The Woodvale Transatlantic Challenge is a grueling event lasting more than 40 days in which pairs of oarsmen in identical kitset boats battle the elements and their own fears to row from La Gomera in the Canary Islands to Barbados, 2,500 nautical miles away. (See [5] for more details.) No outside assistance is available in the race, in which crews must be entirely self-sufficient for food and water.

The Transatlantic Challenge grew out of a long history of transatlantic rowing beginning with the epic journey from East to West across the Atlantic by Norwegians Harbo and Samuelsen in 1896. The modern version of the race really took off in 1997 when Rob Hamil and Phil Stubbs set a record of 41 days and 3 hours for the crossing from La Gomera to Barbados. Since their effort, the race has been dominated by New Zealanders with Matt Goodman and Steve Westlake winning in 2001. (The Woodvale Transatlantic Challenge was changed in 2005 with the race finishing in English Harbour, Antigua, and admitting different classes of entry. See http://www.woodvale-events.com/)

Since Biggar had a limited budget for the Holiday Shoppe campaign, Philpott suggested that he run an undergraduate student project in the Department of Engineering Science at the University of Auckland. This would entail no cost to Biggar and would provide an interesting student experience, but would also carry no guarantees of a successful outcome. Biggar was enthusiastic, and a project was started in March 2003 by student Ivana Stankovich under the supervision of Philpott and Andrew Mason. In July 2003 it became clear that more manpower was required, and so Geoff Leyland from Stochastic Optimization Ltd. joined the team to construct and complete an implementation of some of the findings from the student project.

Since the student project due date was only weeks before the start of the race, Leyland had to work to a strict deadline. To meet the deadline, much of his work was carried out in parallel with Stankovich; while she was providing proof of concepts in her work, Leyland was developing reliable code for optimization which was then tested in simulations of ocean passages.

The Solution


The big challenge in providing a routing solution for the rowers was to offer something that could be used at sea and that complied with the race rules — no outside strategic or routing assistance is allowed. In ocean yacht racing, the crews have on-board computers, weather faxes, radios and high-tech navigational equipment to allow the computation of routing decisions on the fly using commercially available weather-routing software such as MaxSea [6] or Raymarine's RNS racing package [7].

The brief provided by Biggar was to provide them with an optimal route that required little on-board computation. The rowers did in fact carry electronic equipment such as a GPS and a laptop, but so little power was available on board that none of it could be used effectively for running optimization software. Moreover, the environment was so harsh that relying on computer optimization tools to determine routing decisions was likely to be risky. In the absence of backup, a hardware failure could jeopardize the whole venture. Furthermore, there was so little room in the cabin (enough space to comfortably accommodate one of the two rowers) that it was difficult to spend long periods of time on navigation.

The challenge for the weather routing team was to deliver a solution technique that used primitive technology. To do this they appealed to previous work from the 19th century on determining routes for sailing vessels. Using this approach, the rowers were provided with a set of six maps of parts of the course that they could use each day to determine the most effective direction in which to row. A seventh map of the whole course (shown in Figure 1) was also provided. The maps were printed on A4 paper and laminated, so that they could be drawn on with transparency pens.

Figure 1: Isochrone charts for routing from La Gomera to Barbados. Six detailed maps of parts of the course as shown were also provided to the rowers.

The contours on the map in Figure 1 are known as isochrones, a term first coined by Francis Galton in a paper that he submitted to the Proceedings of the Royal Society of London in 1873. They represent the contours of equal minimum expected time sailed by a vessel that follows an optimal course from their location to a given destination. The contours can be computed using dynamic programming. It is remarkable that Galton devised a methodology (and a machine!) for computing isochrones that predates Bellman's work by 80 years (see accompanying story).

The dynamic programming recursion used to compute the contours differs depending on assumptions on the information available to the rowers. The most convenient assumption is to assume that the random variables defining the weather and sea state are independent from the observations of these quantities on the previous day. We call this the climatology assumption.

In practice, the climatology assumption is bound to be false as the wind speed and direction are likely be correlated on consecutive days. This correlation should be accounted for in devising a route. There are several approaches to incorporating this into a routing model. The simplest model (see [3]) models weather conditions as a Markov process that is fitted to wind speed and direction. This can work well in short-course yacht racing, but is problematic to apply in races of long duration where the weather results from large-scale meteorological effects. In this environment, one might use a stochastic programming approach with branching scenarios as described in [3]. The boundary conditions for such a model are typically defined by climatology, but a statistically and meteorologically sound methodology for defining scenarios and blending these together with climatology remains to be worked out.

