2006-10 A Dynamic Decision Model Applied to Hurricane Landfall · A Dynamic Decision Model Applied to Hurricane Landfall EVA REGNIER Defense Resources Management Institute, Naval

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Faculty and Researcher Publications Faculty and Researcher Publications

2006-10

A Dynamic Decision Model Applied to

Hurricane Landfall

Regnier, Eva

Weather and Forecasting, Volume 21, pp. 764 - 780.

http://hdl.handle.net/10945/43581

A Dynamic Decision Model Applied to Hurricane Landfall

EVA REGNIER

Defense Resources Management Institute, Naval Postgraduate School, Monterey, California

PATRICK A. HARR

Department of Meteorology, Naval Postgraduate School, Monterey, California

(Manuscript received 7 April 2005, in final form 24 March 2006)

ABSTRACT

The decision to prepare for an oncoming hurricane is typically framed as a static cost:loss problem, basedon a strike-probability forecast. The value of waiting for updated forecasts is therefore neglected. In thispaper, the problem is reframed as a sequence of interrelated decisions that more accurately represents thesituation faced by a decision maker monitoring an evolving tropical cyclone. A key feature of the decisionmodel is that the decision maker explicitly anticipates and plans for future forecasts whose accuracyimproves as lead time declines. A discrete Markov model of hurricane travel is derived from historicaltropical cyclone tracks and combined with the dynamic decision model to estimate the additional value thatcan be extracted from existing forecasts by anticipating updated forecasts, rather than incurring an irre-versible preparation cost based on the instantaneous strike probability. The value of anticipating forecastsdepends on the specific alternatives and cost profile of each decision maker, but conceptual examples fortargets at Norfolk, Virginia, and Galveston, Texas, yield expected savings ranging up to 8% relative torepeated static decisions. In real-time decision making, forecasts of improving information quality could beused in combination with strike-probability forecasts to evaluate the trade-off between lead time andforecast accuracy, estimate the value of waiting for improving forecasts, and thereby reduce the frequencyof false alarms.

1. Introduction

Over tropical and subtropical latitudes, a tropical cy-clone is one of the most significant weather-relatedthreats to shore- and sea-based locations. Preparationsthat can substantially reduce the impact of tropical cy-clone conditions include evacuation, moving mobile as-sets such as ships and aircraft, and preparing stationaryinfrastructure for flooding and high winds. However,preparations can incur substantial costs. The cost ofcivilian evacuations is usually estimated at approxi-mately $1M per mile of coastline evacuated (White-head 2003), but may be as high as $50M in some areas(Adams and Berri 2004). Baker (2002) summarizesstudies of preparation costs in specific industries, whichrun into tens of millions of dollars. The cost of evacu-

ating U.S. Navy ships from Norfolk, Virginia, duringHurricane Floyd (in 1999) was estimated at $14–$17M,1

and during Hurricane Isabel (in 2003), the direct shipsortie costs were $36M.2 Avoiding unnecessary prepa-rations would therefore be very valuable.

The level of preparation for an oncoming storm de-pends on two factors. One is the availability and accu-racy of forecasts with enough lead time to allow forappropriate preparations. Modern observation, nu-merical models, forecast methods, and coastal infra-structures have significantly reduced the possibility oftragedies such as the Galveston, Texas, hurricane of1900, which caused an estimated 8000 deaths (Jarrell etal. 2001). A second factor is the decision process of thepublic emergency managers, property owners, military

Corresponding author address: Dr. Patrick A. Harr, Dept. ofMeteorology, Naval Postgraduate School, Code MR/Hp,Monterey, CA 93943.E-mail: [email protected]

1 Source: “Navy Meteorologists Recommend how Ships ShouldRespond to Storms,” Daytona Beach News-Journal, 25 June 2001.

2 Estimated cost of $6M, plus $30M in maintenance includingpreparing docked ships to depart. Source: “Navy Costs For Isabelat Least $105.6 Million” Norfolk Virginian-Pilot, 27 September2003.

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commanders, and the public who decide whether,when, and how to prepare. In the face of an approach-ing tropical cyclone, decisions are often made whenthere is still a substantial amount of uncertainty as towhere—and sometimes whether—the storm will makelandfall.

Between 1970 and 1998, 72-h hurricane track forecasterrors declined from a 5-yr average of about 750 km toan average of about 400 km (McAdie and Lawrence2000), and longer-range forecasts have improvedenough to allow for 5-day forecasts that are as accurateas the 3-day forecasts were 15 yr ago. The addition ofstrike probabilities to the National Hurricane Center’s(NHC’s) forecast has given users further useful infor-mation (Jarrell and Brand 1983). Although forecast ac-curacy is improving, there will always be uncertainty atthe lead times required for some types of preparationsuch as mass evacuation and moving ships to sea.Therefore, improving the decision-making process inthe face of uncertainty has the potential to yield sub-stantial value. There is considerable room for extract-ing more value from forecasts without further improve-ments in accuracy by improving decision processes.

Hurricane preparation decisions are usually exam-ined in a static, one-time cost:loss framework, which isa simple decision-analytic approach that has beenwidely used to investigate the value and optimal use ofweather information (for an introduction, see Katz andMurphy 1997, chapter 6). The ultimate impact of atropical cyclone is determined by a series of decisions inwhich weather information and the natural variabilityof tropical cyclones are critical components. To moreaccurately represent the problem of a decision makerfacing an evolving tropical cyclone, the cost:loss sce-nario must be extended to include multiple interrelateddecisions.

This study proposes replacing the static “prepare ordo not prepare” decision-making model with a dynamic“prepare or wait” model that allows for the incorpora-tion of new information in the form of updated fore-casts. Additional value can be extracted by planningdynamically for updated forecasts; however, to makethese decisions optimally, decision makers require in-formation about how the strike probability will changeover time. In this paper, a Markov chain model of hur-ricane travel with a binary weather outcome is devel-oped and its parameters estimated from historicaltracks.3 Combining this storm model with a dynamicdecision model for a stationary target location (which

can be land or sea based), we show that the value ofanticipating improving future forecasts and adjustingearly storm decisions accordingly can reduce the ex-pected total cost associated with a hurricane strike byup to 8%. For some decision makers, the value of an-ticipating improving forecasts is comparable to thevalue of reducing the lead time required for a givenpreparation action by 6–12 h. This added value is inaddition to the value of simply reevaluating a decisionnot to prepare each time a new forecast becomes avail-able.

The savings due to dynamic optimization come froma reduction in false alarms. False alarms may be evenmore harmful than their direct costs indicate because ahigh false alarm rate may reduce a decision maker’swillingness to prepare. Roulston and Smith (2004) haveshown in a theoretical model that the optimal choice ofan action threshold for ordering preparation is sensitiveto a compliance rate that is a function of the false alarmrate in the forecast process. On the other hand, Dowand Cutter (1998) do not observe this “crying wolf”effect in their empirical study, and Baker (2002) reportsthat evacuation rates did not drop after two false alarmsin 1985. The savings due to dynamic optimization comeat the cost of delayed, and therefore more costly, evacu-ations in some cases. The dynamic decision method bal-ances these costs against the benefits to achieve an ex-pected net savings.

