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Raymond K. W. Wong
and
Norman Chidambaram

Abstract

A maximum likelihood approach to the application of the gamma size distribution is described and compared with the method of moments approach suggested by Ulbrich. Estimation of distribution parameters based on the maximum likelihood principle and Ulbrich's estimation method have different weighting characteristics, which are illustrated through the use of quantile-quantile plots. The ability of the gamma size distribution to describe curvature on a semilogarithmic diagram, and the mathematical simplicity of incorporating it in the sampling error model based on the Poisson process make it possible to derive a sampling error model with consideration given to changes in size distribution shape. It is also shown that variations in size distribution shape can have significant effects on the estimation of sampling errors.

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Raymond K. W. Wong

Abstract

Rapidly converging maximum likelihood procedures for estimating and testing Weibull distribution parameters are presented, together with numerical examples of their applications. Goodness-of-fit comparisons based on nine sets of meteorological or hydrological data were made among the gamma, lognormal, three-parameter kappa and Weibull distributions. The Weibull distribution is shown to be a reasonable alternative to the other three distributions.

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Raymond K. W. Wong
and
Keith D. Hage

Abstract

By examining chain rule transformation and tensor transformation results, it is shown that the small slope assumption mentioned in Pielke and Martin (1981) is not required for the validity of the hydrostatic equation in terrain-following coordinates.

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Raymond K. W. Wong
,
Keith D. Hage
, and
Leslie D. Phillips

Abstract

A two-dimensional numerical model is used to simulate nocturnal drainage flow in a small urban valley with light prevailing winds and conditions of supercritical Richardson numbers (Ri). The model uses a hydrostatic and Boussinesq system of equations written in terrain-following coordinates. Radiative transfer is represented by Brunt's method of radiative diffusivity. Eddy diffusivities are specified in the subgrid parameterization for conditions where Ri is supercritical. Tests show the dependence of drainage wind on slope angle, cooling rate, surface drag and prevailing wind speed, and also the insensitivity of wind and temperature to the eddy diffusivities under supercritical Ri conditions. The drainage wind cells are asymmetric, with a shallow surface layer of drainage flow and a thicker upper region of slower return flow. The predicted wind profiles show low-level maxima and the predicted temperature profiles are exponential in shape, in good agreement with observations obtained in Edmonton, Alberta in the summer of 1978. The model is also able to predict the quasi-stationary slope flow observed in the field.

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Raymond K. W. Wong
,
Norman Chidambaram
,
Lawrence Cheng
, and
Marianne English

Abstract

The use of a shifted gamma size distribution for hailstone samples is proposed. This is shown to provide a better fit than the usual exponential form, using time-resolved Alberta data. It is also concluded that there is a dependence of the shape of hailstone size distributions on the duration of sampling time. Such shape variations are associated with the sampling efficiency of the smaller size categories. The importance of the smaller sizes to the common hail integral estimates is also investigated. The minimum sizes required for sampling accuracy of these integral estimates are also obtained.

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Hojun You
,
Jiayi Wang
,
Raymond K. W. Wong
,
Courtney Schumacher
,
R. Saravanan
, and
Mikyoung Jun

Abstract

The prediction of tropical rain rates from atmospheric profiles poses significant challenges, mainly due to the heavy-tailed distribution exhibited by tropical rainfall. This study introduces over-parameterized neural networks not only to forecast tropical rain rates, but also to explain their heavy-tailed distribution. The investigation is separately conducted for three rain types (stratiform, deep convective, and shallow convective) observed by the Global Precipitation Measurement satellite radar over the West and East Pacific regions. Atmospheric profiles of humidity, temperature, and zonal and meridional winds from the MERRA-2 reanalysis are considered as features. Although over-parameterized neural networks are well-known for their “double descent phenomenon,” little has been explored about their applicability to climate data and capability of capturing the tail behavior of data. In our results, over-parameterized neural networks accurately estimate the rain rate distributions and outperform other machine learning methods. Spatial maps show that over-parameterized neural networks also successfully describe spatial patterns of each rain type across the tropical Pacific. In addition, we assess the feature importance for each over-parameterized neural network to provide insight into the key factors driving the predictions, with low-level humidity and temperature variables being the overall most important. These findings highlight the capability of over-parameterized neural networks in predicting the distribution of the rain rate and explaining extreme values.

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