Determining Functional Relations in Multivariate Oceanographic Systems: Model II Multiple Linear Regression

Abstract A method for estimating multivariate functional relationships between sets of measured oceanographic, meteorological, and other field data is presented. Model II regression is well known for describing functional relationships between two variables. However, there is little accessible guidance for the researcher wishing to apply model II methods to a multivariate system consisting of three or more variables. This paper describes a straightforward method to extend model II regression to the case of three or more variables. The multiple model II procedure is applied to an analysis of the optical spectral scattering coefficient measured in the coastal ocean. The spectral scattering coefficient is regressed against both suspended mineral particle concentration and suspended organic particle concentration. The regression coefficients from this analysis provide adjusted estimates of the mineral particle scattering cross section and the organic particle scattering cross section. Greater accuracy and efficiency of the coefficients from this analysis, compared to semiempirical coefficients, is demonstrated. Examples of multivariate data are presented that have been analyzed by partitioning the variables into arbitrary bivariate models. However, in a true multivariate system with correlated predictors, such as a coupled biogeochemical cycle, these bivariate analyses yield incorrect coefficient estimates and may result in large unexplained variance. Employing instead a multivariate model II analysis can alleviate these problems and may be a better choice in these situations.

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Particle sizing calibration with refractive index correction for light scattering optical particle counters and impacts upon PCASP and CDP data collected during the Fennec campaign

Atmospheric Measurement Techniques ◽

10.5194/amt-5-1147-2012 ◽

2012 ◽

Vol 5 (5) ◽

pp. 1147-1163 ◽

Cited By ~ 77

Author(s):

P. D. Rosenberg ◽

A. R. Dean ◽

P. I. Williams ◽

J. R. Dorsey ◽

A. Minikin ◽

...

Keyword(s):

Particle Size ◽

Refractive Index ◽

Cross Section ◽

Scattering Cross Section ◽

Saharan Dust ◽

Particle Scattering ◽

Atmospheric Measurements ◽

Scattering Cross ◽

Highly Nonlinear ◽

The Impact

Abstract. Optical particle counters (OPCs) are used regularly for atmospheric research, measuring particle scattering cross sections to generate particle size distribution histograms. This manuscript presents two methods for calibrating OPCs with case studies based on a Passive Cavity Aerosol Spectrometer Probe (PCASP) and a Cloud Droplet Probe (CDP), both of which are operated on the Facility for Airborne Atmospheric Measurements BAe-146 research aircraft. A probability density function based method is provided for modification of the OPC bin boundaries when the scattering properties of measured particles are different to those of the calibration particles due to differences in refractive index or shape. This method provides mean diameters and widths for OPC bins based upon Mie-Lorenz theory or any other particle scattering theory, without the need for smoothing, despite the highly nonlinear and non-monotonic relationship between particle size and scattering cross section. By calibrating an OPC in terms of its scattering cross section the optical properties correction can be applied with minimal information loss, and performing correction in this manner provides traceable and transparent uncertainty propagation throughout the whole process. Analysis of multiple calibrations has shown that for the PCASP the bin centres differ by up to 30% from the manufacturer's nominal values and can change by up to approximately 20% when routine maintenance is performed. The CDP has been found to be less sensitive than the manufacturer's specification with differences in sizing of between 1.6 ± 0.8 μm and 4.7 ± 1.8 μm for one flight. Over the course of the Fennec project in the Sahara the variability of calibration was less than the calibration uncertainty in 6 out of 7 calibrations performed. As would be expected from Mie-Lorenz theory, the impact of the refractive index corrections has been found to be largest for absorbing materials and the impact on Saharan dust measurements made as part of the Fennec project has been found to be up to a factor of 3 for the largest particles measured by CDP with diameters of approximately 120 μm. In an example case, using the calibration and refractive index corrections presented in this work allowed Saharan dust measurement from the PCASP, CDP and a Cloud Imaging Probe to agree within the uncertainty of the calibration. The agreement when using only the manufacturer's specification was poor. Software tools have been developed to perform these calibrations and corrections and are now available as open source resources for the community via the SourceForge repository.

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Scattering in graphene quantum dots under generalized uncertainty principle

International Journal of Modern Physics A ◽

10.1142/s0217751x19502129 ◽

2019 ◽

Vol 34 (31) ◽

pp. 1950212

Author(s):

Zheng-Xue Wu ◽

Chao-Yun Long ◽

Jing Wu ◽

Zheng-Wen Long ◽

Ting Xu

Keyword(s):

Cross Section ◽

Uncertainty Principle ◽

Scattering Problem ◽

Generalized Uncertainty Principle ◽

Graphene Quantum Dots ◽

Scattering Cross Section ◽

Scattering Coefficient ◽

Scattering Efficiency ◽

Scattering Coefficients ◽

Scattering Cross

In this article, the Dirac electron scattering problem on circular barrier of radius [Formula: see text] is studied under the generalized uncertainty principle (GUP). The expressions of scattering coefficients, scattering cross-section and scattering efficiency of massless Dirac particle are obtained by solving the massless Dirac equation under GUP and discussed by numerical methods. It shows that the scattering coefficient, the scattering cross-section, and the scattering efficiency depend explicitly on the GUP parameter [Formula: see text]. For the scattering coefficient [Formula: see text], GUP may cause slight shift in the oscillation position of [Formula: see text] and make some peaks value of [Formula: see text] smaller. For scattering cross-section and scattering efficiency, GUP may also lead to slight shift in their oscillation position and increase of amplitude when the GUP parameter increases.