The value investigated in this analysis arises from thedecision-making process, not from improving the fore-cast. A model, such as the Markov hurricane model,that allows dynamic optimization will be less skillful atforecasting than existing atmospheric models. In prac-tice, the Markov model should not be utilized as a fore-casting tool. Instead, it could be used to generate mea-sures of information and uncertainty and their expectedevolutions. These measures could serve as an “informa-tion forecast.” Such an information forecast could beused in real time, together with skillful track and prob-ability forecasts for various weather conditions, to ad-just for the value of waiting and to approximate dy-namic optimization.

It is important to note that any decision process thatdelays preparation is only valid if the critical costs areincluded in the analysis. This means that if delaying agiven preparation action would increase risks to life andproperty; the delay must be appropriately balancedwith the risks associated with taking immediate action.Our decision model is conceptual, representing a singletype of preparation action, such as a sortie of ships fromport. It is not intended to indicate that it is ever advis-able to make no preparations in the face of an ap-proaching hurricane.

3 Because the weather outcome is modeled as binary, all spe-cific hazards (wind, storm surge, precipitation, etc.) are encom-passed in a “strike.”

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The meteorology community is increasingly able tomeasure and communicate the uncertainty associatedwith its forecasts. As this study illustrates, the probabili-ties of future events that are conditioned on the currentbest information are not a complete characterization ofthe uncertainty. This has been illustrated previously byMjelde and Dixon (1993), Wilks (1991), and Epsteinand Murphy (1988).

In addition to the quantitative data from observa-tions, atmospheric models, and climatology, meteorolo-gists have the insights gained from experience that af-fect their own assessments of future events. They knowwhen they can anticipate information. For example, At-lantic hurricanes commonly travel west, and then someturn north and eventually east. The timing of the turnslargely determines whether and where the storm willstrike the Atlantic coast. If the storm does not turnbefore some critical time, the storm will strike the coast,and therefore a seemingly large amount of uncertaintyabout landfall will be resolved in the immediate pre-ceding period. Meteorologists and decision makers canexpress this concept explicitly but in a qualitative man-ner. However, current uncertainty estimates are oftenbased on historical error rates, although the NHC is de-veloping a more sophisticated hurricane strike-proba-bility model (Gross et al. 2004). Quantifying uncertaintyin a dynamic framework can help meteorologists com-municate their expertise to decision makers. It can alsohelp to refine hurricane preparation policies to substan-tially reduce the average costs of preparing for storms.

Decision analysis consists of a formal structure foranalyzing problems made difficult by uncertaintythrough modeling the uncertainty that cannot be elimi-nated and its consequences. The decision-analyticframework has been used extensively in the meteoro-logical literature to estimate the value of forecasts(Leigh 1995; Wilks and Hamill 1995; Adams et al. 1995;Brown et al. 1986). A back-of-the envelope calculationof the value of hurricane forecasts in the eastern UnitedStates puts the value at about $7B per year. This as-sumes three landfalls per year (Powell and Aberson2001), with 150 miles of coastline affected per storm,and $17M of damage per mile. The $17M figure is basedon the assumption that NHC warning areas are costeffective up to the limits of the warning area. Assumingthe landfall error is normally distributed along thecoast, and about 460 miles of coastline are warned perstorm (Jarrell and DeMaria 1999), and the cost ofpreparation is $1M per mile (Whitehead 2003), thisyields an estimate of about $17M per mile of damagesif the warning were not issued.

The value of information is measured based on ex-pected cost rather than realized cost because a good

decision can sometimes lead to a poor outcome. Falsealarms are a classic example. It may be a good decisionto evacuate in advance of a hurricane, although thehurricane may never make landfall. Retrospectively,the evacuation was costly and unnecessary (a poor out-come) but the decision minimized expected cost. Theseconcepts are used in a tropical cyclone context by Con-sidine et al. (2004). This framework can also be used togenerate insights about the value of improvements inaccuracy. For example, Considine et al. showed thatincremental increases in accuracy of the forecasts oftropical cyclone track and intensity would produce asignificant increase in forecast value.

In the meteorological literature, decisions have oftenbeen modeled in a one-stage 2 � 2 cost:loss framework(Katz and Murphy 1997, chapter 6). This framework iseasy to understand and to analyze, and may be appli-cable to some real-world decisions. However, it is notnecessarily appropriate for every meteorology-relateddecision. Reverting to decision analysis fundamentalscan greatly expand the range of decisions that can beconsidered in estimating the value of information andfor structuring decision problems. In the context of hur-ricane forecasting, dynamic decision making createsvalue by reducing the frequency of false alarms, andthereby decreasing the expected total cost (cost ofpreparation plus loss of property or life due to hurri-cane strikes at an unprepared target). This additionalvalue will increase the value of existing forecasts.

A dynamic decision includes more than one decisionpoint, such that 1) the alternatives available at laterdecision points, or their consequences, depend on thedecisions at earlier decision points; and 2) relevant in-formation is received between decision points. In a me-teorological context, the information that is receivedbetween decision points is usually a weather forecast.The information can also be the outcome of an earlierdecision.

Several examples in the meteorological literature usedynamic decision models, mostly for the purpose ofestimating the economic value of existing weather fore-casts and hypothetical future forecasts. In all cases, thesequential decisions are interdependent because theconsequences of each decision depend on earlier ac-tions, but generally there are multiple weather eventsthat are not probabilistically related, and there is onlyone forecast per weather event.

The major examples are the fallowing–planting prob-lem, treated in Katz et al. (1987) and Brown et al.(1986), and the fruit–frost problem, treated by Katz etal. (1982) and Katz and Murphy (1990). Stewart et al.(1984) conducted interviews with the decision makers(orchardists) in the fruit–frost problem and found that


they make their decisions on the basis of ongoing moni-toring of temperatures, rather than making a static de-cision once each night. Therefore, each night’s decisionwould be more realistically modeled as a sequence ofdecisions. To our knowledge, Considine et al. (2004) isthe only previous example using a prescriptive decisionmodel for hurricane planning. Their decision rule andvalue-of-forecast estimate depend on just one forecastper storm. However, they repeat their static calculationfor multiple lead times.

Mjelde and Dixon (1993) build anticipation into theirmodel such that the decision maker anticipates thatthere will be a forecast, but does not know what theforecast will be. They observe that the economic valueof longer lead times is overstated when the (unrealistic)assumption is made that the decision maker does notanticipate the forecasts, but acts on them when theyappear. The Mjelde and Dixon framework can accom-modate a stochastic dynamic model of the evolution ofthe climate, but they assume the climate probabilitiesare not conditioned on earlier information.

Wilks (1991) and Epstein and Murphy (1988) are theonly prior examples, to our knowledge, in which twocorrelated forecasts for a single event are used in deci-sion making. Both are based on decisions of the samecost:loss ratio structure as the multiperiod fruit–frostproblem. In this structure, a loss can be incurred atmost once, but the cost of protection may be incurred ineach of many periods. There are multiple weatherevents, and up to two forecasts for each event. Epsteinand Murphy use a conceptual model for forecasts ofadverse weather. Wilks derives his stochastic model forthe probabilistic relationship between the two forecastsand between the forecasts and the outcome of theweather event (in his model, precipitation) on historicalprecipitation and forecast data, and uses dynamic opti-mization to solve for the best decision at each time step.

