Hugh, This looks fun and engaging. I hope to be able to give it good look here in a couple of days. Unfortunately or not, I've had to drop everything to work on this talk for tomorrow. Look forward to seeing you. Dave Letson Hugh Willoughby wrote: Hello David, Thanks for resending. Sorry I flushed the first iteration with the spam. Your rephrasing basically describes what I've done. Here is a bit more detail. It's well established that forecast error increases approximately linearly with lead-time. Thus if we imagine that the coastline is uniformly developed with a fixed coat of say $1M per mile ($621K per km), there is a fixed cost or responding to warnings even at zero lead time equal to the width of the storm [ie twice the radius of gale-force (or hurricane force???) winds ~ 75 nmi]. If we assume a typical 24-h error of 80 nmi, the cost of responding to warnings is given by $1M x (75+ 80t/24) = 75+ 3.333t ($M). I argue that the simplest realistic functional form that describes safeguarding of property is a negative exponential. If with unlimited time to prepare one could prevent 20% of $10B = D0 = $2B in damage, the actual prevented damage might be described by D0 x{ 1 - 2^(-t/T_haf)} = D0{1 - exp[-t ln2/T_haf]}, where a reasonable value for T_haf might be 12 h. Thus, protection of property achieves less effect per hour as lead time becomes longer, i.e. diminishing marginal return. It's sort of a Zeno's paradox of preparation. Half ready in 12 h, three-quarters in 24, seven-eighths in 36, and so on. The reason the curve gets flatter is that enterprises and individuals run out of worthwhile things to so, and (I suppose) would procrastinate if the lead time were very long. Certainly, the curve can have different forms, but the exponential is the simplest that has any connection to experience. If the curve is linear or concave downward (ie constant or increasing marginal effect) , no optimum exists. Of course for long lead times, the cost of responding to warnings actually increases exponentially (in step with track forecast errors) because the hypothesized linear increase in errors is actually an approximation to a slowly growing (e-folding time 7-10 days) exponential. If one considers evacuation costs, as distinct from safeguarding fixed property, the picture changes. For property there is almost always some additional task that will reduce impacts marginally. For example a homeowner who is staying in his house might first put up shutters, then park cars in the lee of the house, then bring in patio furniture, then take down antennas, then.... . By contrast evacuation of people (or mobile property) is a single act. If we consider a threatened population as a whole, after a warning is issued there is some latency while people are taking a decision and assembling supplies. During this time the curve representing value of lives saved is concave downward. Then, during the time when most people are moving, the rate of removal from harm's way is constrained by the capacity of evacuation routes and is approximately linear in functional form. At the end of the evacuation, when the last stragglers are making their way to safety, the curve becomes concave upward approaching the total effect of evacuation asymptotically. Thus, the shape of the curve is like a logistic curve or a hyperbolic tangent. I like to represent it with a polynomial related to the Hermite structure functions that engineers use in finite element models. If one combines evacuation with safeguarding property, assumes that evacuation saves hundreds of lives, and assigns a statistical value of life of several million dollars, the optimum point comes when the evacuation is nearly complete so that the evacuation curve flattens out (i.e.the marginal value of the next hour is decreasing). In this situation safeguarding property plays a secondary role in setting the optimum. This strategy is exactly what NHC tries to do. Post warnings early enough for all but the last stragglers to get to safety. Editorial comment: NHC forecasters are very "death averse." I'm attaching a couple of examples from calculations that I did back when I was first thinking about the problem. Numbers are not what I'd use now, but they illustrate the arguments. One can get some policy insight into the value of accuracy by redoing the calculation different forecast accuracy and evacuation or preparation rates. Do you think that we have a start on something worthwhile? Best regards, Hugh At 09:44 AM 3/22/2005 -0500, you wrote: Hugh, At the FL Hurricane Alliance Workshop three weeks ago, you wrote out a few thoughts about forecast lead time costs and benefits onto a cocktail napkin. Below, I ve tried to express your ideas a bit more formally, as a platform to facilitate discussion. Here are the basic ideas: * Evacuation costs (c) increase with lead time: c >0, c >0; and * Evacuation benefits (b) are avoidable damages to property or life, which also increase with lead time but with diminishing marginal returns: b 0, b <0. Conveniently, this is a constrained optimization problem with a unique and fairly intuitive solution. That problem is to choose the lead time, t, so as to maximize the net benefits of evacuation and preparation. Let b and c be the benefits and costs of evacuation and preparation, e. Here I m interpreting avoided mortality and property losses as expected benefits. (1) maxt b(e) c(e) (2) s.t. e(t) e e If we set this up as a Lagrangian equation and differentiate w.r.t. t, we get the first-order condition for a maximum: lead time should be chosen so that the marginal costs of lead time should just equal the marginal benefits. In other words, the net value of additional lead time to a forecast user (i.e., the demand price) should just equal the value that the additional lead time would contribute by producing evacuations (i.e., the supply price). Geometrically, this solution can be seen in the cocktail napkin drawing either of two ways: (1) by identifying the lead time where the vertical distance is maximized between the evacuation cost and prevented damage functions; or (2) by identifying the lead time where the tangents to the cost and benefits functions are parallel. Nothing earth shattering but it does allow us to stand back and ask a few questions. * First, how sure are we about how the cost and benefit functions look? Might they be steeper or have different curvatures under different circumstances? That might change the problem quite a bit. * Second, are there some important constraints we have left out, e.g., lead time must be adequate to avoid loss of life from storm surge? * Third, do the forecast issuer and local officials necessarily have the same objectives? For example, might some mayors resemble the one portrayed in the movie Jaws, who was more concerned about tourism revenue than public safety? If so, the problem gets more interesting, because the forecaster not only issues the warnings but then also has to use carrots and sticks to motivate the local officials to do the right thing. Have a nice weekend. Dave Letson ---------------------------------------- H.E. Willoughby International Hurricane Research Center Florida International University University Park Campus, MARC 360 Miami, Florida 33199 Phone: 305-348-7096 Fax: 305-348-1761 Hugh.Willoughby@fiu.edu --------------------------------------- dletson11.vcf