Latin Hypercube sampling generates more efficient estimates of. N sample points are then drawn such that a Latin Hypercube is created. When sampling a function of k variables, the range of each variable is divided into n equally probable intervals. A Latin hypercube is the generalisation of this concept to an arbitrary number of dimensions.

(a) A вЂњsquareвЂќ is chosen at random (dark shading) and within it is randomly chosen the first sample point (dot). The PDF has been divided into three вЂњrowsвЂќ i.e., strips of nearly constant r t, and three вЂњcolumns,вЂќ i.e., strips of roughly constant w. With K 1 R 1 1 R 1 1 e (x 2y +x2+y 8x 8y)2dxdy Raw NIM MC uses X comp.A three-point Latin hypercube sample taken from the cloudy portion ( r t > 10) of a bivariate ( w Г— r t) single Gaussian PDF.

How does a hypercube work A hypercube can be defined by increasing the numbers of dimensions of a shape: 0 A point is a hypercube of dimension zero. All squares are associated with equal probability (figure not drawn to scale).In a Latin Hypercube Sampling, one needs to follow the orthogonal rule i.e., in each row&column there can be only 1 sample. (c) The third point is selected randomly from within the only square that remains. (b) The second sample point is chosen, and its row and column are excluded (vertical hatching).

First a point V is chosen randomly from a uniform distribution. My question is rather simple: Can one use LHS for anything other than a random variable following a uniform lawAn illustration of the inverse distribution function method for a single Gaussian PDF. Q Latin Hypercube Sampling.

(37) and (40) (thick solid line), traditional Monte Carlo ( n = 1, n t = 1) (circles), Latin hypercube sampling with ( n = 1, n t = 12) (triangles), and Latin hypercube sampling with ( n = 2, n t = 12) (squares). Estimates are based on analytic Eqs. Here F Gt( s) is a truncated Gaussian, and C denotes cloud fraction.Estimates of grid box average Kessler autoconversion for the BOMEX cumulus simulation.

In this way, the Monte Carlo method acts as an interface between the host modelвЂ™s dynamics and the microphysical parameterization. In this method, the microphysics subroutine is called repeatedly, once with each sample point. Rather, the model needs to determine information about subgrid variability and input it into the microphysics parameterization.One possible solution is to assume the shape of the family of probability density functions (PDFs) associated with a grid box and sample it using the Monte Carlo method. Consequently, there are inaccuracies if microphysics parameterizations are forced with grid box averages of model fields, such as liquid water content. 4, but for the FIRE stratocumulus simulation.One problem in computing cloud microphysical processes in coarse-resolution numerical models is that many microphysical processes are nonlinear and small in scale.

The associated family of PDFs is assumed to be a joint normal/lognormal (i.e., Gaussian/lognormal) mixture. One such method is Latin hypercube sampling, which reduces noise by spreading out the sample points in a quasi-random fashion.This paper formulates a sampling interface based on the Latin hypercube method. If one wishes to mitigate the noise further, one needs a method for reduction of variance. If the family of PDFs is tractable, one can sample solely from cloud, thereby improving estimates of in-cloud processes.

Both stratocumulus and cumulus boundary layer cases are tested. The PDFs that are sampled are extracted diagnostically from large-eddy simulations (LES). As a prototypical microphysical formula, this paper uses the Kessler autoconversion formula. Second, the method is flexible: one can arbitrarily specify the number of hydrometeor categories and the number of calls to the microphysics parameterization per grid box per time step.This paper performs a preliminary test of Latin hypercube sampling. First, the method is general: the same interface can be used with a wide variety of microphysical parameterizations for various processes.

This is because it is also difficult for a host model to provide accurate fields with which to drive the microphysics. This would be difficult even if microphysical formulas were known with perfect accuracy. Introduction: Microphysical parameterization in the presence of small-scale variabilityNumerical models of the atmosphere have difficulty representing small-scale cloud microphysical processes, such as drizzle formation. This paper leaves unanswered the question of whether the Latin hypercube method will work well in a prognostic, interactive cloud model, but this question will be addressed in a future manuscript.1. However, the instantaneous estimates are no less noisy.

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