Katz (1993) also uses a multiperiod version of thesame cost:loss ratio problem, in which the previous pe-riod’s weather serves as a forecast. The forecast is re-lated to the weather outcome via an autocorrelatedMarkov chain model of persistence. Though the persis-tence model would allow for the use of earlier forecastsin decision making, only one forecast for each event isused because the cost of protection is modeled as con-stant, and therefore there is no value to making thedecision before the best—that is, last—forecast be-comes available.

Murphy and Ye (1990) also model successive fore-casts with increasing accuracy for a single event andinvestigate the trade-off between increasing cost andincreasing accuracy as lead time decreases. However,they assume that the decision maker must decide ex

ante the lead time at which he or she will acquire theforecast and make a decision. Only one forecast for theevent is used in decision making.

The current paper is the first to stochastically modeldecision making with respect to a sequence of morethan two forecasts with improving accuracy for a singleevent. The tropical cyclone context is very appropriatefor this approach because in reality, public managers,military commanders, and other decision makers moni-tor the storm’s progress and reevaluate their decisionsevery time a forecast is updated.

2. Markov tropical cyclone model

Optimizing decisions in a dynamic context requires acomplete stochastic model that describes the probabil-ity of every sequence of events and the probability ofevery event, conditional on each possible outcome ofearlier stages.

a. Stochastic modeling

High-resolution meteorological models based onphysical laws may describe the current and future at-mospheric conditions very accurately, but they do notmodel the uncertainty in the evolution of the atmo-sphere. Most methods for adding uncertainty to physi-cal models, including ensemble forecasting, are basedon simulation. In a simulation, a system’s behavior isrecalculated many times, each time with a different setof values for the uncertain parameters. After manyruns, there are many sets of results for the behavior ofthe system, and the assumption is made that the sto-chastic nature of the real system’s behavior, while un-certain, is approximately described by the frequencydistribution of the system’s behaviors in the simulation.A Markov model contains more information about theevolution of the system and can be used as a tool togenerate a simulation, but it can also be used for ana-lytical decision making.

Markov models have been used in decision making,in prediction (especially as a way to model persistenceor other time dependence), and in developing probabil-ity forecasts, especially for precipitation (see referencesin Wilks 1995, p. 296). Wilks (1995, chapter 8) describesMarkov chains and Markov processes that have con-tinuous state variables, and Katz (Murphy and Katz1985, chapter 7) summarizes some applications ofMarkov chains in a meteorological context. Readersinterested in a deeper treatment of decision making onthe basis of Markov models are directed to Puterman(1994).

In a Markov model, the properties of a system at anytime are completely described by its state at that time


and the system’s future evolution is described by ran-dom transitions among these states. The key feature ofa Markov model is the memoryless property, by whichthe evolution of the system (here, the hurricane) de-pends on its history only via its current state. Therefore,according to a Markov model, two hurricanes with thesame atmospheric conditions have the same probabilis-tic future. This is the property that makes Markov mod-els more analytically tractable than complex physicalmodels. In some sense, atmospheric dynamics are natu-rally Markovian. In physical atmospheric models, thefuture evolution of the atmospheric variables is com-pletely determined by their values at any given time.These values may not be known with certainty, andeven the most detailed physical models are necessarilya simplification of the physical state of the atmosphere.However, the temperature, pressure, and other param-eters of every element in the model drive the futureevolution of the conditions of the atmosphere, butwould compose too many state variables for a Markovmodel. The rate of change of atmospheric parametersmay be necessary to model the atmosphere usefully,and can be included as state variables in a Markovmodel. As compared with physical atmospheric modelsand even one based on climatology and persistence(e.g., CLIPER), Markov models are limited in the levelof complexity and detail they can describe, as well as intheir forecast accuracy; however, they contain muchmore probabilistic information.

b. States of the hurricane

We use a discrete-time Markov chain model of thehurricane evolution. In the terminology of Wilks (1995)and Katz (1987), this is a first-order, multistate Markovchain. The state of a hurricane is defined by the locationof its center and its direction of travel. The motion ismodeled according to transitions among these states,which occur at 6-h time steps. Specifically, the hurri-cane location is the 1° latitude � 1° longitude cell con-taining the hurricane center within the region 0°–70°Nand 0°–100°W. For hurricanes in the region 10°–25°Nand 55°–80°W, the direction of travel is also defined,because in this region direction changes are critical. Forexample, whether a hurricane recurves or not will havea profound impact on the potential landfall location.

The direction of travel is categorized as “north” if itsdirection is primarily toward the north, “west” if itsdirection is primarily toward the west, and “other” oth-erwise (see Fig. 1). If a hurricane is stationary, its di-rection is classified as other. The cutoff between north-and west-moving hurricanes is at approximately 0.7 ra-dians west of north, rather than the more natural north-northwest division of �/4. This cutoff was selected to

minimize the occurrence, in the historical database, ofhurricanes that changed direction between west andnorth more than once. The state of the hurricane attime step t is denoted as st � (latt, longt, dirt). Thehurricane dissipates at time T, so a hurricane track canbe denoted s � (s1, s2, . . ., st, st�1, . . ., sT). When hur-ricanes are not in the 10°–25°N and 55°–80°W region,they are not differentiated by direction of travel. There-fore, there are a total of 70 � 100 � 2 � 15 � 25 � 7750possible states.

The Atlantic basin hurricane database (HURDAT)dataset (Jarvinen et al. 1984) contains the tracks forAtlantic hurricanes and tropical storms. AlthoughHURDAT is a best-track dataset in that storm locationand intensity estimates were determined via postanaly-sis for all Atlantic hurricanes from 1851 to 2004, onlythe storms from 1950 through 2002 are used here. Thelocation of a hurricane center as recorded in theHURDAT dataset is latitude and longitude to the near-est tenth of a degree. These values are used to calculatethe hurricane direction of travel. The direction of ahurricane at time t depends on the changes in latitudeand longitude between time t � 1 and time t. The di-rection for the first recorded location of a given hurri-cane is therefore undefined, and is assigned to the samevalue as the direction in the next time step. Each ob-served position and direction of the 538 storms in thedataset are categorized into one of the 7750 states, andeach track is defined as a sequence of observed states.

FIG. 1. Schematic of the method used to assign a hurricane’sdirection of motion in the Markov model.


The historical database contained storm observations inonly 3333 of the 7750 possible states; only these stateswere used in the model, with the addition of j � 0 as thestate that indicates that a storm has terminated. The setof 3334 states is called the state space and is denoted �.

c. Transition probabilities

Hurricane motion is described by transitions amongthe states, which occur at discrete time steps of 6 h. If ahurricane is in a given state st � j at time t, then at timet � 1, it will be in state k with probability qjk � P(st�1

� k | st � j). The value qjk is called a transition prob-ability, and is a function only of j and k, not of time.Speed and direction of travel of a hurricane in state jare reflected in the states to which it can transition (i.e.,states k for which qjk � 0). Although qjk is defined forall j, k pairs, most of these probabilities are equal tozero, as storms rarely move to distant cells in one 6-hperiod. In addition, hurricanes can remain in the samestate for more than one time step; that is, qjj � 0 isallowed. The number of transition probabilities is33342, of which 9445—less than 0.1%—are nonzero.

d. Strike probabilities

A strike is defined to occur at a given geographiclocation (target) if the hurricane center passes throughthe 1° latitude � 1° longitude cell containing the target,or in one of the adjacent cells to the north, south, east,or west of the stationary target, or in the diagonal cellsto the southeast and southwest of the target, but not thecells to the northwest and northeast. This reflects thefact that in general the extent of hurricane force windsis greater on the right-hand side. For a given target, aset � of states are in the strike zone; any storm thatpasses through one of these states strikes the target.The number of states in the strike zone ranges from 7 to21 depending on the location of the target. For targetswhose strike zone is within the region (10°–25°N and55°–80°W), where direction is also a state variable,there are 7 � 3 � 21 cells in the strike zone. For targetswhose strike zone does not overlap with the 10°–25°Nand 55°–80°W region, there are only seven states in thestrike zone.

