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which can be written:

where:

• A_a and A_f are the atmospheric and model variability respectively around the climate

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• cov refers to the covariance. 

Hence the level of forecast accuracy is determined not only by the predictive skill, as reflected in the covariance term, but also by the general variability of the atmosphere, expressed by A_a, and by how well the model simulates this, expressed by A_f.

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Imagine a location where, on average, it rains 3 days out of 10.  Two enterprisesusers, X and Y, each lose €€100 €100 if rain occurs and they have not taken protective action.  X has to invest €€20 €20 for protection, whereas Y has to pay €€60€60.

Thanks to his low protection cost, X protects every day, which costs on average €€20 €20 per day over a longer period.  Y, on the other hand, chooses never to protect due to the high cost, and suffers an average loss of €€30 €30 per day over an average 10-day period, owing to the three rain events (see Fig12.A.17).

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Fig12.A.17: The triangle defined by the expected daily expenses for different costs (c), when the loss (L) is €100€100.  End-users who always protect increase their expenses (yellow), end-users who never protect lose on average €30 per day.  Even if perfect forecasts were supplied, protection costs could not be avoided (blue line). The triangle defines the area within which weather forecasts can reduce the expected expenses.  Note the baseline is not a lack of expenses but the cost of the protection necessary, if perfect knowledge about the future weather is available, in X’’s case €€6 and in Y’’s €€18 .  For user X this is €6 per day; for user Y this is €18 per day.

The Benefit of a Local Weather Service

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Relying on these forecasts over a typical 10-day period, both X and Y protect three times and are caught out unprotected only once.  X is able to lower his loss from €€20 €20 to €€16€16, and Y from €€30 €30 to €€28 €28 (see Fig12.A.18).

Fig12.A.18: The same as Fig12.A.17, but with the expected expenses for end-users served by forecast service A. The red area indicates the added benefits for X and Y from basing their decisions on deterministic weather forecasts from service A.

Note that end-users with very low or very high protection costs do not benefit from A’’s forecast serviceAgency A.

Effect of Introducing further Weather Services

Two new weather agencies, B and C, start to provide forecasts to Users X and Y.  The newcomers newcomer Agencies B and C have forecast performances in terms of H, F, M and Z:

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Agency B heavily under-forecasts rain and agency Agency C heavily over-forecasts.  Both give a distorted image of atmospheric behaviour - but what might seem ““bad”” "bad" is actually ““good””.By following B’’s forecasts, which "good". (See Fig12.A.19).

• User Y has high protection costs.  Agency B heavily under-

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• forecasted rain but User Y reduces his expenses from

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• €28 to

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• €26.
• User X has low protection costs. Agency C

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• heavily over-

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• forecasted rain but User X reduces his expenses from

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• €16 to €12.

Fig12.A.19: The cost-loss diagram with the expected expenses according to forecasts from agencies B and C for different end-users, defined by their cost-loss ratios. Weather service A is able to provide only a section of the potential end-users, the ones with C/L-ratios between 33 and 50%, with more useful forecasts than B and C. The green and yellow areas indicate where X and Y benefit from the forecasts from agencies Agencies B and C respectively.

Agency B has also managed to provide a useful weather service to those with very low protection costs , (C) to those with very high protection costs.  In general, any end-user with protection costs <€€33 benefits from C’’s the services of Agency C, any end-user with protection costs >€€50 benefits from B’’s the services . Only end-of Agency B.  Only users with costs between €€33 €33 and €€50 €50 benefit from A’’s the services of Agency A more than they do from B’’s Agencies B and C’’sC.

There seems to be only two ways in which weather service Agency A can compete with Agencies B and C:

• It can improve the deterministic forecast skill –– this would involve NWP model development, which takes time and is costly.
• It can ““tweak”” "tweak" the forecasts in the same way as Agencies B and C, thus violating its policy of well tuned forecasts.

