The Theory That Would Not Die

by Sharon Bertsch McGrayne

  For Richardson, the fascinating feature of Craven’s probability map was that it was all based on initial information before any searching began. Craven had constructed a rule-of-thumb prior, the first component of Bayes’ rule. Richardson was familiar with Koopman’s search theory, but Craven’s multiple-scenario priors and the promise of Bayesian updating looked intriguing. By assuming that their probabilities would fall into bell shapes, Craven made it possible to use slide rules and desktop electromechanical calculators to develop a map of the bomb’s possible locations based on the prior information available to him. Like Laplace, he assigned different probability weights to each scenario.

  Wagner and Richardson went to work in the company’s headquarters in Paoli, Pennsylvania, verifying and refining Craven’s rough calculations. A coworker, Ed P. Loane, constructed a more precise probability distribution for the H-bomb’s location by punching data into paper tape and feeding it over public telephone lines into an electronic computer in the nearby office of the Burroughs Corporation. Turning typewriter characters into graphical displays on a teletype machine was challenging. A probability map might wind up looking like this:

  ###&

  &&&#

  ###

  where # meant a probability between 0 and .05; was a probability between .06 and .10, and so forth. Loane worked for Wagner, Associates full time while he was a part-time graduate student in applied mathematics at the University of Pennsylvania, and he wanted desperately to go to Palomares in Richardson’s place. Meanwhile, Craven gathered data from the Pentagon for Richardson to take to Spain. The young man was amazed to see Craven and other senior officers milling around, opening doors for him.

  Almost daily planning sessions with the military soon convinced the mathematicians that their goal—using Bayes’ rule and updating to find the H-bomb—was not the reason they were hired. Bayes was window dressing. If the H-bomb was not found, the navy wanted to be able to prove statistically that it was not there. “The general thrust seemed to be to come up with a credible certification to the President that the H-bomb could not be found, rather than proceeding with an expectation that it could be found. Indeed, the former purpose,” Wagner concluded, “is the main reason we were brought into the act.”4

  Richardson agreed: “My recollection of my marching orders was to statistically document the search that was being carried out and, in the event that the bomb was not found, to be able to certify to the President and Congress that everything possible was done and that it was done in a scientifically accurate and careful way. So that was pretty much what I was sent out to do. Having read Koopman’s work, and knowing that there was such a thing as optimal search based on Bayesian ideas, I was hoping to do more.”5

  Richardson was not interested in using Bayes as a mathematical excuse for a failed expedition. He wanted to find the bomb. He flew to Spain with Captain Andrews, who had a physics Ph.D. from Yale and, after the Thresher search, had retired from the navy to join Catholic University’s faculty. Andrews knew the Pentagon had big doubts that the navy search team would ever find the bomb. In addition, he had been warned that, if the bomb was not located, the entire world would know that the search team “had failed professionally.” In short, the navy was on the hot seat, and careers were on the line. “The implication was, of course, that if we did not find the weapon, nobody else could,” Andrews recalled later.6

  During the flight, Richardson briefed Andrews on Bayesian search theory. Andrews exclaimed, “Oh, if we’d only had that during the Thresher search.”7 Once a large search area was divided into small cells, Bayes’ rule said that the failure to find something in one cell enhances the probability of finding it in the others. Bayes described in mathematical terms an everyday hunt for a missing sock: an exhaustive but fruitless search of the bedroom and a cursory look in the bath would suggest the sock is more likely to be found in the laundry. Thus, Bayes could provide useful information even if the search was unsuccessful.

  Arriving in Palomares, the men found an impoverished village so small it had no telephone and did not appear on maps or in Spain’s census. Beginning in 3500 BC, mining and smelting for lead and silver had pockmarked the desert area with open shafts and, cursed with less than eight inches of rain annually and a saline water supply, agriculture was limited to winter tomatoes grown for export. The B-52’s aerial explosion and wind had dusted 558 acres of the town and its fields with radioactive plutonium.

