Thomas Bayes Goes Ridge Soaring!
Something that is extraordinary is that sailplanes can fly hundreds of miles using only the energy of the atmosphere. One of the best energy sources is the Appalachian ridge system. When the wind is blowing hard enough and roughly perpendicular to a ridge, the mass of air is forced up and over the obstacle. The air rises faster than the glider sinks, which lets the glider maintain its altitude while flying along the ridge. So long as the wind keeps blowing and there is a mountain to force the wind to go over the top of it, the glider can keep going forward. Since the Appalachians are such a long ridge system, it is possible for gliders to go quite far along this consistent energy source. Furthermore, since the wind can blow quite hard, we can turn the added energy into forward speed and fly quite fast, sometimes clocking between 100-150 mph for long sections, all 100ft above the treetops. It’s exhilarating and astounding that it is even possible.
However, there is no question that it carries a much greater risk than other types of soaring. The stark reality in that if things go wrong, they can go very wrong very quickly. The ridges we call “mountains” in Pennsylvania are often only 1000ft above their respective valleys. The actual “steep” part of the ridge may only be around 400ft high. This leaves the pilot few options if the ridge stops “working”, if the wind stops blowing hard enough to support the glider in the air. If the ridge quits, it leaves the pilot around four minutes to pick a landing option and execute an approach and landing. This is very little time and leaves very little room for error. Improperly executed, it could lead to catastrophic consequences, such as destroying the glider and possibly getting hurt. A number of pilots have crashed pursuing ridge flying and a number have gotten killed over the years. Ridge soaring is not for the faint of heart.
Furthermore, since some ridges are in fairly remote places and in hilly terrain, certain sections of ridge will not have a field in gliding distance. These sections require a really brave decision to take a gamble that the ridge will work for a given interval of time, until the pilot has made it to a section that has a suitable landing option at the base of the ridge. Certainly, being in an aircraft without an engine, entirely relying on the atmospheric conditions to work is quite an adventurous proposition, but empirically it has worked quite well. Through extensive training, research and experience, pilots learn to recognize what sorts of conditions support this sort of risk assessment. Over the past 50 years, dozens of pilots had flown hundreds of ridge days and experience shows that the ridge has been quite reliable. Most of the accidents related to ridge soaring are “pilot error”, not due to the wind stopping.
Having put a considerable amount of thought about this, I don’t believe I could come up with a reliable calculation as to what the odds really are regarding the gamble we are taking. However, I have come up with a mathematical framework to express the process of decision making while ridge flying. Using this framework, it is possible to compare to other benchmarks of risk so that each person can have a good sense as to what risk they are choosing to take. To do so, let’s meet Thomas Bayes, a theologian and mathematician in 18th century England.
Thomas Bayes came up with a simple formula to express the likelihood of a hypothesis occurring given previous experience. More so, he provided a method for constantly updating one’s hypothesis given newer and better information. Over a given number of observations and subsequent revisions, the observer is then able to revise the hypothesis until achieving near certainty of a continued result.
In The Signal and the Noise, Nate Silver provides an interesting example. Suppose there is a person who born in a bunker and never saw the light of day. Once he ventures out of the bunker, he will observe the sun rising for the first time. He is rather perplexed by this event and observes the sun going across the sky until it finally sets. He makes a hypothesis that indeed the sun will rise again the day. In assessing this, he must weigh his “prior” or how often the situation he is observing has objectively occurred in the past. Furthermore, he must make a subjective assessment of the likelihood of the hypothesis being wrong.
The formula is expressed quite simply as:
x: prior probability
y: likelihood of event occurring if hypothesis is correct
z: likelihood of event occurring if hypothesis is incorrect
xy
p(y|z) = _______
xy + z(1-x)
The beautiful thing about this formula is that it constantly lets you update the math given new information. The result that pops out is called the posterior. So in the case of the sun rising, every following day, the observer would use the posterior of the previous day as the new prior for the following day. After a number of trials and observations, the results will move to near certainty, in this case that it is highly probable that the sun will rise every subsequent day.
How does all of this relate to ridge flying? The way we interpret information on the ridge is inherently Bayesian!
Testing the ridge
In the beginning of the day, we start with a set of expectations for the day given previous forecasts. This is how we set the prior! For instance, when we look at the weather forecasts and listen to the forecasting gurus in our club, we already have a good idea of what the day should be like.
