There is considerable interest in detecting a long-term trend in hurricane intensity possibly related to large-scale ocean warming. This effort is complicated by the paucity of wind speed measurements for hurricanes occurring in the early part of the observational record. Here, results are presented regarding the maximum observed wind speed in a sparsely randomly sampled hurricane based on a model of the evolution of wind speed over the lifetime of a hurricane.
There is considerable interest in detecting and understanding historical variations in the intensity of hurricanes as measured, for example, by maximum wind speed. This interest stems in part from a possible connection between hurricane intensity and large-scale climate change (Emanuel 2005). A potentially serious problem in identifying such variation is that only a few chance wind speed measurements may be available for hurricanes in the earliest part of the observational record (Landsea et al. 2004a). The purpose of this note is to describe some general statistical results concerning the maximum observed wind speed in a hurricane that is sparsely observed at random times during its lifetime. These general results are then specialized to a model of the evolution of wind speed within a hurricane based on Emanuel (2000).
2. The basic result
In this section, the basic statistical result is outlined. This result is specialized to a particular case in section 3.
Consider a hurricane with lifetime (0, T). Let υ(t) be the maximum wind speed of this hurricane at time t. Suppose that this hurricane is observed at random times t1, t2, … , tn during its lifetime and let the random variables W1, W2, … , Wn be the maximum wind speeds observed at these times. To begin with, assume that the error in these observations is negligible, so that Wj = υ(tj). This assumption is relaxed later.
The distribution function of a randomly observed wind speed W is given by
where T(w) is the total time during which wind speed exceeds w. The corresponding probability density function (pdf) is given by
The support of this pdf—that is, the values of υ over which f (w) is positive—has an upper bound at the maximum value υmax of υ(t) over the interval (0, T).
Let W(1) < W(2) < … < W(n) be the observed wind speeds ordered from smallest to largest, so that W(n) is the maximum observed wind speed. It is a standard result that the pdf of W(n) is
(David and Nagaraja 2003). Clearly, the support of g(w) also has an upper bound at υmax. Beyond that, the behavior of g(w) depends on F(w), which depends in turn on υ(t). For large n, the distribution of W(n) will converge to the Weibull extreme value distribution, which is the only extreme value distribution with finite support. However, the situation of interest here is when n is small.
This basic model can be extended to allow for error in observed wind speed. For example, suppose now that observed wind speed Wj is given by the sum
of true wind speed at time tj and a normal observation error with mean 0 and variance σ2. Under this model, the distribution function of a randomly observed wind speed is given by the convolution
where ϕ is the standard normal pdf. Assuming that the errors in the observations are independent, the pdf ge(w) of W(n) has the same form as (3) with f and F replaced by fe and Fe, respectively. In this case, the support of W(n) is unbounded; therefore, its limiting distribution is no longer Weibull.
3. A special case
Emanuel (2000) showed that the evolution of intensity in Atlantic Ocean and western North Pacific Ocean hurricanes whose maximum intensity was not limited by declining potential energy exhibited remarkable regularity, with wind speed increasing linearly by around 12 m s−1 day−1 to maximum wind speed, followed by a linear decay of around 8 m s−1 day−1. Based on Emanuel’s result, suppose that
with β and γ > 0, where υmax = υ0 + βtmax and T = tmax + (υmax − υ0)/γ. Under this model, the lifetime of a hurricane begins when wind speed reaches υ0. Wind speed then increases linearly at rate β until reaching a peak of υmax before declining linearly at rate γ until it again reaches υ0 at time T.
It is straightforward to show that, in the absence of observation error, a randomly observed wind speed W under this model has a uniform distribution over the interval (υ0, υmax) with distribution function
It follows from (3) that the pdf of W(n) is
For n > 1, g(w) increases monotonically with w, becoming increasingly concave as n increases. The expected value of W(n) is given by
The relative underestimation bias in using W(n) as an estimate of υmax is
So, for example, if υ0/υmax = 0.25, then the relative underestimation bias is around 38% for n = 1 and 13% for n = 5.