  In addition to these problems, the village was under siege from a military camp of 750 Americans, complete with field laundries, bakeries, and a movie theater; an offshore fleet of up to 18 ships at a time; a Soviet trawler snooping in international waters; and scores of international reporters incensed by a blackout on news. Applying Bayes’ rule would not be a textbook exercise in abstractions; it would be a high-wire operation conducted under intense scrutiny.

  For four days, the U.S. and Spanish governments refused to admit that the bomber might have carried nuclear weapons of any kind. News of nuclear bombs and radioactivity leaked out only after an American sergeant shouted to the first reporter on the scene, “Hey, buddy, can you speak Spanish?”

  “Sure.”

  “Well, tell that peasant over there to get out of that field, for God’s sake. I can’t make him understand a damned thing. There’s radioactivity there and we’ve got to keep the people cleared out.”

  A public relations catastrophe was in the making. This was the first accident involving the widespread dispersal of radioactive material and the first that attracted widespread, highly critical scrutiny from the world’s media. Within three days, reporters in Palomares knew that a nuclear bomb was missing, but six weeks passed before the U.S. Department of Defense would confirm it. Censorship by Spain’s dictator, Francisco Franco, kept news of the radioactivity off local radio stations, while broadcasts from Communist Eastern Europe spread the word. Radio Moscow announced that “the bomb is still in the sea, irradiating the water and the fish,” and the Soviet government complained that the United States had broken the nuclear test-ban treaty of 1963. The on-site press corps fumed at being scooped by Radio Moscow, Pentagon-based reporters, and even Stars and Stripes.

  Locals were understandably terrified. Tourism and exports of Spanish fruit and tomatoes collapsed. Demonstrators in Mexico City, Frankfurt, and the Philippines updated a popular song from My Fair Lady, “The bomb in Spain lies mainly in the drain.”8 Adding to the pressure, the Vietnam War was escalating and U.S. military bases around the world were at stake. President Johnson phoned the Department of Defense every day demanding news on the search.

  Arriving in this hotbed of tension, Captain Andrews immediately introduced Richardson to RAdm. William S. Guest, commander of the navy task force looking for the bomb. Guest was celebrated as the first U.S. aircraft carrier pilot to sink an enemy ship during the Second World War. He was notoriously stubborn and, behind his back, people called him Bull Dog. Guest understood airplanes and budgets, but not Bayes. However, he did understand Washington’s message: “You will listen to Dr. Richardson and we will listen to him, so basically . . . for that reason you should pay attention yourself.” Expecting an august authority, Guest had assigned the mathematician a captain’s stateroom and steward. When he met Richardson, who looked even younger than his 26 years, Bull Dog harrumphed, “I didn’t think we were getting a teenager.”

  The first thing Guest told Richardson was tongue-in-cheek—but not really. The mathematician was to prove that the missing bomb was on land because Guest’s job was to look in the sea, and if it was on land, finding it would be someone else’s job. Right off, Richardson declared, “I don’t think I have the ability to do that.”

  Guest commanded 125 swimmers and scuba divers eyeballing the shallow coastline, minesweepers cruising deeper water in heavy surf, 3,000 navy personnel, 25 navy ships, 4 research submersibles, and a host of civilian researchers and contractors. The entire search, called Aircraft Salvops Med, would cost 12 million in 1966
dollars.

  Guest wanted to save money by using the equipment where it was best suited and then returning it as soon as possible. This meant he wanted to search some areas that were actually unlikely sites for the H-bomb.

  Craven’s initial hypotheses were based on prevailing winds, so the early hunt focused on a large rectangular area called Alpha II off the beach of Palomares. Guest ordered his swimmers, divers, and minesweepers to search there over and over again.

  Richardson began work immediately by combing the charts of the search to date. The first weakness he saw was that, although there were tracks of where the ships had gone back and forth, there was no mention of effectiveness. “Just going back and forth wouldn’t do you much good if you couldn’t see the bottom of the ocean,” he said. “And that in fact was the case. Some of their sensors couldn’t penetrate deep water, so they were basically out there running around but not contributing anything to the effectiveness of the search. . . . None of this is criticism. It was just a horrible situation to be in. With the whole world looking at you, you can’t tie boats up in a dock and say they’re useless.” So, inspired by a conversation with Andrews, Richardson coined the term “Search Effectiveness Probability” (SEP).

