Alexander B. Kostinski

On foundations of radar polarimetry

Polarization aspects of the radar target scattering problem are reexamined. The optimization problem of radar polarimetry is formulated and Kennaughs method of finding optimal polarizations is modified and extended to nonreciprocal and bistatic cases. Our approach does not necessitate diagonalization of the target scattering operator and therefore, a change-of-basis is not required. The change-of-polarization-basis is motivated by the comparison of experimental data taken with different antenna sets. Unitary matrix algebra is used to derive proper transformation formulas for scattering operators and bilinear voltage forms.

Scale-dependent droplet clustering in turbulent clouds

The current understanding of fundamental processes in atmospheric clouds, such as nucleation, droplet growth, and the onset of precipitation (collision–coalescence), is based on the assumption that droplets in undiluted clouds are distributed in space in a perfectly random manner, i.e. droplet positions are independently distributed with uniform probability. We have analysed data from a homogeneous cloud core to test this assumption and gain an understanding of the nature of droplet transport. This is done by examining one-dimensional cuts through clouds, using a theory originally developed for x-ray scattering by liquids, and obtaining statistics of droplet spacing. The data reveal droplet clustering even in cumulus cloud cores free of entrained ambient air. By relating the variance of droplet counts to the integral of the pair correlation function, we detect a systematic, scale-dependent clustering signature. The extracted signal evolves from sub- to super-Poissonian as the length scale increases. The sub-Poisson tail observed below mm-scales is a result of finite droplet size and instrument resolution. Drawing upon an analogy with the hard-sphere potential from the theory of liquids, this sub-Poisson part of the signal can be effectively removed. The remaining part displays unambiguous clustering at mm- and cm-scales. Failure to detect this phenomenon until now is a result of the previously unappreciated cumulative nature, or ‘memory,’ of the common measures of droplet clustering.

On the polarimetric contrast optimization

The problem considered is one of finding the polarization state of an antenna such that a power ratio due to two different objects is optimized. It is reduced to an optimization problem of two Hermitian forms which can be simultaneously diagonalized using a well-known linear-algebraic construction. The corresponding polarization vectors are found. The calculations are performed directly on the expression for the energy density of the reflected wave as a function of the transmitter polarization and are, therefore, decoupled from the receiving antenna parameters. Such decoupling is particularly convenient in bistatic applications.

A Simple Necessary and Sufficient Condition on Physically Realizable Mueller Matrices

Abstract Development of simple tools to test physical realizability of measured or computed Mueller matrices is the subject of this paper. In particular, the overpolarization problem, i.e., the problem of ensuring that the output degree of polarization does not exceed unity is solved by finding an easily implementable necessary and sufficient condition. With G being the Lorentz metric, it states that a given matrix M is not overpolarizing if and only if the spectrum of GM T GM is real and an eigenvector associated with the largest eigenvalue is a physical Stokes vector. This result is used to characterize some M classes of special interest, and is used to test several examples from recent literature.

Fluctuations and Luck in Droplet Growth by Coalescence

After the initial rapid growth by condensation, further growth of a cloud droplet is punctuated by coalescence events. Such a growth process is essentially stochastic. Yet, computational approaches to this problem dominate and transparent quantitative theory remains elusive. The stochastic coalescence problem is revisited and it is shown, via simple back-of-the-envelope results, that regardless of the initial size, the fastest one-in-a-million droplets, required for warm rain initiation, grow about 10 times faster than the average droplet. While approximate, the development presented herein is based on a realistic expression for the rate of coalescence. The results place a lower bound on the relative velocity of neighboring droplets, necessary for warm rain initiation. Such velocity differences may arise from a variety of physical mechanisms. As an example, turbulent shear is considered and it is argued that even in the most pessimistic case of a cloud composed of single-sized droplets, rain can still for...

On the Spatial Distribution of Cloud Particles

Abstract Recent studies have led to the statistical characterization of the spatial (temporal) distributions of cloud (precipitation) particles as a doubly stochastic Poisson process. This paper arrives at a similar conclusion (larger-than-Poissonian variance) via the more fundamental route of statistical physics and significantly extends previous findings in several ways. The focus is on the stochastic structure in the spatial distribution of cloud particles. A new approach for exploring the stochastic structure of clouds is proposed using a direct relation between number density variance and the pair correlation function. In addition, novel counting diagrams, particularly useful for analyzing counts at low data rates, demonstrate droplet clustering and striking deviations from Poisson randomness on small (centimeter) scales. These findings are shown to agree with pair correlation functions calculated for droplet counts obtained from an aircraft-mounted cloud probe. Time series of the arrival of each dro...

Comparison of TOMS and AVHRR volcanic ash retrievals from the August 1992 eruption of Mt. Spurr

On August 19, 1992, the Advanced Very High Resolution Radiometer (AVHRR) onboard NOAA-12 and NASAs Total Ozone Mapping Spectrometer (TOMS) onboard the Nimbus-7 satellite simultaneously detected and mapped the ash cloud from the eruption of Mt. Spurr, Alaska. The spatial extent and geometry of the cloud derived from the two datasets are in good agreement and both AVHRR split window IR (11–12µm brightness temperature difference) and the TOMS UV Aerosol Index (0.34–0.38µm ultraviolet backscattering and absorption) methods give the same range of total cloud ash mass. Redundant methods for determination of ash masses in drifting volcanic clouds offer many advantages for potential application to the mitigation of aircraft hazards.

On the extinction of radiation by a homogeneous but spatially correlated random medium.

Exponential extinction of incoherent radiation intensity in a random medium (sometimes referred to as the Beer-Lambert law) arises early in the development of several branches of science and underlies much of radiative transfer theory and propagation in turbid media with applications in astronomy, atmospheric science, and oceanography. We adopt a stochastic approach to exponential extinction and connect it to the underlying Poisson statistics of extinction events. We then show that when a dilute random medium is statistically homogeneous but spatially correlated, the attenuation of incoherent radiation with depth is often slower than exponential. This occurs because spatial correlations among obstacles of the medium spread out the probability distribution of photon extinction events. Therefore the probability of transmission (no extinction) is increased.

Optimal reception of partially polarized waves

Various aspects of the physics of partially polarized waves are discussed with applications to optical (lidar) reception problems. We focus on the issue of the optimal intensity reception of partially polarized waves scattered off a fluctuating object (ensemble of scatterers) of known polarization properties (measured Mueller matrix). Expressions for total available intensity and adjustable (polarization-dependent) intensity are derived in a clear and novel manner by using the coherency matrix approach. A general numerical technique is developed and illustrated for the optimization of adjustable intensity as a function of transmitted polarization. Closed-form expressions are derived for two important subcases, and numerical illustrations for the general case are discussed in detail, including the use of relevant experimental data.

Towards quantifying droplet clustering in clouds

Droplet positions in atmospheric clouds are random but possibly correlated on some scales. This ‘clustering’ must be quantified in order to account for it in theories of cloud evolution and radiative transfer. Tools as varied as droplet concentration power spectrum, Fishing test, and fractal correlation analysis have been used to describe the small-scale nature of clouds, and it has been difficult to compare conclusions systematically. Here we show, by using the correlation-fluctuation theorem and the Wiener–Khinchin theorem, that all of these measures can be related to the pair-correlation function. It is argued that the pair-correlation function is ideal for quantifying droplet clustering because it contains no scale memory and because of its quantitative link to the Poisson process. Copyright