scholarly journals Statistical Theory on the Functional Form of Cloud Particle Size Distributions

2018 ◽  
Vol 75 (8) ◽  
pp. 2801-2814 ◽  
Author(s):  
Wei Wu ◽  
Greg M. McFarquhar
Keyword(s):  
Particle Size ◽  
Gamma Distribution ◽  
Relative Entropy ◽  
Analytical Form ◽  
Size Distributions ◽  
Particle Size Distributions ◽  
Cloud Particle ◽  
Generalized Gamma Distribution ◽  
Generalized Gamma ◽  
Principle Of Maximum

Abstract Several functional forms of cloud particle size distributions (PSDs) have been used in numerical modeling and remote sensing retrieval studies of clouds and precipitation, including exponential, gamma, lognormal, and Weibull distributions. However, there is no satisfying theoretical explanation as to why certain distribution forms preferentially occur instead of others. Intuitively, the analytical form of a PSD can be derived by directly solving the general dynamic equation, but no analytical solutions have been found yet. Instead of a process-level approach, the use of the principle of maximum entropy (MaxEnt) for determining the theoretical form of PSDs from the perspective of system is examined here. MaxEnt theory states that the probability density function with the largest information entropy among a group satisfying the given properties of the variable should be chosen. Here, the issue of variability under coordinate transformations that arises using the Gibbs–Shannon definition of entropy is identified, and the use of the concept of relative entropy to avoid these problems is discussed. Focusing on cloud physics, the four-parameter generalized gamma distribution is proposed as the analytical form of a PSD using the principle of maximum (relative) entropy with assumptions on power-law relations among state variables, scale invariance, and a further constraint on the expectation of one state variable (e.g., bulk water mass). The four-parameter generalized gamma distribution is very flexible to accommodate various type of constraints that could be assumed for cloud PSDs.

scholarly journals Calibration of the Cloud Particle Imager Probes Using Calibration Beads and Ice Crystal Analogs: The Depth of Field

2007 ◽  
Vol 24 (11) ◽  
pp. 1860-1879 ◽  
Author(s):  
Paul J. Connolly ◽  
Michael J. Flynn ◽  
Z. Ulanowski ◽  
T. W. Choularton ◽  
M. W. Gallagher ◽  
...  
Keyword(s):  
Particle Size ◽  
Calibration Method ◽  
Depth Of Field ◽  
Strong Positive Correlation ◽  
Size Distributions ◽  
Ice Crystal ◽  
Particle Size Distributions ◽  
Object Plane ◽  
Cloud Particle ◽  
Sample Mean

Abstract This paper explains and develops a correction algorithm for measurement of cloud particle size distributions with the Stratton Park Engineering Company, Inc., Cloud Particle Imager (CPI). Cloud particle sizes, when inferred from images taken with the CPI, will be oversized relative to their “true” size. Furthermore, particles will cease to be “accepted” in the image frame if they lie a distance greater than the depth of field from the object plane. By considering elements of the scalar theory for diffraction of light by an opaque circular disc, a calibration method is devised to overcome these two problems. The method reduces the error in inferring particle size from the CPI data and also enables the determination of the particles distance from the object plane and hence their depth of field. These two quantities are vital to enable quantitative measurements of cloud particle size distributions (histograms of particle size that are scaled to the total number concentration of particles) in the atmosphere with the CPI. By using both glass calibration beads and novel ice crystal analogs, these two problems for liquid drops and ice particles can be quantified. Analysis of the calibration method shows that 1) it reduces the oversizing of 15-μm beads (from 24.3 to 14.9 μm for the sample mean), 40-μm beads (from 50.0 to 41.4 μm for the sample mean), and 99.4-μm beads (from 103.7 to 99.8 μm for the sample mean); and 2) it accurately predicts the particles distance from the object plane (the relationship between measured and predicted distance shows strong positive correlation and gives an almost one-to-one relationship). Realistic ice crystal analogs were also used to assess the errors in sampling ice clouds and found that size and distance from the object plane could be accurately predicted for ice crystals by use of the particle roundness parameter (defined as the ratio of the projected area of the particle to the area of a circle with the same maximum length). While the results here are not directly applicable to every CPI, the methods are, as data taken from three separate CPIs fit the calibration model well (not shown).

scholarly journals What Is the Maximum Entropy Principle? Comments on “Statistical Theory on the Functional Form of Cloud Particle Size Distributions”

2019 ◽  
Vol 76 (12) ◽  
pp. 3955-3960 ◽  
Author(s):  
Jun-Ichi Yano
Keyword(s):  
Maximum Entropy ◽  
Relative Entropy ◽  
Functional Form ◽  
Maximum Entropy Principle ◽  
The Other ◽  
Standard Entropy ◽  
Size Distributions ◽  
Particle Size Distributions ◽  
Cloud Particle ◽  
Entropy Principle