  Looking at the map of the ocean bottom broken into a grid of little squares, Richardson computed for each of the squares the probability that, if the bomb were there, it would have been found by the amount of search effort applied to that area. “If the Search Effectiveness Probability came in at 95%, you could say to the Admiral, ‘This area has been searched pretty thoroughly, and maybe you want to go somewhere else,’” Richardson said.

  By then he probably knew as much as anyone about the ongoing search. Quarantined from the curious reporters on land, he worked nights in the ship’s accounting office. His luggage, filled with reference books and Reber’s declassified tables for minesweepers, had been lost in Madrid, so he painstakingly recreated some of the tables, overlaying bits of paper to superimpose curves. He had no alternative. Portable computers did not exist, and even IBM mainframes had only 32 kilobytes (not gigabytes or even megabytes) of memory. Armed with his paper cutouts, his slide rule, and an adding machine that could also multiply, he computed the effectiveness of each day’s operations. Each morning he greeted Bull Dog Guest with new probabilities. The admiral enjoyed joking about Richardson’s boyish appearance, but the probabilities unsettled him.

  “I began computing SEPs—the probability you’d have found the bomb if it were there—and a lot of zeros showed up indicating that, even if it had been there, you probably wouldn’t have seen it because your capabilities weren’t up to the task.” At the other end of the SEP scale, a “one” would have signified that the bomb would have been found, had it been there. Richardson was calculating very few ones: “All those zeroes. When Guest saw them—remember this is some young teenager talking to him—the minute he saw zeroes, he was quite outspoken in his questions. ‘Why are you giving me zeroes when we’ve been out there for two weeks?’”

  Guest began using the search effectiveness evaluations as quantitative guides for moving equipment. He wanted to document that his equipment had conducted their investigations thoroughly; he was not interested in using Bayesian updating to find new, more probable places to inspect. Even when a more likely site for the H-bomb appeared, Admiral Guest stuck to his “plan of squares.”

  Years later, Craven complained that “the least informed and knowledgeable was Admiral Guest, the on-scene commanding officer.” The admiral was furious “because he thinks we’re out of our noggin.” Richardson is more forgiving. Guest “had other concerns. Bayes was a little bit high-falutin’. SEP was understandable. But you start getting into Bayesian updating and these funny words like priors and posteriors, admirals tend not to be patient with this stuff.” As a result, evaluating the search’s effectiveness became the focus of the H-bomb search. The idea of using effectiveness data to update the first Bayesian component—Craven’s presearch scenarios—faded into the background.

  Meanwhile, the eyewitness testimony of the veteran fisherman Francisco Simo Orts was gaining credibility fast. The morning of the crash Orts had watched a large parachute pass over his boat and splash down 100 yards away. He called it “half a man, with the insides trailing.” Despite the odd description, his report sounded authentic. Strangely stiff in air, the object sank fast, within 30 seconds, parachute and all. Moreover, Orts said the chute was grayish; air force chutes were orange and white for personnel but gray-white for bombs. Navy personnel had interviewed Orts shortly after the crash but had discounted him because he did not use standard procedures to triangulate the spot. Having fished those waters all his life, he could make a seaman’s eye calculation of familiar mountains and villages along the shore and identify the location.

  Lt. Cdr. J. Brad Mooney, assistant operations officer for deep submersibles, thought Orts might know what he was talking about. Mooney, later promoted to commander and chief of naval research, came from New Hampshire, where lobstermen used similar methods to find their submerged pots. He and Jon Lindberg, a commercial diving consultant, commandeered a jeep, found Orts in a bar, and took him to sea. When Orts twice pointed minesweepers to the same spot in the Mediterranean, Moony believed him.