Turning to the case in which wind speed is observed with normal error, it is straightforward to show that, for the wind speed model in (7), the pdf of a randomly observed wind speed W is given by
No closed form expressions for Fe(w), ge(w), or Ee[W(n)] are available, but it is straightforward to evaluate these numerically. For example, Fig. 1 shows ge(w) for the case υmax = 1, υ0 = 0.25, n = 4, and σ = 0.1. For comparison, Fig. 1 also shows g(w) for the same values of υmax, υ0, and n. In Fig. 2, Ee[W(n)] is plotted against n for these values of υmax, υ0, and σ. Again, for comparison, Fig. 2 also shows E[V(n)] for the same values of υmax and υ0. It is notable that, by opening the possibility of overestimation, the presence of measurement error actually reduces estimation bias.
4. A practical application
The theoretical results presented in the previous section provide the basis for a rough practical method to correct for underestimation bias. The expression in (11) can be rearranged as
Thus, a rough correction for underestimation bias can be made by inflating the observed value of W(n) by the factor of (n + 1)/n, leading to the estimator υ̂max = W(n)(n + 1)/n. This requires knowledge of only n and W(n).
A simulation experiment based on the reanalyzed National Hurricane Center North Atlantic basin best-track hurricane database (known as “HURDAT”) wind speed measurements for Hurricane Andrew reported in Table 1 of Landsea et al. (2004b) was conducted to assess the performance of υ̂max. For this analysis, υ0 was taken to be 35 kt (1 kt ≈ 0.5 m s−1). The data consist of 39 6-hourly wind speed measurements covering a 228-h period with a maximum of 150 kt. These data were taken to represent the true evolution of maximum wind speed for Hurricane Andrew, with values between observation times reconstructed by linear interpolation. This profile is shown in Landsea et al. (2004b, their Fig. 4) and also in Emanuel (1999, his Fig. 2).
The simulation experiment proceeded in the following way. For each value of n between 2 and 10, wind speeds were sampled at n random times from this profile, and both the maximum W(n) of these n wind speeds and the estimate υ̂max based on it were recorded. The procedure was repeated a total of 10 000 times. In Fig. 3, the averages of the 10 000 values of W(n) and υ̂max simulated in this way are plotted against n. In this case, υ̂max performed very well, on average, underestimating υmax by only around 3 kt for n between 2 and 10. In contrast, on average, W(n) underestimated υmax by more than 50 kt when n = 2 and almost 20 kt when n = 10.
The experiment was repeated for a small number of other hurricanes listed in Table 1. Wind speed data for these hurricanes were available online at the HURDAT Internet site, and the corresponding wind speed profiles are shown in Emanuel (1999). Table 1 reports the value of υmax and the average values over 10 000 simulated random wind speed samples of W(n) and υ̂max for n = 3. With the exception of Hurricane Dean, the absolute relative bias of υ̂max is less than 10%, with υmax underestimated in half the cases and overestimated in the other half. In contrast, by necessity, W(n) always underestimates υmax with a relative underestimation bias of up to 32%. It is only for Hurricane Dean that the absolute bias of W(n) is smaller than that of υ̂max. The reason is that the wind speed profile for this hurricane is far from the model in (7) and, in particular, exhibits a plateau just below its peak. In overall terms, given its extreme simplicity, υ̂max appears to perform well at correcting the underestimation bias of W(n).
The purpose of this note has been to outline and illustrate a statistical formalism for exploring the effect of sparse random sampling on the maximum observed wind speed in a hurricane. The model considered here is clearly stylized and can be extended in a number of ways. In particular, realism could be gained through an explicitly spatial model of hurricane wind fields and their sampling. In some situations, it may be reasonable to assume that observers seek to avoid the highest winds. This would have the effect of exacerbating underestimation bias. On the wind speed side, the model for υ(t) in (7) can be extended to include a random component, so that υmax is itself a random variable.
The focus here has been on maximum wind speed. In some situations, interest centers on a function of it. For example, the power dissipation index of Emanuel (2005) depends on the cube of maximum wind speed. The underestimation bias in estimating by is worse than that in estimating υmax by W(n). For example, in the absence of measurement error, when υ0/υmax = 0.25, the relative bias in estimating is 0.51 when n = 2 and 0.30 when n = 5. As reported earlier, the corresponding values in estimating υmax are 0.38 and 0.13.
The comments of two reviewers are acknowledged with gratitude. Support for this work was provided by NOAA Grant NA17RJ1223.
Corresponding author address: Andrew Solow, Woods Hole Oceanographic Institution, Woods Hole, MA 02543. Email: email@example.com