Seafaring men for centuries have appreciated the enormous influence which weather has played in their lives, and the outcome of more than one naval campaign has been decided by wind and sea.
We need not retrace history even 20 years to understand the continuing respect which military men pay to the vicissitudes of nature. Thousands of men still living will recall the occasion in June 1944 when, on the beaches of Normandy, they entrusted their lives to no more than one tenuous weather forecast. Many examples can be found of naval operations which depended for their successful accomplishment on a particular set of weather and oceanographic conditions.
Yet, despite the general agreement that exists as to the importance of weather, meteorological and climatological information is often badly used by commanders and staffs in their planning and decision-making functions. There seem to be at least two reasons for this state of affairs. First, no one really knows the precise effect which a given weather condition will produce on a given operation. Most naval personnel will agree, for example, that refueling a destroyer in a 15-foot sea is much more difficult than in a five-foot sea. But how much more difficult— twice as difficult, three times as difficult? In what units is difficulty measured, and might not the apparent difficulty vary from ship to ship and from time to time, depending on the experience of the commanding officer, the crew, their morale, or other intangibles? Second, a weather forecaster usually phrases the weather forecast as a categorical statement—it will rain or will not rain, and that the temperature, wind speed, sea height, and humidity, will each have some stated value—without any further information as to the degree of confidence he has in his forecast.
Let us consider a hypothetical example of an incorrectly phrased forecast: A commander is planning an important operation for the following day, which can be successfully carried but only in dry weather. The staff weather officer provides the commander with a forecast of “no rain” and the commander accordingly proceeds with the operation. But the forecast is wrong; rain it does; and the operation is a failure. The commander ascribes the failure of the operation to what he regards as an incorrect forecast. Moreover, the commander is right; the forecast was incorrect, but not because it led him to the wrong decision. It was incorrect because it failed to convey to him any indication of the uncertainty in the forecast, and it was thus only an incomplete statement of the weather outlook. In this respect, the forecast would have been no better even if it had led him to the correct decision.
This statement requires clarification. When the meteorologist, in an example such as the one above, delivers his forecast, he knows that such forecasts are not always correct. Perhaps he has kept a record of the accuracy of earlier forecasts he has made and therefore has some idea of his “batting average.” He has noticed that on those occasions in the past on which he had forecast no rain for the following day, rain did in fact occur a certain percentage of the time, say 20 per cent. He therefore knows that when he now calls for no rain, there is actually a 20 per cent chance that rain will occur, although he may wish to adjust this percentage up or down depending on his confidence in his present forecast. At any rate, if he now delivers a no rain forecast, without mentioning the possibility of rain, he is implying an undue optimism, and any decision the commander makes is made in partial ignorance. Such a decision can be dangerous and is particularly unfortunate when we realize that the commander could just as easily have acted after considering all the weather information, not merely a portion of it.
Other problems remain. Suppose the weather officer had offered the commander what he knew to be the correct forecast, “20 per cent rain, 80 per cent no rain.” Would the commander have accepted such a forecast, or would he have thought to himself, “Now what kind of a hedged forecast is this supposed to be? Will it rain or won’t it?” Would the commander have altered the forecast in his own mind to a categorical “no rain” forecast in view of the low probability of rain? Or would he have accepted the forecast as it was stated and used it to arrive at the “best” possible decision?
The term “best” demands further definition, but first we must introduce the simple concept of mathematical expectation. Most persons are already familiar with the concept intuitively and need only become acquainted with the nomenclature, as a simple example will show: Two friends, A and B, are walking along together and both simultaneously notice a silver dollar lying on the ground. To settle the question as to who will keep the dollar, they agree to flip the coin and decide accordingly. In this case, we say that A’s expectation, prior to tossing the coin, is 50 cents, which is also B’s expectation. This means that if the episode were repeated many times, A would sometimes win a dollar and sometimes nothing, but his average gain, over a period of time, would be 50 cents on each occasion. Even if the episode occurs only once, as in the present example, it is still possible to say that A’s expectation is 50 cents, even though it is impossible for him to win this amount on any single toss of the coin.
Consider a second example: A customer in a gambling casino plays a game in which he throws a die onto a table, and then receives from the casino an amount of money, in dollars, equal to the amount shown on the die. We can see that over a period of time his average gain per throw, or expectation, will be (1+2-(-3+4+5+6) divided by six, or $3.50. This means that if the casino charges him more than $3.50 each time he plays (as it surely will do), then the game is “stacked” against him. It is evident that expectation is found by multiplying each of the possible payoffs by the probability with which that payoff will occur, and then summing. Thus in the example of the coin, the expectation, .5X$1.00+.5X$0.00 = $.50. A similar procedure was used to find the expectation of $3.50 in the die example, since the probability that any one face of the die will appear is one- sixth.
We have already attempted to show that for purposes of making a tactical decision, a categorical all-or-nothing forecast is inadequate and even dangerous. Let us now consider an example of a forecast properly phrased and properly used to arrive at a dichotomous decision.
A task force commander and his staff are planning an amphibious assault on an enemy island. The decision to commit forces to the attack must be made a day in advance, but weather is the critical factor. A high surf will increase landing craft casualties and may even result in disaster. Moreover, if the attack is attempted and fails, the enemy will be alerted and the element of surprise lost. If the surf is light, however, chances of success are considered excellent; the chances diminish rapidly with increasing surf. If the attack is not made on the following day, tidal conditions will not be acceptable again for another month and in the interval the enemy will have improved his defenses. The commander has two choices, to attack the next day, or to delay a month.
