Analytical solutions to sampling effects in drop size distribution measurements during stationary rainfall: Estimation of bulk rainfall variables

International audienceA stochastic model of the microstructure of rainfall is used to derive explicit expressions for the magnitude of the sampling fluctuations in rainfall properties estimated from raindrop size measurements in stationary rainfall. The model is a marked point process, in which the points represent the drop centers, assumed to be uniformly distributed in space. This assumption, which is supported both by theoretical and by empirical evidence, implies that during periods of stationary rainfall the number of drops in a sample volume follows a Poisson distribution. The marks represent the drop sizes, assumed to be distributed independent of their positions according to some general drop size distribution. Within this framework, it is shown analytically how the sampling distribution of the estimator of any bulk rainfall variable (such as liquid water content, rain rate, or radar reflectivity) in stationary rainfall converges from a strongly skewed distribution to a (symmetrical) Gaussian distribution with increasing sample size. The relevant parameter controlling this evolution is the average number of drops in the sample ns. For a given sample size, the skewness of the sampling distribution is found to be more pronounced for higher order moments of the drop size distribution. For instance, the sampling distribution of the normalized mean diameter becomes nearly Gaussian for ns > 10, while the sampling distribution of the normalized rain rate remains skewed for ns not, vert, similar 500. Additionally, it is shown analytically that, as a result of the mentioned skewness, the median Q50 as an estimator of a bulk rainfall variable always underestimates its population value Qp in stationary rainfall. The ratio of the former to the latter is found to be View the MathML source, where b is a constant depending on the drop size distribution. For bulk rainfall variables this constant is positive and therefore the median always underestimates the population value. This provides a theoretical confirmation and explanation of previously published simulation results. Finally, relationships between the expected number of raindrops in the sample ns and the rain rate are established for different parametric forms of the raindrop size distribution. These relationships are first compared to experimental results and then used to provide examples of sampling distributions of bulk rainfall variables (in this case rain rate) for different values of the average rain rate and different integration times of the disdrometric device involved (in this case a Joss–Waldvogel disdrometer). The practical relevance of these results is (1) that they provide exact solutions to the sampling problem during (relatively rare) periods of stationary rainfall (e.g., drizzle), and (2) that they provide a lower bound to the magnitude of the sampling problem in the general situation where sampling fluctuations and natural variability co-exist

Properties of resonant activation phenomena

The phenomenon of resonant activation of a Brownian particle over a fluctuating barrier is revisited. We discuss the important distinctions between barriers that can fluctuate among up and down configurations, and barriers that are always up but that can fluctuate among different heights. A resonance as a function of the barrier fluctuation rate is found in both cases, but the nature and physical description of these resonances is quite distinct. The nature of the resonances, the physical basis for the resonant behavior, and the importance of boundary conditions are discussed in some detail. We obtain analytic expressions for the escape time over the barrier that explicitly capture the minima as a function of the barrier fluctuation rate, and show that our analytic results are in excellent agreement with numerical results

Mean exit times for free inertial stochastic processes

We study the mean exit time of a free inertial random process from a region in space. The acceleration alternatively takes the values +[ital a] and [minus][ital a] for random periods of time governed by a common distribution [psi]([ital t]). The mean exit time satisfies an integral equation that reduces to a partial differential equation if the random acceleration is Markovian. Some qualitative features of the behavior of the system are discussed and checked by simulations. Among these features, the most striking is the discontinuity of the mean exit time as a function of the initial conditions

Continued fraction solution for the radiative transfer equation in three dimensions

Starting from the radiative transfer equation, we obtain an analytical solution for both the free propagator along one of the axes and an arbitrary phase function in the Fourier-Laplace domain. We also find the effective absorption parameter, which turns out to be very different from the one provided by the diffusion approximation. We finally present an analytical approximation procedure and obtain a differential equation that accurately reproduces the transport process. We test our approximations by means of simulations that use the Henyey-Greenstein phase function with very satisfactory results

Absorbing boundary conditions for inertial processos

A recent paper by J. Heinrichs [Phys. Rev. E 48, 2397 (1993)] presents analytic expressions for the first-passage times and the survival probability for a particle moving in a field of random correlated forces. We believe that the analysis there is flawed due to an improper use of boundary conditions. We compare that result, in the white noise limit, with the known exact expression of the mean exit time

Persistent random walk model for transport trough thin slabs

We present a model for transport in multiply scattering media based on a three-dimensional generalization of the persistent random walk. The model assumes that photons move along directions that are parallel to the axes. Although this hypothesis is not realistic, it allows us to solve exactly the problem of multiple scattering propagation in a thin slab. Among other quantities, the transmission probability and the mean transmission time can be calculated exactly. Besides being completely solvable, the model could be used as a benchmark for approximation schemes to multiple light scattering

Bistability driven by dichotomous noise: A comment

Two recently reported treatments [J. M. Porrà et al., Phys. Rev. A 44, 4866 (1991) and I. L¿Heureux and R. Kapral, J. Chem. Phys. 88, 7468 (1988)] of the problem of bistability driven by dichotomous colored noise with a small correlation time are brought into agreement with each other and with the exact numerical results of L¿Heureux and Kapral [J. Chem. Phys. 90, 2453 (1989)]