The key question facing the team was, Should we attempt to model correlation in the weather? To estimate the amount of correlation in the weather is not easy: it requires a large dataset of historical observations of Atlantic weather conditions that was not available to the team. Even with these data, some meteorological scenario generation technique would be required, typically using an on-board computer. Moreover, it appeared to be impossible to include this feature without requiring a large amount of user input into navigation. For example, even if the route choice was dependent in some sense on a small number of state variables reflecting the history of weather observations, then many more laminated charts would have to be provided to define the optimal policy. The risks of making a mistake here, amplified by the amount of energy the rowers could devote to navigation, dictated that they adopt the climatology assumption.

The climatology was based entirely on past weather data taken from U.K. Hydrographic Office routing charts. These charts are based on observations taken over 20 years by ships crossing the Atlantic, and show what wind direction and strength each ship observed. The chart also shows (using blue arrows) the prevailing ocean current at each point in the ocean. Since ships tend to take the same routes across the ocean, some areas have many thousands of observations, while others have very few. Ships also tend to cross the oceans more at some times of the year than others, but luckily, a chart is published for every month, so any bias is manageable. The chart for October is shown in Figure 2.

Figure 2: October routing chart published by U.K. Hydrographic Office.

One of the circular diagrams (called a wind rose) on this chart is shown blown up in Figure 3. Each wind rose gives the total number of weather observations corresponding to its location, as well as the proportion of observations at each wind strength in each direction. The length of each red or white bar sticking out of the rose gives the proportion of observations in each of five wind ranges: 0-10 knots, 11-16 knots, 17-27 knots, 28-34 knots and above 34 knots. The wind rose shown has been computed from 2,430 observations as shown by the number in the circle. The two numbers in the circle below this figure give the probability (as a percentage) of variable winds and the probability of no wind respectively.

Figure 3: Wind rose.

Given a look-up table defining how fast the rowers could travel, and ocean current and climatological information from the UKHO charts, the calculation of isochrones is relatively straightforward. It was accomplished by "discretizing" the ocean into a fine grid of points, and then solving a backwards dynamic programming recursion. The isochrones were then computed from the "time-to-go" values at each grid point by a contouring program.

Our approach was to compute an optimal policy rather than a fixed optimal route. The policy is conditioned on the particular observation of ocean current, weather and sea state made by the rowers at their current location, and so the expected minimum time to the destination is computed by taking expectations after an optimal heading is chosen for each possible weather realization. This is often called a wait-and-see policy, as opposed to the here-and-now policy that chooses the optimal heading after computing expected speeds. Though familiar to mathematicians, the use of a policy rather than an optimal route was a novel idea for the rowers, and it took some persuading to convince them that this provided them with much greater flexibility.

To compute the isochrones, one needs a method for determining the speed of a rowing boat in a given set of weather and sea-state conditions. For sailing boats this information is often encapsulated into a polar diagram that plots (in polar coordinates) the velocity of the boat against the angle it makes to the wind direction. Polars are typically computed by designers of ocean-racing yachts using velocity prediction software, and are usually restricted to speed in flat water.

The development of an analytical model for rowing speed proved to be very difficult. Expert estimates of rowing efficiency were highly variable (ranging from 25 percent to 65 percent). The effect of waves on vessel speed also proved difficult to estimate. The Holiday Shoppe team spent some hours gathering data on the water to estimate speeds, but this was used only as a guide to verify estimates of velocity obtained from position and meteorological data recorded in the previous race. It was realized early in the data analysis that rowing speed itself was not going to be the sole factor in achieving a fast time, as it was relatively small in comparison to the speed added by ocean currents and a following sea. Using isochrones during the race that accounted realistically for these effects would be a key to a successful campaign.