At the commander’s request, a member of the staff, perhaps the meteorologist, has prepared a tabular summary based partly on prior experience with landing craft, estimating the relative military success of the assault in various surf conditions. He has used an arbitrary scale ranging from plus ten for highly successful to minus ten for disastrous. He has also prepared a. probability forecast, using a forecasting system based on appropriate statistical or climatological studies, showing the probability with which he believes any of a number of different surf conditions will occur on the following day. The information is combined into a payoff table; Table I, below.
Table I indicates, for example, that if the surf height is 5-7 feet, a high degree of success (payoff 7) is expected, and that moreover the weather officer considers the probability of such a surf to be 12 per cent.
The meteorologist has no hope of making a forecast a month in advance. He does, however, have some climatological data about the enemy island, giving the relative frequencies with which various surf conditions during the month in question have been observed in past years. With the aid of other staff members, he prepares a second table relating the various surf classes to the expected payoffs, and showing the climatological probability of each class of surf. The payoffs in Table II, below, are somewhat smaller than in the first to reflect the enhanced enemy capability that is expected to exist a month later.
We see from the table, for example, that in a surf of 8-10 feet, an almost indifferent degree of success (payoff 1) is expected, and that such a surf occurs 35 per cent of the time during the month in question.
There now remains only the problem of computing the expectation for an attack on the following day, and for an attack a month later. This is done, as described earlier, by multiplying each payoff by the probability with which it will occur, and summing. For an attack on the following day we obtain from Table I: E1=.06X10 + . 12X7 + .40X3 —.20X2 —. 15 X7 —.07X10 = 0.49. From Table II we obtain the expectation for an attack delayed a month: E2= .16X8 + .20X5 + .35 XI — .15 X4 —.10X9 —.04X10 = 0.73. These two values can be thought of as measures of the relative military desirability of attacking on the following day, and of delaying the attack for a month. We see that in this example, the advantage which the commander feels he might enjoy by an immediate attack, before the enemy has augmented his defenses, is more than offset by the disadvantage of the unusually poor surf conditions expected on the following day.
Table I |
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Surf Height (feet) |
4 or less |
5-7 |
8-10 |
11-13 |
14-16 |
17 or more |
Forecast probability |
.06 |
.12 |
.40 |
.20 |
.15 |
.07 |
Payoff (success) |
10 |
7 |
3 |
-2 |
-7 |
-10 |
Table II |
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Surf Height (feet) |
4 or less |
5-7 |
8-10 |
11-13 |
14-16 |
17 or more |
Climatic probability |
.16 |
.20 |
.35 |
.15 |
.10 |
.04 |
Payoff (success) |
8 |
5 |
1 |
-4 |
-9 |
-10 |
It is imperative to note that there is no guarantee against error in this system. It is quite possible that the commander would wait a month only to find the surf then even worse than on the first occasion. There can never be guarantees against error until and unless weather forecasts are completely accurate. The method we have described, however, is adequate to restrain the commander from following a course of action in which he would be wrong more often than he would be right. Hence his “best” decision is to delay the attack. This decision is best not because it is infallible, but rather because in arriving at it, the commander has considered all the surf possibilities, not merely the one that the forecaster happens to mention. He has attached to each possibility the weight that it deserves by considering the probability of its occurrence. His decision is dichotomous; he attacks or does not attack on the following day. But, in his reasoning, he has made the best possible use of weather information as it can now be generated.
The meteorologist, in this example, could have given the commander a categorical forecast. Noting that the probability of a surf of 8-10 feet is 40 per cent, higher than that of any other single class, he could have simply offered 9 feet as the forecast surf height. Such a forecast has the specious justification that 9 feet is more probable than any other single value. But had the meteorologist done this, without mentioning other possibilities or indicating the degree of uncertainty in the forecast, he would have conveyed the implication that a 9-foot surf was 100 per cent probable. The commander’s implied expectation in such a case would be E= 1.00X3 = 3.00, a much higher value than Ei computed earlier. The commander therefore could have been misled an unjustifiably optimistic estimate of his chances, and the forecaster would have been guilty of presenting distorted and erroneous information. Even if the commander had suspected that a nine-foot surf was not absolutely certain, without information about the probability of other surf conditions, he would have no rational basis for considering them.
With the regard to the payoffs shown in Tables I and II, they are merely estimates of the likely success, ease of operation, damage or loss to the enemy, or some other measure of the military desirability of an attack in each of the possible surf categories. They are at best partly subjective, and at worst completely so, yet this fact does not seriously impair their usefulness. Any operation ultimately will be based on someone’s subjective evaluation of the merit of the operation. Even the desirability of winning a war in the first place is a subjective emotion, since there are no equations describing the disadvantage of being conquered by a foreign power. To the extent that a commander can describe numerically his estimate of the various possibilities in the payoffs, the computed expectations can serve as an over-all estimate of the situation, and hence are meaningful to him.
Completely accurate categorical forecasts —forecasts that are correct 100 per cent of the time—are not now possible, nor is there any prospect of their becoming so in the foreseeable future. Although a categorical forecast may be esthetically satisfying, the fact remains that it is hardly useful and may even be dangerous when a forecast is needed in arriving at a decision. When a meteorologist offers his commander a probability forecast, he is not trying to hedge or “weasel out” of committing himself. He is committing himself. The ability to make an accurate probability forecast can be obtained only by adequate climatic or statistical studies, some of which may require weeks or even months to conduct. No commander should expect a weather officer to produce meaningful probability forecasts until the latter has had an opportunity for developing a system by which to produce such forecasts. Reliable probability forecasts are possible, nonetheless, and it is through their generation and use that naval commanders will realize the full benefit and usefulness of meteorological information.