For each state j ∈ �, the instantaneous strike prob-ability is denoted pj and defined as the probability thata hurricane passing through state j will eventually strikethe target. For a given target, the values of pj are thesolutions to a set of simultaneous equations,

�j∈�, pj � 1 and �j∈�, j∉�, pj � k∈�

qjkpk.

1�

Within the framework of the Markov model, all the

information available at time t is contained in the stateof the hurricane, and therefore the probability that ahurricane in state j will eventually strike, conditional onthe information at time t, is pj. For information to beconsidered good in a given state, which indicates accu-racy is high and uncertainty is low, the state strike prob-ability should be close to zero or close to one.

As reflected in Eq. (1), the value pj depends on thestrike probabilities in the next time step, which in turnreflect strike probabilities in the following time step.However, pj compresses the future probabilities into asingle, scalar value. A value of pj � 0.5 could reflectthat in the next 6 h, all uncertainty will be resolved andeither pk � 1 or pk � 0 � k such that qjk � 0. Alter-nately, it is possible to have pj � 0.5 in a state j fromwhich information will not improve in the next 6 h (pk

� 0.5 � k such that qjk � 0), or something in between.The Markov model completely describes probabilisticevolutions among states, through all the possible termi-nal states of the hurricane. Based on the Markovmodel, the way uncertainty will be resolved can be usedquantitatively in decision making.

As the hurricane evolves through many states, itsinstantaneous strike probability also evolves; ps1

,ps2

, . . . , psT. Sometimes the strike probability will in-

crease (decrease) monotonically. For example, a hurri-cane may form and then move directly toward (awayfrom) the target such that its strike probability is in-creasing (decreasing) throughout its progress. On theother hand, a hurricane may evolve through states withincreasing strike probability, then change course andhead away from the target, so that its strike probabilitydeclines. As will be shown in section 5, this nonmono-tonicity can lead to a high rate of false alarms if thevalue of waiting is neglected.

e. Data fitting and calibration

The transition probabilities were derived from theclimatological data in the HURDAT dataset (Jarv-inen et al. 1984) using hurricane positions at 6-h inter-vals for the 538 tropical cyclones between 1950 and2002. The transition probability between two states jand k is denoted as qjk, and is set equal to the fractionof all hurricanes in the database that passed throughstate j that then moved to state k in the next observa-tion. The probability distribution of hurricane forma-tion across states is denoted as r, where rj is the relativefrequency (fraction) of the historic hurricanes in thedatabase that formed in state j.

Forecast tracks are not defined in this model. How-ever, the probability distribution about the most likelytrack that results from the forward propagation of thestorm state using the Markov transition probabilities is


well calibrated to the NHC strike-probability forecasts.Table 1 compares the maximum strike probability at12-, 24-, 36-, 48-, and 72-h lead times for the NHC fore-casts and for the Markov model. A simulation based onthe model can be used to develop a probability distri-bution of its future locations that implies a most likelyfuture track. An example of such a 72-h simulation isgiven in Fig. 2a and compared with an NHC forecasttrack and strike-probability ellipses (Fig. 2b) for Hur-ricane Isabel. The most likely tracks generated by ourmodel are not necessarily close to the NHC forecasttracks, because as discussed earlier the Markov modelis not highly skillful, as it is designed for its descriptionof uncertainty rather than for forecast accuracy.

3. Modeling tropical cyclone preparations

The real-world problem that we model occurs eachtime an Atlantic tropical cyclone forms. Each tropicalcyclone is treated as an independent event. The prob-lem is viewed from the perspective of a single decisionmaker with assets at a fixed geographical location,which we call the target. The decision maker can makepreparations that will reduce the damage caused by thehurricane if it strikes at the target location. For ex-ample, given enough lead time and a good forecast,ships and aircraft can be moved from the path of thehurricane, homeowners can board up their doors andwindows, and people can evacuate. However, prepara-tion is costly and/or its effectiveness depends on thelead time at which it is initiated. The decision to initiatepreparation must be made on the basis of incompleteinformation, that is, the forecast, which is the best in-formation available at the time.

a. The alternatives

Usually, analysis of the value of forecasts is based onthe assumption that the lead time required to completea preparation is fixed, and/or that there is only onepossible preparation action. However, many decisionmakers have more flexibility than this assumption re-

flects. For example, when a tropical cyclone threatens anaval installation, a set of predetermined disaster pre-paredness actions are implemented. These “conditionsof readiness” are based on the anticipated arrival ofsustained 50-kt winds. To avoid unnecessary prepara-tions as much as possible, decision makers would like towait until the last possible opportunity to initiate apreparation. However, the timing of the last possibleopportunity is not precisely determined. First, the leadtime remaining before a strike, or before conditionsthat will hamper further preparation, is uncertain. De-laying increases the risk that there will not be enoughlead time to complete a preparation. Second, theamount of lead time required to complete a preparationmay be flexible. For example, in 2004, Hurricane Char-ley intensified rapidly immediately before landfall,

TABLE 1. A comparison of maximum strike probabilities at 12-,24-, 36-, 48-, and 72-h lead times for the NHC forecasts (middlecolumn; http://www.nhc.noaa.gov/HAW2/english/forecast/probabilities_printer.shtml) and for the Markov model (rightcolumn).

Forecast leadtime (h)

Max probability (%)

NHC probability method Markov model

72 10–15 1248 20–25 2036 25–35 2824 40–50 4412 75–85 86

FIG. 2. (a) Strike probability for 72-h positions based on theMarkov model of a hurricane in the 1° lat � 1° lon box centeredat 68°N, 25°W. (b) As in (a) but showing the official NHC strike-probability forecast for Hurricane Isabel (information online athttp://www.nhc.noaa.gov).


causing the U.S. Navy to order a sortie of ships fromMayport, Florida, with less lead time than they wouldusually allow. This is evidence that, if necessary, a par-tial evacuation preparation can be completed in ashorter time at greater cost (or lesser effectiveness).Moreover, decision makers can reevaluate their deci-sions every time a forecast is updated, and decide toprepare, abandon previous preparations (if the deci-sions are staged), or delay further.