Using the Isochrone Map


The isochrone map shown in Figure 1 is actually quite simple to understand. Each isochrone can be thought of as a finish line for a much shorter race than the one actually being rowed — getting to the next finish line first means we'll be in the best position to get to the subsequent line, and so on until we finish the race. The hard problem of how to row from La Gomera to Barbados is reduced to a series of simpler problems of how to row over a day, this reduction being possible because we have worked out in advance how long (on average, if we follow the right strategy) it will take to row from each point in the ocean to Barbados. The "daily" race isn't quite as simple as it could be — each day, instead of racing to a point (any real finish line looks like a point from far enough away), one has to race to cross a line.

A first cut at the problem is to work out which point on the line is the closest to the boat's current position. However, because wind, currents and waves mean that the boat's speed is not the same in every direction, the real problem is to work out which point on the line can be reached the quickest. To do this at sea, the crew plots their position on the map and then guesses roughly where the nearest (in time) point on the next isochrone is, and takes a bearing on it. They then row as best they can in that direction, and after five minutes use the GPS to see their actual course and speed over the five minutes. They then change their course a little to the left and repeat the process, and then to the right, and try once again.

In practice, however, doing even one iteration of the above algorithm is difficult to do too often. The transatlantic rowing race is one of the most grueling sporting events in the world. Energy output of the top contestants (and input — how much they are eating) is comparable to cyclists in the Tour de France. However, cyclists in the Tour de France cycle for about 100 hours, and have large support teams. Even though they take turns at rowing in shifts, the contestants in the Transatlantic race row for more like 600 hours each, are alone in the middle of the ocean, and eat dried food. Consequently, in times of high stress, even what seem like simple tasks become unusually difficult. To quote Biggar:

"We tended to use the maps every morning and sometimes in the evening also as we would get updates on our progress relative to the rest of the fleet every 12 hours. ... The expanded scale was very helpful psychologically because a day's progress was a couple of centimetres. It was also helpful that the isochrone lines had been marked out at day intervals corresponding to a 42-day crossing — then the record. So we could quickly tell if we were on track to break the record or not.

"Typically I would update our direction just after the end of a shift. During the shift we would adjust our course and identify what changes it made to our speed. Once our position was marked on the map, then we could use this information to work out the fastest course to the next isochrone line. Sometimes I would call out to Jamie to try a different bearing and tell me what speed he was getting. This worked particularly well when we had the autohelm going because then it was possible to steer on a quite accurate heading.

"Most of the time changing our course was simply a matter of small tweaks. But I think that the method was most useful when we faced sudden adverse weather, as we did on Day 2 and Day 32. In particular on Day 2, with winds coming directly out of the southwest, we could only head either south-southeast or west-northwest (or go on sea anchor).

"There was a strong inclination to head south as you knew that you had to get 'down' to Barbados, but the isochrones clearly showed that it would be no man's land down there. So we tried to hold a westerly route with the result that we stole quite a lead on the rest of the fleet.

"Again on Day 32 we faced a similar situation with the result that we made up 20 miles on the race leaders. They were probably trying to steer to waypoints for their routing and so lacked the flexibility to deal with sudden strong changes in wind direction."

After the race, Kevin Biggar declared that the isochrone map was the "most useful piece of technology on the boat"; it gave them constant guidance as to where to head, how they were doing in the race (it proved an accurate predictor of time and distance to go, as well as a way of estimating leads), and in two cases inspired them to keep rowing when the rest of the fleet was on sea anchor. By adhering to the policy recommended by the map, the Holiday Shoppe team won the race in record time: 40 days, 5 hours and 31 minutes — nearly 22 hours faster than the previous world record.

Back in New Zealand, the weather routing team used the map to analyze the race as it unfolded. We were able to provide information on who was leading the race and estimates of when the boats would finish, and the total distance they would row. These estimates were far more accurate than those published by the race organizers — our estimated distances to go were within 5km from about four days into the race. Estimates of the race finish time were less accurate, as we did not have a good model of how fast the boats would go, but they were still significantly better than other estimates.

The Protest


The Holiday Shoppe team was so successful in their campaign that the second-place finishers, Team CRC, formally protested shortly after the race on the grounds that Holiday Shoppe had gone faster than physically possible. They offered little evidence to support their protest apart from some speculation that Holiday Shoppe may have used their solar panel as a sail, and a report from a naval architect explaining that the energy required to row was proportional to the cube of the boat's speed. In particular, they claimed that in the last two weeks of the race, Holiday Shoppe had inexplicably sped up, and that this could only be the result of cheating. Because we had been following and analyzing the race very closely, we had proof that the claims were untrue. Our analysis of the race was used to argue Holiday Shoppe's innocence when the protest went before the race committee in London. We showed that:

• With respect to allegations of use of a sail, that Holiday Shoppe made their biggest gains into headwinds. They won because they did not go on a sea anchor when CRC did. CRC made their large gains when winds were favorable.
  