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Particle sizing calibration with refractive index correction for light scattering optical particle counters and impacts upon PCASP and CDP data collected during the Fennec campaign

Atmospheric Measurement Techniques Discussions ◽

10.5194/amtd-5-97-2012 ◽

2012 ◽

Vol 5 (1) ◽

pp. 97-135 ◽

Cited By ~ 5

Author(s):

P. D. Rosenberg ◽

A. R. Dean ◽

P. I. Williams ◽

A. Minikin ◽

M. A. Pickering ◽

...

Keyword(s):

Particle Size ◽

Refractive Index ◽

Cross Section ◽

Scattering Cross Section ◽

Saharan Dust ◽

Particle Scattering ◽

Atmospheric Measurements ◽

Scattering Cross ◽

Highly Nonlinear ◽

The Impact

Abstract. Optical particle counters (OPCs) are used regularly for atmospheric research, measuring particle scattering cross sections to generate particle size distribution histograms. This manuscript presents two methods for calibrating OPCs with case studies based on a Passive Cavity Aerosol Spectrometer Probe (PCASP) and a Cloud Droplet Probe (CDP), both of which are operated on the Facility for Airborne Atmospheric Measurements BAe-146 research aircraft. A method is also provided for modification of OPC bin boundaries when the scattering properties of measured particles are different to those of the calibration particles due to differences in refractive index or shape. This method provides mean diameters and widths for OPC bins based upon Mie-Lorenz theory or any other particle scattering theory, without the need for smoothing, despite the highly nonlinear and non-monotonic relationship between particle size and scattering cross section. By calibrating an OPC in terms of its scattering cross section the optical properties correction can be applied with minimal information loss and full propagation of uncertainty. Analysis of multiple calibrations has shown that for the PCASP the bin centres differ by up to 30% from the manufacturer's nominal values and can change by approximately 20% when routine maintenance is performed. The CDP has been found to differ from the manufacturer's specification by 15–64% and over the course of the Fennec project in the Sahara the variability of calibration was always less than the 2-σ calibration uncertainty or 10%. As would be expected from Mie-Lorenz theory the impact of the refractive index corrections has been found to be largest for absorbing materials and the impact on Saharan dust measurements made as part of the Fennec project has been found to be up to a factor of 3 for the largest particles which could be measured by CDP. Software tools have been developed as part of this work and are now made available as open source resources for the community via the SourceForge repository.

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Mass Determination by Direct Measurement of Electron Scattering

Proceedings, annual meeting, Electron Microscopy Society of America ◽

10.1017/s0424820100114827 ◽

1975 ◽

Vol 33 ◽

pp. 240-241

Author(s):

M. K. Lamvik ◽

A. V. Crewe

Keyword(s):

Cross Section ◽

Electron Scattering ◽

Mass Density ◽

Variable Mass ◽

Scattering Cross Section ◽

Mass Determination ◽

Number Density ◽

Cross Sectional ◽

Thin Slice ◽

Scattering Cross

If a molecule or atom of material has molecular weight A, the number density of such units is given by n=Nρ/A, where N is Avogadro's number and ρ is the mass density of the material. The amount of scattering from each unit can be written by assigning an imaginary cross-sectional area σ to each unit. If the current I0 is incident on a thin slice of material of thickness z and the current I remains unscattered, then the scattering cross-section σ is defined by I=IOnσz. For a specimen that is not thin, the definition must be applied to each imaginary thin slice and the result I/I0 =exp(-nσz) is obtained by integrating over the whole thickness. It is useful to separate the variable mass-thickness w=ρz from the other factors to yield I/I0 =exp(-sw), where s=Nσ/A is the scattering cross-section per unit mass.

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Absolute inner-shell cross section determination for EELS analysis

Proceedings, annual meeting, Electron Microscopy Society of America ◽

10.1017/s042482010010473x ◽

1988 ◽

Vol 46 ◽

pp. 534-535

Author(s):

P.A. Crozier

Keyword(s):

Cross Section ◽

Cross Sections ◽

Absolute Cross Section ◽

Specimen Thickness ◽

Mass Thickness ◽

Scattering Cross ◽

Before And After ◽

Estimated Error ◽

Scattering Cross Sections ◽

Electron Microscopist

Absolute inelastic scattering cross sections or mean free paths are often used in EELS analysis for determining elemental concentrations and specimen thickness. In most instances, theoretical values must be used because there have been few attempts to determine experimental scattering cross sections from solids under the conditions of interest to electron microscopist. In addition to providing data for spectral quantitation, absolute cross section measurements yields useful information on many of the approximations which are frequently involved in EELS analysis procedures. In this paper, experimental cross sections are presented for some inner-shell edges of Al, Cu, Ag and Au.Uniform thin films of the previously mentioned materials were prepared by vacuum evaporation onto microscope cover slips. The cover slips were weighed before and after evaporation to determine the mass thickness of the films. The estimated error in this method of determining mass thickness was ±7 x 107g/cm2. The films were floated off in water and mounted on Cu grids.

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