Numerical models of weather and climate need to compute grid-box-averaged rates of physical processes such as microphysics. These averages are computed by integrating subgrid variability over a grid box. For this reason, an important aspect of atmospheric modeling is spatial integration over subgrid scales.

The needed integrals can be estimated by Monte Carlo integration. Monte Carlo integration is simple and general but requires many evaluations of the physical process rate. To reduce the number of function evaluations, this paper describes a new, flexible method of importance sampling. It divides the domain of integration into eight categories, such as the portion that contains both precipitation and cloud, or the portion that contains precipitation but no cloud. It then allows the modeler to prescribe the density of sample points within each of the eight categories.

The new method is incorporated into the Subgrid Importance Latin Hypercube Sampler (SILHS). The resulting method is tested on drizzling cumulus and stratocumulus cases. In the cumulus case, the sampling error can be considerably reduced by drawing more sample points from the region of rain evaporation.

Coarse-resolution atmospheric models of weather and climate do
not solve differential equations; they solve

Mathematically, the problem is to evaluate integrals of the form

To carry out this integration (i.e., “quadrature”), researchers
have proposed several methods. First, the
integral (Eq.

To improve the convergence of Monte Carlo integration, many methods
have been proposed. Two broad strategies are stratified sampling
and importance sampling

Some sampling methods combine stratified and importance sampling.
For instance, a prior version of the Subgrid Importance Latin
Hypercube Sampler (SILHS) placed sample points preferentially in
cloud, and also stratified the within-cloud sample points using
Latin hypercube sampling

This paper proposes a new importance sampling method that is highly
flexible. It divides the domain of integration into

This paper will introduce nCat sampling and evaluate it in an
idealized, single-column setting. Section

SILHS does not

CLUBB's PDF is multivariate. It includes several variates (i.e.,
variables) that are useful inputs to thermodynamical and
microphysical calculations. One of the PDF's variates is the
extended cloud (liquid) water mass mixing ratio (

The functional form of CLUBB's PDF is a compromise between realism
and mathematical simplicity. CLUBB's PDF may be written as

In the notation used
above,

In both the new and prior versions of SILHS, sample points are drawn from CLUBB's PDF and fed into subroutines that compute microphysical process rates. To reduce the noise associated with the random sampling of processes, both versions of SILHS incorporate stratified sampling (specifically, Latin hypercube sampling) and importance sampling.

The Latin hypercube algorithm is described in many sources

Importance sampling is useful when a process rate is particularly
large and variable within a small portion of the sample space.
For instance, autoconversion of cloud droplets occurs only within
cloud, which in cumulus cases often occupies a small fraction of
the domain. Without importance sampling, the density of sample
points in the sample space is given by the PDF

The prior version of SILHS

In both the new and prior versions of SILHS, a sample is first
drawn from a starting grid level. This grid level is the only grid
level where SILHS explicitly performs importance sampling; it is
called the “importance sampling level” in this paper.
The importance sampling level is chosen at each timestep to be the height
level with the maximum within-cloud cloud water mass mixing ratio. To
represent vertical overlap, sample points at other height levels
are drawn such that they are correlated with adjacent levels
according to a correlation coefficient that decreases
exponentially with increasing height

For each of the eight categories, this table lists (1) the category
number; (2) whether the category is cloudy, in mixture component 1 or 2, or
precipitating; (3) what inequalities must be satisfied for a sample point to
lie within the category; and (4) the original probability mass associated
with the category,

The nCat flexible importance sampling method is a generalization of the original SILHS importance sampling method described above. It is designed to give the modeler finer control over which parts of the subgrid PDF are preferentially sampled, that is, regarded as “important”.

First, the domain of the PDF is split into a set of disjoint
categories,

In this paper, eight categories are used. The definitions of the
categories are based on the following three criteria: in/out of
cloud, in mixture component 1/2, and in/out of
precipitation. A sample point lies in cloud if and only if

Each category

In general, the

The notation introduced so far in this section relates to the PDF
itself, rather than importance sampling per se. In order to
implement importance sampling, we sample what we regard as the
“important” categories preferentially. To do so, we introduce
for each category,

To draw sample points from the

Importance sampling allows the modeler to concentrate sample points in areas of the sample space that are considered important. But sample points given to important areas are taken from unimportant areas. Therefore, if importance sampling is applied overzealously, the less important processes can become excessively noisy.

In SILHS, we wish to employ an importance sampling scheme that improves results for important processes (e.g., certain microphysical processes) while still producing reasonably accurate estimates of other “less important” or perhaps less variable processes. One reason that we wish to avoid overdoing the importance sampling is that a favorable sampling distribution at one grid level (altitude) may be unfavorable at another.