  
• With respect to the naval architect's claims that it was very difficult to row as fast as Holiday Shoppe had, we showed that it was in fact CRC that had had the highest peak speed of the two, and that CRC had at one stage had the highest speed advantage over Holiday Shoppe when they were rowing in roughly the same part of the ocean.
  
  
• With respect to the claim that Holiday Shoppe had sped up inexplicably in the last two weeks of the race, we showed that both boats had, on average, slowed down. CRC had slowed more.

The race committee dismissed all claims in CRC's protest, and Holiday Shoppe was finally declared the official winners and world record holders more than two months after they crossed the finish line.

Conclusions


This project was a very successful application of stochastic optimization techniques, and it is worth speculating on why it was so successful. Team Holiday Shoppe won the race and still hold the world record for a double-handed rowing passage from the Canaries to Barbados. Would they have won without the routing chart? In the weather they experienced, probably not — the key stage of the race was in on Day 32 when Holiday Shoppe rowed during a storm in which other teams rested. This decision was made on the basis of the isochrone map. Without this storm, it is likely that Holiday Shoppe would have finished second. But, of course, bad things do occasionally happen, and having a policy that adapted to this was significant in this case.

Why did Biggar and Fitzgerald use the isochrone map so confidently? The weather-routing team spent a lot of time thinking of how to present the results of a reasonably complex calculation in a tangible and usable manner. Once we had decided on how to do this, we explained the use of the map carefully to them so they understood the idea of a policy as compared with a fixed plan. Educating them on this feature was critical. Once this was grasped, the map and its use were very simple. No instruction manual was necessary. Of course the data encapsulated in the isochrones requires a lot of modeling and computation, but this did not need to be provided to the rowers.

Though the weather-routing team performed a limited amount of testing using simulations before the race, there was not time in this project to verify to Biggar and Fitzgerald that the policies using the isochrone maps were as good as we expected. Nevertheless, they used the map confidently at the race start, and went into an early lead. (The main competition, CRC, rowed a similar route and overtook Holiday Shoppe in mid-Atlantic.) This early lead gave them the confidence to believe in the map, and to be prepared to think carefully about routing decisions as the race progressed.

To illustrate the importance of confidence in optimization tools, it is interesting to contrast the Holiday Shoppe experience with a related one in the 1989-90 Whitbread around-the-world yacht race. In that event the skipper of Fisher and Paykel had invested in an on-board weather routing system on a (then) state-of-the-art DEC workstation platform. In the first leg of the race from Portsmouth to Punta del Este the navigation team missed an important weather announcement, and so the routing model recommended a suboptimal course. After arriving second into Punta after Steinlager (who had heard the missing weather report), the skipper of Fisher and Paykel appointed a new navigator and dumped the computer equipment for the remaining five legs of the race.

• Read the accompanying story: Francis Galton, Sailboat Routing and Dynamic Programming

References

1. Galton, F., 1873, "On the employment of meteorological statistics in determining the best course for a ship whose sailing qualities are known," Proceedings of the Royal Society, Vol. 21.
2. Philpott, A.B., 2005, "Stochastic optimization in yacht racing," in "Applications of Stochastic Programming," S. W.Wallace and W.T. Ziemba (ed.), SIAM.
3. Philpott A.B. and Mason, A.J., 2001, "Optimizing yacht routes under uncertainty," in Proceedings of the Chesapeake Bay Yacht Symposium, 2001, pp. 89-98.
4. Stankovich, I., 2003, "Routing for the Trans-Atlantic Rowing Race," Year 4 Project Report, Department of Engineering Science.
5. http://www.woodvale-events.com
6. http://comfr.maxsea.fr/MaxSea/default.aspx
7. http://www.raymarine.com

Andy Philpott is a professor and the head of the Department of Engineering Science at the University of Auckland in Auckland, New Zealand.

Geoff Leyland, previously with Stochastic Optimization Ltd, is currently a director of Incremental Ltd, an Auckland-based software company.

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