To model this flexibility, we expand the decisionmaker’s alternatives so that he or she has a sequence ofdecisions at discrete time steps. We model the decisionfor one type of preparation action. Many types ofpreparation available to a single decision maker (e.g.,to sortie ships from port, and evacuate personnel fromthe port) would be modeled separately. Preparation istherefore modeled as binary, as in many cost:loss prob-lems, including in Considine et al. (2004). At each timestep, the chosen action is denoted as a, where a � 1 ifpreparation is chosen, and a � 0 if delay is chosen. Theaction will be a function of the state of the hurricane, st

at time t, defined in the next section, and will thereforebe denoted as. Because of the memoryless property ofthe Markov model, the decision for a hurricane in agiven state will not depend on t, and therefore the sub-script t is suppressed. However, preparation can bemade no more than once per hurricane. The hurricanepreparation decision is now framed as an optimal tim-ing problem—a decision of when, not whether, to pre-pare.

b. Preparation cost profile

The static cost:loss framework is equivalent to as-suming that before a certain point, which we call thecritical lead time �crit, the cost of preparation has aconstant value Ccrit, and after that point, no preparationis possible. Adding flexibility to the model implies thateven after �crit passes, there are still preparation actionsavailable that would reduce the amount of damage sus-tained if a hurricane struck the target. However, it isfair to assume that these actions are more costly and/orless effective than preparation at �crit. For example, re-moving boats from the water, but not from the threat-ened region would reduce damage, but not as much assailing them entirely out of the way of the hurricane.4

The cost of preparation depends on the lead time re-maining at the time the preparation is initiated, which istaken to be immediately following the decision. There-fore, we model cost as a function of the minimum pos-sible remaining time before a hurricane strikes the tar-get, denoted �, which captures the increase in the costof preparation if the action is taken with urgency, evenif the actual lead time turns out to be longer than theminimum.5 Each decision maker has only one cost func-tion for a given preparation action, which depends onthe parameter �crit.

6

Costs and losses are normalized such that the miti-gable portion of the damage caused by a hurricanestriking an unprepared target is L � 1. The value of Lincludes all damage that could be reduced or avoidedby preparation, including loss of life and injuries. Vis-cusi and Aldy (2003) cite estimates of the value of astatistical life (used in analysis of the value of reducingfatality risks) that range from $0.9M to $20.8M in year2000 dollars, which is enough to pay for a considerable,but not unlimited, amount of preparation. The Envi-ronmental Protection Agency used a baseline value of$6.1M (in 1999 dollars) per statistical life in a study ofwater contamination (Stedge 2000). At $6.1M per lifesaved, a forecast leading to an evacuation for theGalveston hurricane of 1900 alone would have beenworth almost $50B (in 1999 dollars). Lumping and bal-ancing economic damages with risks to life and health ismorally and practically difficult, but it is an unavoidableresponsibility of government decision makers, not onlyin emergency management and planning, but in envi-ronmental protection, homeland security, occupationalsafety, and many other public functions.

The preparation cost C(�) is a fraction of the maxi-mum mitigative loss, and is equal to Ccrit at � � �crit, andstrictly increases, to approach L as � declines to zero.This reflects the assumption that there is always a wayto mitigate the effects of a strike at least somewhat, andno preparation that is more costly than hurricane dam-age will be considered.

4 This suggests that there might be flexibility to prepare for ahurricane at less cost if the preparation were undertaken with � ��crit. For example, the U.S. Navy could prepare its ships to sortieat less cost without paying overtime, and perhaps steaming atlower, more fuel-efficient speeds if they decided to order the sor-tie earlier. We assume that the preparation cost at �crit is theminimum preparation cost.

5 Cost of preparation could alternatively be modeled as a func-tion of actual lead time, in which case its value will be uncertainat the time of the decision. That choice would reflect that the costof the preparation action may be due to partially complete or lesseffective protection such that some portion of the mitigative losswould be incurred. This portion of the cost of the protective actionwould be a function of the time until the hurricane strikes.

6 Because �crit reflects the lead time required to complete apreparation action before a storm strikes, it would include thelead time required to implement the action plus any additionalbuffer necessary; for example, hurricane force winds arrive about10 h before the storm’s center (Powell and Aberson 2001).


The shape of the cost function is specific to a decisionmaker’s vulnerability, alternatives, and costs. The alter-natives may vary by context; for example, there may bepreparations that cannot be initiated at night, as a givenpreparation might have different cost curves dependingon the time of day. To conceptually illustrate the valueof dynamic optimization, we examine the performanceof dynamic and static policies using both a linear func-tion and an exponential function, with each increasingfrom C(� � �crit) � Ccrit to C(� � 0) � L. Murphy andYe (1990) use a similar, exponential cost function. Wefurther assume that the cost:loss ratio at the critical leadtime (Ccrit � Ccrit/L) is 0.1 (i.e., that the cost of prepar-ing is 10% of the mitigative portion of the loss). Theactual ratio depends on the decision maker’s context,and each independent preparation action by a singledecision maker has its own cost:loss ratio. The 10%value was selected because Considine et al. (2004) es-timated the cost:loss ratio for the oil rig evacuationdecision at approximately 9%, and 0.1 is approximatelythe ratio implied by the average length of NHC coast-line warnings together with the 24-h cross-track fore-cast errors. Wilks (1991) also used 10% as the minimumcost:loss ratio. To illustrate how the value of dynamicoptimization depends on the cost profile, we vary �crit

(Fig. 3).The decision maker’s objective is to minimize the

expected total cost of any hurricane. The expected totalcost is a function of the hurricane path (i.e., whether itstrikes the target) and of the decisions (aj). The totalcost for a given hurricane is equal to the preparationcost if preparation is ordered, the mitigative loss if there

is no preparation and the hurricane strikes the target, orzero if there is no preparation and no strike.

c. The forecast

The information available to the decision maker ateach decision point t is pj, which is simply a strike prob-ability conditional on information available to thattime, as contained in the state st � j. For the dynamicpolicy, the decision maker also has more informationabout the future evolution of the hurricane, conditionalon each state of future information. This information iscontained in the Markov model. Track forecasts are notparameters in the decision rules presented here, al-though in practice they determine the strike-probabilityforecasts (Crutcher et al. 1982; Gross et al. 2004).

4. Dynamic decision making with the Markovmodel

We combine the Markov model with the dynamicdecision model to show how the static and dynamicframeworks can be used to generate decisions, andcompare the performance of policies under the twoframeworks.

a. The policies

A policy, denoted as �, is a complete description ofthe action that a decision maker will take in any pos-sible scenario. Each scenario corresponds to a state inthe Markov hurricane model. Therefore, a decisionmaker following policy � will take action aj � �( j) ∈ {0,1} from each state j.

Our reference policy uses a strike-probability fore-cast in a static decision framework and is, therefore,denoted �S. The strike probability pj, defined in section2, is a function of the state j and a given stationarytarget location. In the static framework, the decisionrule for each state j is to prepare if the cost of preparingC(�j) is less than the expected loss. The static decisionrule, which defines the policy �S, is therefore

Prepare if and only if C�j� � pjL for �j � �crit.

2�

Although we refer to this policy as “static,” and thedecision rule in Eq. (2) does not depend on any otherdecision points, multiple opportunities exist to preparefor each hurricane by taking advantage of the flexibilityof late preparations. Beginning with �crit, the static de-cision rule is reapplied at each decision point. If thepreparation has not already been undertaken, prepara-tion can be accomplished at a cost determined by the

FIG. 3. Sample cost profiles showing exponential and linearfunctions with 24- and 72-h critical lead times.


minimum remaining lead time according to the functionC(�j). The policy is called static because the decisionrule does not account for future updated forecasts andopportunities to prepare.

There are two reasons that late preparations mayoccur under the static policy. First, hurricanes may formsuch that their lead time is already less than �crit. Sec-ond, a hurricane may have a low strike probability at�crit, and later transition into a state whose strike prob-ability triggers a preparation under the static policy. Ineach case, the preparation is allowed at a cost of C(� �crit).

