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Article Excerpt
1. INTRODUCTION
Since its introduction, phase Doppler interferometry (PDI) has been used to characterize sprays in a wide variety of areas, including hazardous waste incineration, internal combustion engines, spray coatings, agricultural pesticide sprays, fire suppression, and others. PDI, which is an extension of laser Doppler velocimetry that measures droplet size as well as velocity (Durst and Zare 1975; Bachalo and Houser 1984a,b), involves creating an interference pattern in the region where two laser beams intersect. The region where the laser beams intersect is called the probe volume, or sample volume. Due to the interference pattern, a droplet passing through the probe volume scatters light, exhibiting an angular and temporal intensity distribution that is characteristic of the size, refractive index, and velocity of the droplet. For a droplet with known refractive index, the droplet size and velocity can be determined by analyzing the scattered light collected with several photomultiplier tubes. Additional details on PDI are available elsewhere (e.g., Bachalo 1994).
At the National Institute of Standards and Technology, the diffusion rate of the droplets in a methanol spray flame was studied by Widmann et al. (2000). For this purpose, the arrival times of droplets at the PDI probe volume were used. The measurement of droplet arrival times has also been used to investigate droplet clustering in sprays and to identify measurement artifacts like burst splitting (Lazaro 1991; Van den Moortel et al. 1997; Widmann and Presser 2001, 2002; Widmann et al. 2001a,b). However, in the study by Widmann et al. (2000), not all of the arrival times could be recorded, due to the presence of dead time. Dead time in PDI measurements, first reported by McDonnell and Samuelsen (1995), refers to the periods of inactivity of the PDI processor during which data are not recorded. This is illustrated by the experimental data presented in Figure 1. In this article we address estimation of the diffusion rate with incomplete arrival-time data caused by such dead times.
Understanding spray characteristics is of critical importance in many areas of science and technology. From numerous investigations of atomization and spray processes, a host of statistical issues have arisen, among which is the problem of dead time. Following the spray literature (Edwards and Marx 1995, 1996), we assume that droplets arrive at a PDI probe volume according to a homogeneous Poisson process, {N(t):t [greater than or equal to] 0}, with intensity [lambda], where N(t) represents the number of droplets that passed through the probe volume (but not necessarily recorded) in the time interval (0, t]. The presence of dead times renders the observed arrival times of the droplets an incompletely observed Poisson process.
Dead time occurs throughout the experiment in the following way. The PDI continuously records the arrival times of droplets for [W.sub.1] amount of time, then becomes inactive (dead) for [Y.sub.1] amount of time. It resumes recording for [W.sub.2] amount of time, then becomes inactive for a duration of [Y.sub.2], and subsequently alternates between W and Y until termination of the experiment. As such, the total recording time of the PDI may be viewed as an alternating sequence of nonnegative random variables.
[W.sub.1], [Y.sub.1], [W.sub.2], [Y.sub.2], [W.sub.3], [Y.sub.3],...,
where their exact values are not observable.
It is important to point out that characteristics of dead time vary with measurement instruments. The dead time considered here is different from the two kinds of dead time studied in the statistics literature, the paralyzable counter (electron multipliers) and nonparalyzable counter (e.g., Geiger counters), pioneered by Feller (1948), Takacs (1956), Pyke (1958), and others. In the former, a true event is recorded if and only if no other true event has occurred within the preceding time interval, [tau], known as the dead time. In the latter, the counter is inactive for a time interval [tau] after each recorded event.
In the case of PDI, multiple events can occur during an observation interval as well as during a dead time. Little is known about the physical cause of dead time (see, e.g., McDonell and Samuelsen 1995), except its presence as manifested by the gaps in the PDI recordings. These gaps are prominently displayed in Figure 1, which plots the total number of arrivals of the droplets recorded in the time interval (0, t] against the time t.
[FIGURE 1 OMITTED]
Without dead time, the estimation of [lambda] can be easily carried out by, for instance, using the exponential interarrival times. With dead time, the interarrival times of the observed process are no longer exponentially distributed. In fact, as noted by Widmann et al. (2000), the interarrival times have a multimodal (oscillatory) distribution (Fig. 2.) In this example the average duration of the dead time is about 5.2 milliseconds (ms). The effect of multimodality dampens as the average duration of the dead time decreases, as can be seen in the simulated results illustrated in Figure 3. Ignoring dead time would introduce bias in the estimation.
In the (spray) literature, the diffusion rate is typically studied by constructing the interparticle (interarrival) time distribution (see, e.g., Edwards and Marx 1995). However, if the counting process {N(t):t [greater than or equal to] 0} in question is not homogeneous Poisson or a renewal type, then the interarrival times need not be identically distributed. It would then be very difficult to interpret the interparticle time distribution. The utility of such a distribution is doubtful. To understand what causes this multimodality, we went back to the raw data by looking at the observed counts (without summarizing them into a interparticle time distribution), as shown in Figure 1. There we can clearly see the presence of dead times, which, however, is not visible in the interarrival time distribution (see Fig. 2). This shows that an important feature is well hidden by summarizing the data! It would be hard to link the multimodality to dead times without examining the raw data.
[FIGURE 2 OMITTED]
[FIGURE 3 OMITTED]
Widmann et al. (2000) seem to be the first authors to look at counting process itself (see Fig. 1) and to attribute the occurrence of multimodality to dead times. Their finding is important in that it shows that the modulation in interarrival time is not an intrinsic property of the spray, but rather is a limitation of the measurement instrument. They used simulation method to estimate the distribution of dead time Y and [lambda]. They found that a normal distribution (with most of its mass located on the positive part of the real line) for Y provides a good fit to the data that supports the Poisson process as a model for the spray process. Having demonstrated the adequacy of normal distribution for Y, they found the estimate of [lambda] by trial and error.
In this article we investigate the estimation problem analytically. We construct consistent estimators for [lambda] under very minimal conditions on [Y.sub.1] for i = 1, 2,.... Essentially, only two assumptions are needed, the [Y.sub.i]'s are independent and the expected length of the observation interval E[W.sub.i] is larger than 2/[lambda].
The success of our estimation procedure relies in...
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