There is, however, a third way, which will enable weather service Agency A to quickly outperform Agencies B and C with no extra cost and without compromising its well tuned forecasts policy.

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Local forecast office A in its competitive battle with B and C starts to make use of this insight. It offers a surprising change of routine service: it issues a categorical rain or no-rain forecast only when the forecast is absolutely certain. If not, a ““don"don't know”” forecast is issued. If such a ““don"don't’ know”” forecast is issued about four times during a typical ten-day period, the contingency table might look like this (assuming ““don"don't’ know”” equates to ““50"50-50”” 50" or 50%):

This does not look very impressive, rather the opposite, but, paradoxically, both Users X and Y benefit highly from this special service.  This is because they are now free to interpret the forecasts in their own way. (see Fig12.A.20).

• User X,

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• has low protection costs

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• and can afford to interpret the

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• "don't know”” forecast as if it could rain and therefore

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• decides to take protective action.  By doing so, User X drastically lowers his costs to

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• €10 per day

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• .  This is €20 cheaper than following

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• the forecasts of Agency C.
• User Y,

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• has expensive protection

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• costs and will prefer to interpret

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• "don't know”” forecast as if there will be no rain and decides not to

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• take protective action.  By doing so, User Y lowers his costs to

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• €26 per day

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• .  This is similar to following the forecasts of Agency B.

Fig12.A.20: The expected daily expenses when the end-users are free to interpret the ““don"don't’ know”” forecast either as ““rain””"rain", if they have a low c/L ratio, or as ““no rain””"no rain", if their c/L ratio is high.

So what might appear as ““cowardly”” "cowardly" forecasts prove to be more valuable for the end-users! If  If forecasters are uncertain, they should say so and thereby .  In this way forecasters can gain respect and authority in the longer term.

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However, service A can go further and quantify how uncertain the rain is. This is best done by expressing the uncertainty of rain in probabilistic terms.  If ““don"don't know”” is equal to 50% then 60% and 80% indicate less uncertainty, 40% and 20 % larger uncertainty. Over a 10-day period the contingency table might, on average, look like this, where the four cases of uncertain forecasts have been grouped according to the degree of uncertainty or certainty:

Note: A ““do "do not know”” know" forecast does not necessarily mean ““50"50-50””50".  It could mean the climatological probability.  In fact, unless the climatological rain frequency is indeed 50% a ““50"50-50”” 50" statement actually provides the non-trivial information that the risk is higher or lower than normal.

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The use of probabilities allows other end-users, with protection costs different from X’’s and Y’’sfrom user X or user Y, to benefit from A’’s forecast service A.  They should take protective action if the forecast probability exceeds their cost/loss ratio (P > c / L).  Assuming possible losses of €€100€100, someone with a protection cost of €€30 €30 should take action when the risk >30% probability, someone with costs of €€75, should take action when the risk >75% probability (see Fig12.A.21).

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Fig12.A.21: The same figures but with the expected expenses indicated for cases where different end-users take action after receiving probability forecasts. The general performance (diagonal thick blue line) is now closer to the performance for perfect forecasts.

User X lowers his expenses to €€10 €10 and User Y lowers his expenses to €€24€24.

Towards more Useful Weather Forecasts

What looks ““bad”” "bad" has indeed been ““good””"good".  Using vague phrasing or expressing probabilities instead of giving a clear forecast is often regarded by the public as a sign of professional incompetence. 

““Unfortunately"Unfortunately, a segment of the public tends to look upon probability forecasting as a means of escape for the forecaster”” forecaster" (Lorenz, 1970).

Instead, it has been shown that what looks like ““cowardly”” "cowardly" forecast practice is, in reality, more beneficial to the public and end-users than perceived ““brave”” "brave" forecast practice.

““What the critics of probability forecasting fail to recognize or else are reluctant to acknowledge is that a forecaster is paid not for exhibiting his skill but for providing information to the public,

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