The performance of the dynamic policy, relative tothe static policy, reflects the value of anticipating moreaccurate future forecasts, over and above the value ofmonitoring updated forecasts and taking appropriateaction. The dynamic policy �D takes advantage of thestochastic hurricane model by planning for the oppor-tunity to take action later, and quantifying the value offuture scenarios that can arise conditional on each pos-sible updated forecast. Each state in the Markov modelis associated with a value, denoted Vj, and its valuedepends on the values for all other states, and throughthis value, on the actions in other states. Therefore, Vj

is a function of the policy �D. Specifically,

• � j ∈ �, Vj � 1, which reflects that if the hurricanereaches this state without a prior preparation, thenthe mitigable loss is incurred;

• � j ∉ �, such that aj � �D( j) � 1, Vj � C(�j), whichreflects the cost of preparation; and

• � j ∉ �, such that aj � �D( j) � 0, Vj � k∈�qjkVk.

For these states, Vj reflects the expected total cost tothe decision maker of a hurricane in that state, includ-ing both the possibility of a strike on an unpreparedtarget and the costs of possible preparations.

The dynamic rule is prepare if and only if C(�j) is lessthan the expected total cost associated with the state ofthe hurricane at the next time step; that is,

prepare if and only if C�j� � k∈�

qjkVk for �j � �crit.

3�

Although a backward induction is usually the solutionmethod for optimizing Markov decision problems, inthis model the optimal action does not depend on thetime step t, measured either from the hurricane’s for-mation or from its terminal state. Therefore, a compu-tationally less demanding policy iteration method wasused, as follows:

Start, let aj � �S( j) � j;step 2, solve the system of simultaneous equations

represented by Vj � k∈�pjkVk � j ∉ �, aj � 0, withthe boundary conditions Vj � 1 � j ∈ �, and Vj �C(�j) � j ∉ �, aj � 1;

step 3, � j ∉ �, let aj � 1 if C(�j) � Vj � k∈�pjkVk

and aj � 0 otherwise; andstep 4, check whether aj has changed for any j ∈ � in

this iteration. If not, the optimal dynamic policy�D(j) � aj, �j ∈ �. Otherwise, repeat steps 2–4.

b. Expected total cost

The expected total costs (Figs. 4a–d) of the static anddynamic policies are computed for targets at Norfolkand Galveston using both the linear and exponentialcost functions. The value of �crit is also varied from 6 to120 h to represent varying degrees of flexibility. Thevalue of a forecast is equal to the difference betweenthe expected total cost using the forecast and the no-skill expected total cost. The value decreases with �crit

because even with perfect information at the time ahurricane forms, the cost of preparation is higher for acost function based on a long critical lead time.

The white area in Fig. 4 represents the reduction inexpected total cost due to the dynamic framing. Thissavings is also expressed as a percentage improvementrelative to the static policy’s performance, and is plot-ted as a solid line. As an additional reference, the ex-pected total costs under perfect information and undera no-skill rule, which is defined as limited to climato-logical information, are also shown. Under perfect in-formation, the decision maker knows whether the hur-ricane will strike the target as soon as the hurricaneforms [i.e., the expected total cost under perfect infor-mation � j∈�rjpjC(�j)]. The no-skill rule compares thestate-specific cost of preparation with the historicalprobability of a hurricane striking the target, not con-ditional upon the state. The expected total cost underno skill is j∈�rj min[C(�s), pL], where p is simply themean of pj over all j. The savings resulting from thedynamic optimization vary from 0% to 6% for Norfolkand 0% to 8% for Galveston, depending on the shapeof the cost function and the value of �crit. In both loca-tions, the largest percentage improvement can be ex-tracted by decision makers whose �crit is in the range of24–48 h. For Norfolk, there is a secondary increase inpercentage improvement between 72 and 96 h. Thiscontributes to a jagged appearance of the percentageimprovement curve, which is partly attributed to thefact that at this time interval from the target cell ofNorfolk, many tropical cyclones that have tracked west-ward across the tropical North Atlantic will either begina turn toward the north (i.e., recurve) and toward Nor-folk or move straight westward and away from Norfolk.Delaying preparation in this interval therefore yields a


substantial improvement in information regarding land-fall. This secondary maximum in the percent improve-ment curve does not appear in the curves for Galveston(Figs. 4c,d) as there is generally no bifurcation point inthe track where a recurve or straight track follows thataffects landfall at Galveston. A jagged appearance isalso caused by the discretization of the model, as dis-cussed further below.

The high-value periods reflect a stage of hurricaneevolution during which information about the relevantevent—landfall at the target—is improving quickly. Fordecision makers with �crit � 24 h, forecasts are alreadyquite accurate at �crit and there are few remaining op-portunities to reevaluate a preparation decision. Deci-sion makers with �crit � 72 h do not have very muchflexibility to respond to improving forecasts, becausetheir costs of preparation become prohibitive beforethe time forecast accuracy is high. By contrast, decisionmakers with �crit in the 24–60-h range can wait an extra

6 or 12 h and gain a large benefit in terms of improvedaccuracy. They can gain substantially from planning fortheir future opportunities to prepare after the nextforecast update. For some decision makers (at Norfolk,with �crit � 96, 102, and 108 h) the value of framing thedecision dynamically exceeds the value of reducing �crit

by 6 h. The additional analysis required to anticipateupdated forecasts is likely to be less expensive thaninvestments to reduce preparation lead time.

These results are dependent on the model specifica-tion, which is relatively simple, though well calibratedwith the NHC strike-probability ellipses. The magni-tude of the results also depends on the estimation of themodel parameters, and in particular on the transitionprobabilities qjk, which were estimated from a finitehistorical database. As the model is formulated, how-ever, the additional value derived from dynamic opti-mization is necessarily nonnegative, and depending onthe cost function, the lead time, and the location of the

FIG. 4. (a) Expected total cost of a hurricane under the Markov model, following each policy, for a target at Norfolk, VA, with anexponential cost profile. (b) As in (a) but using a linear cost profile. (c) Expected total cost of a hurricane under the Markov model,following each policy for a target at Galveston, TX, with an exponential cost profile. (d) As in (c) but using a linear cost profile.


target, could exceed the maximum savings achieved inour numerical examples. The difference between dy-namic and static optimization is of similar magnitude tothe differences found by Wilks (1991) with the sameminimum cost:loss ratio, though he modeled a dif-ferent decision process and a different meteorologicalevent.

Under our cost assumptions, no value is gained inexpectation from using later forecasts for decision mak-ers at Galveston with �crit � 72 h. These decision mak-ers are better off waiting, unprepared, for the hurricanebecause only about 4.5% of hurricanes in the historicaldatabase strike at Galveston. Given that the minimumcost:loss ratio assumed for the preparation action in thisexample (10%) is higher than the overall strike rate, thequality of forecasts at long lead times is too low to makeresponding to them cost effective. In the model, for adecision maker with a critical lead time of 72 h, the costof preparation is 70% as large as a loss by the time leadtime has declined to 24 h. Even with perfect informa-tion, very little flexibility exists for delayed prepara-tions. This result does not imply that in reality decisionmakers at Galveston do not benefit from forecasts,because generally some preparation is available evenwith very short lead times that can mitigate lossessomewhat. The cost function should be interpreted ascorresponding to a single preparation type. For ex-ample, it might be true that the only way to sortie shipswhen the lead time is 24 h is to hire tugboats at emer-gency rates to take ships upriver, reflected in a high costC(� � 24), but there may be other preparatory actionsavailable at 24-h lead times that can reduce loss sub-stantially (i.e., for many decision makers there are atleast some actions whose cost function is pushed to theright).