Abstract The basic idea of the maximum entropy principle is presented in a succinct, self-contained manner. The presentation points out some misunderstandings on this principle by Wu and McFarquhar. Namely, the principle does not suffer from the problem of a lack of invariance by change of the dependent variable; thus, it does not lead to a need to introduce the relative entropy as suggested by Wu and McFarquhar. The principle is valid only with a proper choice of a dependent variable, called a restriction variable, for a distribution. Although different results may be obtained with the other variables obtained by transforming the restriction variable, these results are simply meaningless. A relative entropy may be used instead of a standard entropy. However, the former does not lead to any new results unobtainable by the latter.

scholarly journals Cloud particle size distributions measured with an airborne digital in-line holographic instrument

2009 ◽  
Vol 2 (1) ◽  
pp. 259-271 ◽  
Author(s):  
J. P. Fugal ◽  
R. A. Shaw
Keyword(s):  
Particle Size ◽  
Three Dimensional ◽  
Sample Volume ◽  
Equivalent Diameter ◽  
Size Distributions ◽  
Particle Size Distributions ◽  
Holographic Method ◽  
Cloud Particle ◽  
Data Set ◽  
Two Dimensional Image

Abstract. Holographic data from the prototype airborne digital holographic instrument HOLODEC (Holographic Detector for Clouds), taken during test flights are digitally reconstructed to obtain the size (equivalent diameters in the range 23 to 1000 μm), three-dimensional position, and two-dimensional image of ice particles and then ice particle size distributions and number densities are calculated using an automated algorithm with minimal user intervention. The holographic method offers the advantages of a well-defined sample volume size that is not dependent on particle size or airspeed, and offers a unique method of detecting shattered particles. The holographic method also allows the volume sample rate to be increased beyond that of the prototype HOLODEC instrument, limited solely by camera technology. HOLODEC size distributions taken in mixed-phase regions of cloud compare well to size distributions from a PMS FSSP probe also onboard the aircraft during the test flights. A conservative algorithm for detecting shattered particles utilizing their depth-position along the optical axis eliminates the obvious ice particle shattering events from the data set. In this particular case, the size distributions of non-shattered particles are reduced by approximately a factor of two for particles 15 to 70 μm in equivalent diameter, compared to size distributions of all particles.

scholarly journals Cloud particle size distributions measured with an airborne digital in-line holographic instrument

2009 ◽  
Vol 2 (2) ◽  
pp. 659-688 ◽  
Author(s):  
J. P. Fugal ◽  
R. A. Shaw
Keyword(s):  
Particle Size ◽  
Three Dimensional ◽  
Sample Volume ◽  
Equivalent Diameter ◽  
Size Distributions ◽  
Particle Size Distributions ◽  
Holographic Method ◽  
Cloud Particle ◽  
Data Set ◽  
Unique Method

Abstract. Holographic data from the prototype airborne digital holographic instrument HOLODEC (Holographic Detector for Clouds), taken during test flights are digitally reconstructed to obtain the size (equivalent diameters in the range 23 to 1000 μm), three-dimensional position, and two-dimensional profile of ice particles and then ice particle size distributions and number densities are calculated using an automated algorithm with minimal user intervention. The holographic method offers the advantages of a well-defined sample volume size that is not dependent on particle size or airspeed, and offers a unique method of detecting shattered particles. The holographic method also allows the volume sample rate to be increased beyond that of the prototype HOLODEC instrument, limited solely by camera technology. HOLODEC size distributions taken in mixed-phase regions of cloud compare well to size distributions from a PMS FSSP probe also onboard the aircraft during the test flights. A conservative algorithm for detecting shattered particles utilizing the particles depth-position along the optical axis eliminates the obvious ice particle shattering events from the data set. In this particular case, the size distributions of non-shattered particles are reduced by approximately a factor of two for particles 15 to 70 μm in equivalent diameter, compared to size distributions of all particles.

Comparison of cirrus clouds in naturally and anthropogenically influenced regions of the atmosphere

2020 ◽  
Author(s):  
Maximilian Dollner ◽  
Josef Gasteiger ◽  
Charles A. Brock ◽  
Manuel Schöberl ◽  
Christina Williamson ◽  
...  
Keyword(s):  
Particle Size ◽  
Field Experiments ◽  
Anthropogenic Impacts ◽  
Dispersion Model ◽  
Cirrus Clouds ◽  
Size Distributions ◽  
Particle Size Distributions ◽  
Cloud Particle ◽  
Microphysical Properties ◽  
Aerosol Particle Size