  Soon Orts’s testimony formed the basis for a high-likelihood hypothesis: with one parachute deployed, the bomb had plunged into a steep, deep-water canyon filled with tailings from an old lead mine. Mooney drew a one-mile radius around Orts’s spot and named it Alpha I.

  As Craven recalled, “We didn’t find the bomb for a long period of time because the place of highest probability is a place we can’t get to. It’s in a narrow crevasse, too deep.” Much of the military’s equipment needed for a deep-sea search was inadequate: navigational charts dated from the early 1900s; detectors were “grossly inaccurate” with errors up to 1,000 yards; and many of the most useful devices were available only from commercial or research sources. They included three small submarines: the mini Alvin from Woods Hole Oceanographic Institution, the Aluminaut from the Reynolds Aluminum Company, and a little yellow sub called the Perry Cub.

  Of Guest’s entire squadron, only the two- or three-man sub Alvin could penetrate the rugged depths of the highest probability site. But Alvin’s battery was fading, and to rejuvenate its power the submersible had to be lifted out of the water and docked for long periods.

  Six weeks after the plane crash, Captain Andrews hitched a ride down the crevasse with the Alvin crew. Peering through its five-inch portholes, they suddenly spied a strange track leading down a slope, “totally different from anything there,” Andrews recalled, “basically like somebody dragging a heavy log or barrel down the slope.” Alvin’s battery was running low again, so they had to abandon the skid marks and surface. Then, for two weeks a large storm circled in, grounded, and thwarted the Alvin.

  During all this time President Johnson was telephoning the Department of Defense every day, only to be told, “We cannot tell you when we’ll recover the bomb. We can only tell you the probability of when we’ll recover it.” Fuming, LBJ replied that he did not want a probability; he wanted a date. Privately, Craven added, “I’m sure his response was profane.”

  Finally, Johnson’s volcanic temper exploded: “I want you to get a series of top-level academics to look at this search plan and tell me what’s wrong with it. I don’t want this probability stuff. I want a plan that tells me exactly when we’re going to find this bomb.”

  Craven convened a committee of Cornell, Harvard, and MIT professors to come to the Pentagon on the morning of March 15, 1966. Number crunchers from Wagner, Associates presented “a mathematical model whose complexity defied understanding by mere mortals.”9 The professors endorsed the Bayesian plan and adjourned for lunch.

  On their return they learned that Alvin’s crew, during its nineteenth dive off Palomares, had just spotted the bomb with its enormous parachute strewn over the seafloor rocks. Alvin had telephoned the surface that the
H-bomb looked “like a ghost down there . . . like a big body in a shroud.”10 It had hit ground in 1,300 feet of water and been dragged by currents down a steep slope almost 2,850 feet deep. It lay within a mile of where Orts had pointed.

  After the bomb was safely retrieved, the fisherman sued for 5 million in salvage prize money. At the government’s request, Richardson again used optimal search theory based on Bayes’ rule to estimate the value of Orts’s testimony: he had saved the government at least a year’s hard work. In 1971 an admiralty court in New York awarded Orts 10,000. The United States had already settled 600,000 on Palomares residents and given the town a 200,000 desalting plant.

  Just as the RAND Corporation’s Bayesian study had warned eight years earlier, SAC’s crash over Palomares diminished the authority of the U.S. Air Force. Military flights over Spain were forbidden, the number of SAC air-alert missions was halved, and responsibility for American air bases in Spain was transferred from SAC to the U.S. Tactical Air Command in Germany. In return for allowing the United States to keep its bases, Franco demanded American help in getting Spain into NATO and the Common Market.

  SAC’s next accident involving nuclear weapons, two years after Palomares, was the last straw for Operation Chrome Dome. The accident occurred when a B-52 loaded with four nuclear bombs crashed onto sea ice outside a U.S. air base at Thule, Greenland. The weapons were destroyed by fire, but, as at Palomares, radioactivity contaminated the area. As a result of the two accidents, the rising cost of keeping SAC’s planes in the air, and the advent of intercontinental ballistic missiles, Secretary of Defense Robert McNamara ended SAC’s airborne alert program in 1968.