In some cases (although not in the results shownhere), the static decision rule can produce counterin-tuitive results when the static performs worse than theno-skill policy. The reason is that repeatedly reevaluat-ing a decision to take an irreversible action in a staticframework can be worse than making a decision andsticking with it regardless of future information. As ahurricane evolves, its strike probability can both in-crease and decrease, and the changes will not necessar-ily be monotonic. If the trigger for irreversible andcostly preparation is set at pj � C(�j)/L, then the prepa-ration is likely to be undertaken when the hurricanestrike probability is higher than it is through most of itstrack. By repeatedly applying this decision rule with theaction trigger set at a point that is applicable for a one-time decision, the decision maker will tend to preparetoo often. If the rule is reapplied, a tight trigger that is

optimal for a one-time decision will lead to overprepa-ration.

The estimates of the expected total cost of eachpolicy and of the percent improvement of the dynamicapproach are functions of the storm model. Like anymodel, its formulation is an inexact representation ofthe real system, and the parameter estimates are de-pendent on the dataset used to fit the model. To ex-plore the impact of sampling variability on our results,we run a bootstrapping process. For each of 100 itera-tions, we generate a sample from among the 538 stormsin the dataset. The sample size is 200, which representsa balance between preserving sampling variability byhaving a small sample size relative to the entire data-base, while keeping the sample large enough that thedataset is not too sparse to generate useful transitionprobabilities.

At each iteration, the parameters qjk, pj, and rj arecalculated as described in section 2, based on the 200storms in the sample. Once the parameters are fitted,we reproduce the thick line in Fig. 4a. The static anddynamic policies are applied for each �crit, from 6 to 120h, using Ccrit � 0.1, normalized to L � 1 as in sections3 and 4a, using an exponential cost function, for a targetat Norfolk. Then the percentage savings, or reductionin expected total cost of the dynamic policy relative tothe static policy, is calculated.

The mean, median, and 5th and 95th percentiles ofthe savings are shown in Fig. 5a as a function of �crit.The overall shape of the curve is very similar to thecurve calculated using all the storm tracks in the pa-rameter calculations, shown as the thick line in Fig. 4a.The greatest savings are 7.2% for the mean in Fig. 5aand 6.2% for the results in Fig. 4a. In Fig. 5a, the largestsavings are for decision makers with �crit in the range of36–48 h, whereas in Fig. 4a, the peak savings are fordecision makers with �crit � 30 h. A difference betweenthe two figures is that the dynamic savings drop tonearly zero (0.02%) in Fig. 4a, but in Fig. 5a we see thatalthough the savings drop off, they are still 2%–3% for�crit � 120 h.

A second noticeable difference between the twocurves is that in Fig. 4a, the dynamic savings curve isquite jagged. In particular, there is a major dropoff insavings for �crit � 72 h. The jaggedness is a simple ar-tifact of the discretization of the model: each �crit

changes the cost curve, and changes the preparationcost and optimal action in many states under eachpolicy. The jaggedness is averaged away in Fig. 5a, asthe mean, median, and percentiles are all taken for eachvalue of �crit. Figure 5b shows the percentage improve-ment for each �crit for 10 of the samples, which retainthe jaggedness of the thick line in Fig. 4a.


c. Simulation

The expected total cost results reflect a balancing oftwo effects. On the positive side, the dynamic policyprevents false alarms when preparation is delayed andupdated forecasts show that preparation is not neces-sary. The negative effect of the dynamic policy ariseswhen updated forecasts show that a strike is more likelythan it appeared earlier. This leads to either a delayed,

and therefore more costly, preparation, or a greater riskof a strike at an unprepared target.

To examine the contributions of these positive andnegative outcomes to the expected total cost, the staticand dynamic policies are also evaluated on the basis ofa simulation. Ten thousand hurricane tracks were gen-erated by Monte Carlo simulation using the historicaldistribution of the location of hurricane formations andthe Markov transition probabilities. Of the simulatedhurricanes, 8.3% strike at Norfolk, and 4.5% atGalveston, as compared with 10.0% and 4.6% of his-torical hurricanes, respectively. The expected total costfor the static and dynamic policies for Norfolk using anexponential cost function (Fig. 6) is broken down by thetype of cost: necessary preparations (for hurricanes thateventually strike the target), false alarms (preparationsfor hurricanes that do not strike), and unpreparedstrikes.

The frequency of outcomes for each policy (Fig. 7) isexamined for the simulated hurricanes. The number offalse alarms is about 1500 (about 15% of all hurricanes)under the dynamic policy and about 2000 (about 20%of hurricanes) under the static policy; that is, the dy-namic framing averts about a quarter of all false alarms.The number of false alarms drops off for short criticallead times (less than 24 h) because the relevant fore-casts are more accurate, and for long critical lead timesbecause more often no preparation is optimal undereither policy.

This savings is partly offset by a slightly greater num-ber of unprepared strikes and delayed—and thereforemore expensive—preparations under the dynamicpolicy. Although the expected total cost is lower under

FIG. 5. (a) Percentage improvement of the dynamic model overthe static model for 100 iterations of the Markov model in whichthe model parameters for each iteration are based on a subsampleof 200 tropical cyclones from the dataset, chosen at random.Mean, median, and 5th and 95th percentiles of the 100 iterationsare shown for each critical lead time. The results are for a targetat Norfolk, VA, using an exponential cost profile, with criticallead times ranging from 6 to 120 h. (b) As in (a) butt showing thepercent improvement in a subset of 10 of the 100 iterations usedto construct (a).

FIG. 6. Average total cost breakdown by hurricane based on asimulated set of hurricane tracks, for a target at Norfolk, VA,using an exponential cost profile.


the dynamic policy, unprotected strikes make up alarger portion of the expected total cost, as the savingsin reduced false alarms are partially offset by an in-crease in delayed preparations and unprepared strikes.Unprepared strikes make a larger contribution to theexpected total cost (Fig. 6) than their numbers (Fig. 7)indicate because each unprepared strike costs morethan a preparation. The number of storms that strikeunprepared targets (the dashed lines in Fig. 7) is slightlyhigher under the dynamic policy (a maximum 1.23%higher, and typically K 1% higher, expressed as a per-centage of storms). In addition, the dotted line showsthe number of delayed preparations. For these hurri-canes, both policies call for a preparation, but the dy-namic policy delays the preparation, usually incurring ahigher cost (sometimes the cost is not higher becausethe minimum possible lead time does not increase).

d. Discussion

Although it is quite simple, the static decision rule isnot an unrealistic straw man. First, it is consistent withprescriptive decision modeling in the literature. Consi-dine et al. (2004) use a static decision rule in their pre-scriptive model of oil rig evacuation and shutdown de-cisions based on hurricane forecasts. To the extent thattheir decision making is quantitatively based on fore-casts, decision makers are likely to be following a rulesimilar to the static rule, and in fact this is what issuggested by Jarrell and Brand (1983). The most de-tailed information officially available is the NHC strike-probability forecast, which would support a static deci-

sion rule, but it would not support a dynamic decisionprocess. Some decision makers may intuitively adjustfor the fact that they anticipate a significant reductionin uncertainty, but they would have to be very familiarwith tropical cyclones to be able to do this effectively.

