Integration of the Trivariate Normal Distribution Over an Offset Sphere and an Inverse Problem

The backbone modeling in ground-motion characterization (GMC) is a useful methodology to describe the epistemic uncertainty in median ground-motion predictions. The approach uses a backbone ground-motion model (GMM) and populates the GMC logic tree with the scaled and/or adjusted versions of the backbone GMM to capture the epistemic uncertainty in median ground motions. The scaling and/or adjustment should represent the specific features and uncertainties involved in source, path, and site effects at the target site. The identification of the backbone model requires different considerations specific to the nature of the ground-motion hazard problem. In this article, we present a scaled backbone modeling approach that considers the magnitude- and distance-scaling predictors as well as their correlation to address the epistemic uncertainty in median ground-motion predictions. This approach results in a trivariate normal distribution to fully define a range of epistemic uncertainty in a model sample space. The simultaneous consideration of magnitude and distance scaling while defining the epistemic uncertainty and the methodology followed for the simplified representation of trivariate normal distribution in ground-motion logic tree are the two important features in our procedure. We first present the proposed approach that is followed by a case study for Central and Eastern North America (CENA) stable continental region. The case study discusses the underlying assumptions and limitations of the proposed approach.

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Prediction in a trivariate normal distribution via a linear combination of order statistics

Statistics & Probability Letters ◽

10.1016/j.spl.2009.07.029 ◽

2009 ◽

Vol 79 (21) ◽

pp. 2289-2296 ◽

Cited By ~ 9

Author(s):

A. Jamalizadeh ◽

N. Balakrishnan

Keyword(s):

Normal Distribution ◽

Order Statistics ◽

Linear Combination ◽

Trivariate Normal

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A Probabilistic Model for the Determination of Acid Rain Levels

Water Quality Research Journal ◽

10.2166/wqrj.1993.016 ◽

1993 ◽

Vol 28 (2) ◽

pp. 337-354 ◽

Cited By ~ 5

Author(s):

Serge B. Provost ◽

Rajesh K. Barnwal

Keyword(s):

Normal Distribution ◽

Probabilistic Model ◽

Concentration Distribution ◽

Three Dimensional ◽

Elliptical Cylinder ◽

Gaussian Fields ◽

Rain Drops ◽

Trivariate Normal ◽

Elliptical Cylinders

Abstract This article gives, in closed form, exact representations of the probability content of elliptical cylinders lying in three-dimensional Gaussian fields. The concentration of a pollutant present in the air, in a given neighborhood, is assumed to follow a trivariate normal distribution. Noting that the amount of a certain pollutant absorbed by a cluster of clouds—whose volume is approximated by an ellipsoid—is proportional to the pollutant’s integrated concentration distribution over the elliptical cylinder that the cluster sweeps out, and that the trajectories of the rain drops coming from this cluster also fill an elliptical cylinder, one could, for example, use the derived results to determine the amount of sulphuric acid deposited on a given area of the earth’s surface during rainfall.

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