But to be more specific, let’s use a simpler parameter. Let’s say that when we are gridded and ready to launch, we dial in to Mount Pocono Airport weather and listen to the conditions. Most pilots will agree that if they are calling winds at 15 knots or more, with a wind direction between 310-350 degrees, the ridge should be working quite well.
Using this information, we look back at our previous experiences and those of others in the club in setting our prior in the formula. Let’s presume that 95% of the time that KMPO dictates the parameters listed above, the ridge does indeed work. When we first start our tow, we are unsure whether today will work, but we are confident that it will based on previous experience. Let’s say that just starting out, we assess the likelihood of this specific day working at 50 percent, a coin toss.
So on tow, the probability of the ridge working is:
x = .95 (Prior)
y = .5 (likelihood of event occurring if hypothesis is correct)
z = .5 (likelihood of event occurring if hypothesis is incorrect)
.95 * .5
__________ = ..95
(.95 *.5) + .5(1-.95)
The result that we get is 95%. The odds are good, but a 1 in 25 gamble of falling off the ridge is still substantial. As a result, we will be quite careful getting down on the ridge and will look for more information before committing to getting out of gliding distance from the home airport.
Now, as you are towing over the upper reservoir, you look down and you see the waves rippling across the lake. You release and do a couple turns to establish the drift. You see that the wind is blowing well and the conditions look favorable. Now, you reassess and update the formulation with a more confident hypothesis:
x = .95
y = .75
z = .25
.95 * .75
_________ = .98
(.95*.75) + .25(1-.95)
The new information really helped our situation and we assess the situation favorable. However, while the odds of the ridge working are good, we should be very cognizant of a landing option, in case the ridge does not work.
Once we choose to “get down on the ridge”, now we will have much better data to work with. So let’s suppose we drop down to 100ft above ridge top and find that the ridge is supporting the glider, going at 80 knots (say LS3 or LS4). Pretty good! Now we revise our formula once more:
x = .98
y = .99
z = .01
.98 * .99
________ = .999
(.98 * .99) + .05(1- .98)
Through this process, we have achieved pretty much near certainty that the ridge is working. We may test the ridge a few more times and go through this process some more, but it is after following this process of testing and inference that we come to the conclusion that we could now start moving along.
Flying cross country on the ridge
Content that the ridge is “working”, we are now going to head cross country on the ridge! Now we throw in two more elements into the risk calculus. The first is that we need to assess the likelihood of the ridge working ahead. The second is that given the ridge does NOT work, what is the likelihood of falling off and landing out or worse, crashing.
An Educated Guess: Will the ridge work?
By the time we are going cross country on the ridge, the pilot may feel quite confident that the ridge is working, due to his really solid prior. Once cruising on the ridge, the pilot will need to continually assess his surroundings and make decisions and assessments about the conditions and the risk being taken. Sometimes, the pilot will find himself approaching places that require a decision regarding tactical or safety risk. Tactical risk is the possibility of landing out and not completing the mission. Safety risk is the possibility of crashing. From every given decision point, the pilot will look ahead and make an educated guess as to whether the ridge will keep working. Some things that would influence a pilot’s hypothesis include:
- Topographic features:
- Does the shape and height of the ridge get better or worse?
- Is the ridge alignment favoring the wind better or worse?
- Are there ridges upwind that may block or funnel the flow of wind?
- Atmospheric factors:
- Are there obvious wave features that may cause wave suppression?
- Could over-development limit mixing of the wind down to the surface?
- Is the wind predicted to change speed or direction over the course of the day?
- Will you be flying far enough to get to the frontier of the pressure system, where the wind speed would steeply decay?
- Transitions: Gaps and transition points to other ridges instill a greater degree of uncertainty if the ridge will work and should downgrade the hypothesis accordingly.
- Duration until the next decision point: It is much easier to make a bet that the ridge will work for the next 2 minutes than the next 20 minutes and the hypothesis should be downgraded accordingly.
Weighing all of the variables and information possible, the pilot would then assign a hypothesis of the ridge working until the next decision point, to be repeated continually during the flight.
