, this approach is limited in two aspects: the normalization of wind intensity and the blindness to short temporal variability.

Fig. 6: Sample diversity score function outputs.

Diversity score function output for a selection of informative wind columns to understand the diversity metric.

First, abstracting away wind intensity will be most consequential in regions that have powerful, persistent jet streams. In previous climatological analyses¹⁴, this is found to happen for the mid-latitudes in the winter, predominantly in austral winter, through strong easterlies in the relevant operating altitudes, although these seasonal and geographic conditions already do not allow for appreciable navigation. This is a general trend that high wind speeds are often tied to strong zonal means, which tend to have limited steerability. In cases with diverse winds with a single, strong layer, the agility of the system (how fast the balloon can transit up and down) will determine how well it can balance the varying wind intensities.

Second, the wind fields in the upper atmosphere are not constant in time. High-frequency studies have been conducted using balloons and infrasound measurements, and identified sub-daily and hourly wind fluctuations due to inertial or gravity waves^28,29,30. This temporal variability is specifically relevant when wind speeds are of low magnitude and more easily deviate from their previous or predicted course. Sufficient agility is required to capitalize on favorable wind layers before they change. While the temporal shifts may make certain layers unfavorable for the specific navigational objective, they may also improve previously unfavorable layers. Therefore, important work must be done to develop effective control methods and algorithms that can sufficiently navigate through this rapid temporal variability.

This method of computing diversity is both intuitive and interpretable, as well as computationally effective. The area formed by the polygon can be computed incredibly fast, with time complexity of ${{{\mathcal{O}}}}(n\log n)$, and can effectively take advantage of just-in-time (JIT) compiling. For the study, each global computation required over 1 million evaluations of the diversity function. This was then done for each of the 5 years of weekly data and 338 altitude band configurations, for nearly 100 billion evaluations. Thus, efficient computation is similarly essential for both the study and practical predictive use cases of analyzing forecasts. For a given band, the monthly averages are visualized in Fig. 7.

Fig. 7: Monthly global wind diversity (2020).

Global maps of mean wind diversity for each month in 2020 for ERA5 reanalysis levels 44–64, the approximate beginning of the Goldilocks region. Colors represent the diversity percentage, with 100% indicating maximum potential diversity and lower percentages indicating more uniform wind directions.

Validation against Simulation

Previous work¹⁴ investigated the viability of steering a balloon using the winds through mass trajectory simulation. The study modeled optimal balloon flight to maintain close proximity to a target region (station-keeping). The results demonstrated the viability of achieving this form of navigation and its variability (as dependent on the geographic location and season). The public dataset³¹ includes the trajectories and associated data of each of the approximately 10,000 flights. This dataset was additionally analyzed to investigate trends in operating altitude. In the study, the algorithm was allowed to use the ERA5 model level reanalysis data going from 15 km to 32 km, and the decisions were made solely based on what would maximize the system’s time on target (time spent within 50 km of the target station). A frequency histogram of each of the possible target altitudes is shown in Fig. 8 as looking at both the number of occurrences and time spent at each altitude. We see a substantial over-density in the altitudes between 16 km and 21 km, aligning with the results of this study. The navigational algorithm opted to have the system spend a large fraction of its time within this altitude range, aligning with our finding that these altitudes are critical for effective navigation.

Fig. 8: Station-keeping target altitude distribution.

A Frequency of each target altitude as outputted by the navigational model. B Time spent at each target altitude.

Labeled Graphic

To aid in clear visual understanding, a labeled version of the floor/ceiling plots is given in Fig. 9. This describes both the Goldilocks region and optimal configurations for both zero-pressure and super-pressure systems.

Fig. 9: Labeled diversity contour plot.

Altitude Diversity plot with Goldilocks zone, and optimal altitude configurations for zero-pressure and super-pressure balloons. Zero-pressure balloons benefit most from increasing their altitude ceiling, due to their large feasible operating range. Super-pressures are constrained in operating range, and thus are optimized to fly at the “elbows” of the contours.