<p>Cirrus clouds are an important contributor to the uncertainty of future climate prediction, especially due to the weak understanding of anthropogenic impacts on cirrus clouds.</p><p>We investigate aerosol and cloud microphysical properties of the remote atmosphere over the Pacific and Atlantic Oceans from about 80°N to 86°S and the region in the Mediterranean using airborne aerosol and cloud measurements of the entire atmospheric column up to approx. 13 km from the ATom (Atmospheric Tomography; 2016-2018) and the A-LIFE (Absorbing aerosol layers in a changing climate: aging, lifetime and dynamics; 2017) field experiments, respectively. Aerosol microphysical properties are retrieved from in-situ measurements of aerosol particle size distributions between 0.003 and 50 µm, single particle mass spectrometry as well as simulations with the Lagrangian transport and dispersion model FLEXPART. The microphysical properties of cirrus clouds are obtained from size distribution measurements covering the range between 3 and 930 µm.</p><p>In this study we show microphysical properties of aerosols and cirrus clouds in regions with high mineral dust concentrations as well as pristine and anthropogenic influenced regions in order to advance the knowledge of the natural and anthropogenic impact on cirrus clouds.  We present comparisons of ice crystal number concentrations, aerosol and cloud particle size distributions, and meteorological conditions of cirrus clouds in the above-mentioned regions of the atmosphere.</p>

scholarly journals The Modified Gamma Size Distribution Applied to Inhomogeneous and Nonspherical Particles: Key Relationships and Conversions

2011 ◽  
Vol 68 (7) ◽  
pp. 1460-1473 ◽  
Author(s):  
Grant W. Petty ◽  
Wei Huang
Keyword(s):  
Particle Size ◽  
Gamma Distribution ◽  
Nonspherical Particles ◽  
Size Distributions ◽  
Power Law Distribution ◽  
Particle Size Distributions ◽  
Equivalent Radius ◽  
Atmospheric Particle ◽  
Gamma Distributions ◽  
Special Cases

Abstract The four-parameter modified gamma distribution (MGD) is the most general mathematically convenient model for size distributions of particle types ranging from aerosols and cloud droplets or ice particles to liquid and frozen precipitation. The common three-parameter gamma distribution, the exponential distribution (e.g., Marshall–Palmer), and power-law distribution (e.g., Junge) are all special cases. Depending on the context, the particle “size” used in a given formulation may be the actual geometric diameter, the volume- or area-equivalent spherical diameter, the actual or equivalent radius, the projected or surface area, or the mass. For microphysical and radiative transfer calculations, it is often necessary to convert from one size representation to another, especially when comparing or utilizing distribution parameters obtained from a variety of sources. Furthermore, when the mass scales with Db, with b < 3, as is typical for snow and ice and other particles having a quasi-fractal structure, an exponential or gamma distribution expressed in terms of one size parameter becomes an MGD when expressed in terms of another. The MGD model is therefore more fundamentally relevant to size distributions of nonspherical particles than is often appreciated. The central purpose of this paper is to serve as a concise single-source reference for the mathematical properties of, and conversions between, atmospheric particle size distributions that can expressed as MGDs, including exponential and gamma distributions as special cases. For illustrative purposes, snow particle size distributions published by Sekhon and Srivastava, Braham, and Field et al. are converted to a common representation and directly compared for identical snow water content, allowing large differences in their properties to be discerned and quantified in a way that is not as easily achieved without such conversion.

scholarly journals Reply to “What Is the Maximum Entropy Principle? Comments on ‘Statistical Theory on the Functional Form of Cloud Particle Size Distributions’”

2019 ◽  
Vol 76 (12) ◽  
pp. 3961-3963
Author(s):  
Wei Wu ◽  
Greg M. McFarquhar
Keyword(s):  
Particle Size ◽  
Maximum Entropy ◽  
Functional Form ◽  
Maximum Entropy Principle ◽  
Size Distributions ◽  
Particle Size Distributions ◽  
Cloud Particle ◽  
Functional Relations ◽  
Entropy Principle ◽  
Gibbs Entropy

Abstract We welcome the opportunity to correct the misunderstandings and misinterpretations contained in Yano’s comment that led him to incorrectly state that Wu and McFarquhar misunderstood the maximum entropy (MaxEnt) principle. As correctly stated by Yano, the principle itself does not suffer from the problem of a lack of invariance. But, as restated in this reply and in Wu and McFarquhar, the commonly used Shannon–Gibbs entropy does suffer from a lack of invariance for coordinate transform when applied in continuous cases, and this problem is resolved by the use of the relative entropy. Further, it is restated that the Wu and McFarquhar derivation of the PSD form using MaxEnt is more general than the formulation by Yano and allows more constraints with any functional relations to be applied. The derivation of Yano is nothing new but the representation of PSDs in other variables.

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