First, let’s assess how certain we are that the ridge is working. By the time we have settled down on a good ridge, we have had a solid history of the ridge working up to this point. So, given the following:
x = .999 (prior)
y= .990 (hypothesis that the ridge will work)
z = .010 (hypothesis that the ridge will fail)
This means that having a ridge that has been completely solid up until this point and having a 99 percent certainty that the ridge will work beyond this point, we have subjectively assessed that there is a 1/1000 of falling off the ridge.
To get a baseline, take a look at these rather depressing statistics when it comes to mortality risk. According to these statistics, the odds of a driver getting killed in a motor vehicular accident in a given year are about 1 in 9000. Certainly this risk is affected by many other factors, but nonetheless it could serve as a base line. Let me stress this once more: If you assess that the ridge has a 99 percent chance of working, with a near certain prior, by your own subjective assessment, you have a nine times greater chance to fall off the ridge than getting into a car accident! This is not to suggest that this is the actual risk. It could be much greater or much less. However, it is something to consider when driving along on the ridge.
So what happens when we fall off?
When falling off a ridge, the risk calculus incorporates several more elements. Included are the possibilities of hitting a thermal prior to landing, the quality of the field selection and the likelihood of crashing given the landing choices. Since each event is dependent on the previous one, this suggests that we can follow the associative property, which means we can multiply each probability to get the overall risk of crashing.
Note: These numbers are subjective. Each pilot assesses incorporates his own risk calculus. An “aggressive” pilot assigns lower risk to this calculation, a “conservative” pilot assesses the risk higher.
Let’s consider a couple scenarios. For the first, let’s presume that the pilot is cruising along Mahantango Mountain, which consists of many landable fields at its base. We will use the 1/1000 risk of falling off as our subjective assessment. At a given decision point, the pilot assesses that his risk consists of:
The risk of falling off- 1/1000
Likelihood of hitting a thermal prior to landing: 1/50
Likelihood of crashing* into field: 1/100
Likelihood of getting hurt, given crashing into field: 1/10
(*- Crashing is defined as an event that causes substantial damage to the glider that would render it unusable and necessitating major structural repair. I am not counting breaking off gear doors, skids, damaging fabric, etc, things that are “routine” land-out damage)
If I got the math right, the risk of getting hurt in this scenario is 1/3,500,000. Since the fields are quite good, the likelihood of a field causing major damage to the glider is quite low and the sort of damage that could happen is less likely to hurt the pilot, such as an unexpected ground-loop.
Sounds great, doesn’t it? However, let’s consider the next scenario, wherein a pilot decides to fly the section of ridge from Wind Gap to the Ski Area, an area considered to be unlandable.
The risk of falling off- 1/1000
Likelihood of hitting a thermal prior to landing: 1/50
Likelihood of crashing if you fall off the ridge- 1
Likelihood of getting hurt, given crashing into the trees: 1/2
This gives us a risk of 1/5,000 of crashing in the trees and 1/10,000 of getting hurt, substantially higher than driving along the Mahantango.
Let’s take an even more extreme example, flying Sharp Mountain between the two gaps at Pottsville:
The risk of falling off- 1/1000
Likelihood of hitting a thermal prior to crashing: 0
Likelihood of crashing if you fall off the ridge- 1
Likelihood of getting hurt, given crashing into the city- 1
Now the odds are 1/1000 of crashing and it’s a given you are going to get hurt because that area is unlikely to generate a thermal and the pilot would be crashing in a residential area, an unsuitable landing area indeed.
The critical observation
Regardless of what the actual risk is relative to the assessment above, it’s the relationship that counts. When committing to fly in an unlandable area, the relative risk goes up once the pilot leaves a landable choice behind him. Given these scenarios, the risk can go up anywhere from 350 to 3,500 times! As ridge pilots, we do not feel this change as the odds of failure are inherently low to begin with. However, the relationship exists and it should weigh heavily in a pilot’s mind, particularly if there is greater uncertainty expressed in their hypothesis going into an unlandable area.
Conclusion
Ridge flying is an amazing and exciting pursuit in soaring and one that incorporates a fascinating and complex structure of decision making. The pilot must constantly assess and reassess how well the ridge is working, the weather ahead and the risk he is taking given his landing options. This risk can be expressed using some probabilistic reasoning, particularly using Bayesian methods. However you assess the risk is up to you, but be aware of the risk you are taking and how it could affect your decision making.
Soaring Economist 