5. Real-time decision making

The previous section indicated that there is value inplanning for future forecasts using a dynamic policy,over and above the value of monitoring and respondingto updated forecasts using the static decision rule. How-ever, dynamic optimization could not be widely imple-mented in real time, partly because the decision modelis specific to the decision maker’s cost profile. It is notas general as a strike-probability forecast, which appliesto every decision maker with assets at a given location.A second reason this process would be difficult toimplement in practice is that the stochastic model re-quired for dynamic optimization is not designed forforecasting, and would not have nearly the accuracyprovided by current numerical weather prediction mod-els. Ideally, it would be possible to create a highly de-tailed atmospheric model that was fully stochastic andtherefore supported dynamic optimization and at thesame time gave highly accurate predictions with longlead times. In a single model, a trade-off must be madebetween atmospheric detail and stochastic information.

How then can the value of planning for future fore-casts be extracted in real-time hurricane preparationdecisions? One approach is to develop an informationforecast that quantifies how information about a rel-evant weather event can be expected to improve in thefuture or, equivalently, how uncertainty will be re-solved. The information forecast could take the form ofa time profile of information quality or of a measure ofuncertainty over future forecast updates.

A real-time information forecast could be used incombination with track and strike-probability forecastsand a decision maker’s specific alternatives and costprofile to develop a decision rule that approximatesdynamic decision making. An individual decision mak-er’s choices would be a function of the following:

• The available alternatives, at each lead time, andtheir costs;

• the remaining lead time for a given hurricane;• the strike probability and, ideally, the probability of

each intensity or wind speed at the target location;and

• the anticipated information improvement that is rep-resented in the information forecast.

In general, the necessity of costly preparation in-

FIG. 7. Frequency of each outcome for 10 000 simulated hurri-canes using both static and dynamic policies. Hurricanes that arenot shown either had a necessary preparation at the same time forboth policies, or did not strike the target. The target is at Norfolk,VA, with an exponential cost profile.


creases as lead time declines, and increases with strikeprobability for the target location. The desirability ofimmediate preparation also decreases when informa-tion quality is expected to improve in the near future.For example, it might be optimal for a given decisionmaker to prepare with a 72-h lead time when the strikeprobability is 50% for low anticipated information, butoptimal to delay in the same situation, if an informationforecast indicates high anticipated information. De-pending on the form of the forecast, a decision makerwould use it together with the other relevant factors tobalance the value of waiting for improving forecastswith the value of undertaking preparation at a long leadtime.

As in the design of any forecast, the goal is to take alarge amount of information and distill it into a formthat is accessible and understandable to users and si-multaneously make it as valuable to them as possible.These information forecasts would represent a reduc-tion of the information from a complete stochasticmodel, but they would be more informative than a sca-lar instantaneous strike probability.

In designing an information forecast, several trade-offs must be considered. One trade-off is its degree ofreliance on historical information, such as track errors,versus measurable characteristics of individual hurri-canes including speed, intensity, location, and even con-sistency—all of which are related to forecast error. Ahurricane-specific measure could even quantify some ofthe information available in consensus forecasting (Go-erss 2000) and in the systematic approach introduced byCarr et al. (2001). For example, the measure could de-pend on the level of certainty as reflected in the agree-ment among tracks resulting from multiple atmosphericmodels.

A second dimension in the design of an informationforecast is its level of specificity to a decision maker’scontext. At one extreme, an information forecast couldbe as general as an accuracy profile applicable to allhurricanes, which is no more informative than plottingthe average track error as a function of lead time. Fora given decision maker’s cost profile, this could be usedto achieve a rough understanding of the trade-off be-tween lead time and accuracy. An information forecastthat was designed to take different values for differenttarget locations would have the potential to be morevaluable in approximating dynamic decisions. It is natu-ral to think about information quality as dependent onthe geographic location of interest. For example, accu-rate information about landfall at Caribbean locationsis available earlier than accurate information aboutlandfall along the Gulf coast.

An information forecast could even be designed as a

function of a specific decision’s objective. For example,it could reflect the probability of the best decision in agiven context changing in the next 6 or 12 h. An infor-mation forecast specific to a given decision maker couldinclude economic information by quantifying the valueof waiting. There is a trade-off between the portabilityof a low-specificity forecast and the potential value thatcan be extracted by each decision maker. Users with agood understanding of their cost profile and a lot offlexibility in the period during which forecast informa-tion improves rapidly would benefit from tailored in-formation forecasts and from tailored decision rules.

The NHC method for generating strike-probabilityellipses is an example of a compromise in these dimen-sions. The hurricane track is specific to the hurricane,but the probability distribution around each track pointis based on purely historical parameters (Crutcher et al.1982; Sheets 1985). It is not specific to the decisionmaker’s cost profile, but it is specific to each targetlocation.

6. Conclusions

The current paper models decision making with re-spect to a sequence of up to 20 interrelated forecasts.We have developed and integrated a climatology-basedMarkov storm model with a dynamic decision model,and estimated the value of dynamic decision making.This framework allows for the explicit anticipation ofimproving, updated forecasts.

The results indicate that a decision maker who hasthe flexibility to wait for updated hurricane forecastscan extract a substantial value from adopting a dynamicapproach. For some decision makers, the value of fram-ing the decision dynamically exceeds the value of re-ducing �crit by 6 h. Improving the decision process tocapture this value is likely to be less expensive thaninvestments to reduce preparation lead time.

The frequency and predictability of storms in thewestern North Pacific suggest that a dynamic approachto anticipating improving forecast accuracy would beeven more valuable for typhoons. The value of the dy-namic framework depends on the decision maker’s lo-cation, preparation alternatives, and cost profile. Weestimate the added value for a multiperiod cost:lossframework, with a single preparation action and binaryweather outcomes. The approach can be expanded toinclude multiple weather conditions, such as varyingwind speeds, as well as staged preparation actions.

The insights gained in this work could be utilized inan operational setting by elaborating upon the alterna-tives and cost profiles of individual decision makers,such as fleet commanders who must decide whether to


sortie ships, and expanding the state space of theMarkov model appropriately, for example by includingintensity.

Another way to adapt this approach to real-time de-cision making is to develop forecasts of improving in-formation quality that could be used in combinationwith strike-probability forecasts to evaluate the trade-off between lead time and forecast accuracy, estimatethe value of waiting for improving forecasts, andthereby reduce false alarms. An information forecastwould complement the increasingly accurate track fore-casts and the new NHC strike-probability product thatwill include multiple weather conditions (Gross et al.2004).

Acknowledgments. This research has been sponsoredin part by the Office of Naval Research, Marine Me-teorology Program. The authors acknowledge valuablecomments from Prof. R. Elsberry, Prof. C. Wash, Prof.K. Wall, and the anonymous reviewers.

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2006-10 A Dynamic Decision Model Applied to Hurricane Landfall · A Dynamic Decision Model Applied to Hurricane Landfall EVA REGNIER Defense Resources Management Institute, Naval