The change in accuracy for a given category due to importance
sampling can be assessed by noting the weight of sample points in
that category. The inverse weight of a sample point in category

In order to mitigate the negative impact of importance sampling, we
now propose a simple method to impose a maximum weight,

Formally, the algorithm is constructed as follows. We compute the
difference between the category's modified probability,

In SILHS, we currently set

The success of the nCat method depends on knowing how to allocate
sample points to the categories. In some cases, it is easy to see
how to allocate points. For example, if it is known that the
process(es) of interest are active in only one of the

For a given process rate,

One could prescribe the modified probabilities

More generally, prescribing each

Specifically, we prescribe the following normalized standard
deviation of the process rate for each category

Given the

The prescription (Eq.

As compared to the previous version of SILHS, the chief advantage
of the new nCat method is its flexibility. In particular, the
user can individually prescribe the sampling density per unit
probability (

This flexibility is made possible in part by the fact that the
nCat method imposes no restriction on the number of sample points
used per timestep. The previous version of SILHS required an even
number of sample points per timestep, because one point was placed
in cloud and the other was placed outside cloud. Generalizing this
method to eight categories would have required a multiple of at
least eight sample points per timestep, and would not allow much
flexibility in prescribing the relative importance of
categories. Instead, the nCat method uses a probabilistic approach
to picking a category for each sample point. This allows any
number of sample points to be used at each timestep, including the use of fewer than

In summary, to implement the new importance sampling method, the
following steps should be taken.

Pick a set of categories,

Pick a set of sampling fractions,

Compute, from the fractions

Compute the weight in each category,

Feed the (unweighted) sample points one by one into a physical parameterization (e.g., a microphysics scheme).

Compute a weighted average of the function of interest using
Eq. (

Notable configuration settings for the RICO and DYCOMS-II RF02 simulations performed in this paper.

In order to evaluate how well the new importance sampling scheme
simulates multiple cloud types, we have simulated two cloud
cases. The first is a drizzling shallow cumulus case: Rain in
shallow Cumulus over the Ocean (RICO), configured as in the
intercomparison of

The following four configurations of SILHS were used for
comparison.

“LH-only”. This configuration uses only Latin hypercube sampling. No importance sampling is performed. The nCat method is not used.

“2Cat-Cld”. This configuration is functionally equivalent to the old
version of SILHS that placed one point in cloud and one point out of cloud.
This configuration uses two categories: in cloud, and out of cloud. The
categories (c,p,1), (c,p,2), (c,np,1), and (c,np,2) are all lumped together
into the “cloud” category, and the other four categories are analogously
lumped into the “clear” category. That is, a point that is in cloud belongs
to the cloud category regardless of whether it is in precipitation or which
mixture component it is in, and similarly for points in clear air. When cloud
fraction is between 0.5 and 50 %, it places 50 % of sample points in each
of the two categories. (That is,

“2Cat-CldPcp”. This configuration also uses two categories. The first
consists of points that are either in cloud or in precipitation, and the
second consists of the complement, namely, points that are neither in cloud
nor in precipitation. That is, (nc,np,1) and (nc,np,2) are lumped into the
no-cloud-or-precipitation category, and the others are lumped into the
cloud-or-precipitation category. Since no microphysical processes act in the
area of the domain outside of cloud and precipitation, the sample points are
initially prescribed such that all points fall in the cloud-or-precipitation
category (i.e., the first category). (That is,

“8Cat”. This configuration uses all eight categories listed in
Table

The microphysics scheme
used in the simulations is that of

Each SILHS configuration was evaluated on its ability to estimate the
following three microphysical processes.

Autoconversion: the conversion of cloud water to rain water. This process occurs within cloud, both inside and outside of precipitation (rain).

Accretion: the growth of rain droplets by collection of cloud water. This process occurs when both cloud and precipitation are present.

Evaporation: the conversion of rain water to water vapor. This process occurs in areas outside cloud but within precipitation.

In this section, we present results obtained using the new importance sampling method.

Prescribing the

Estimated optimal sampling fractions (

We see that in both cases, the optimal

Figure

RICO: the root-mean-square error (RMSE) at the importance sampling level of SILHS simulations as a function of the number of sample points, for the RICO cumulus case. The error is time-averaged over the entire simulation. The 2Cat-CldPcp and 8Cat methods show a large improvement over the 2Cat-Cld and LH-only methods in the estimate of evaporation, but not for autoconvesion and accretion, which are in-cloud processes. Nevertheless, the 8Cat and 2Cat-CldPcp methods both impove the estimate of the sum of the three processes.

The largest improvement of the 8Cat and 2Cat-CldPcp methods over the (old) 2Cat-Cld method is in sampling evaporation. In fact, even the LH-only method (no importance sampling at all) results in a better estimate of evaporation than the 2Cat-Cld method. The reason that evaporation is so poorly sampled in the 2Cat-Cld method is that the 2Cat-Cld method performs importance sampling only within cloud. Indeed, for in-cloud processes, such as autoconversion and accretion, the 2Cat-Cld method equals or improves upon both the 2Cat-CldPcp method and the 8Cat method in the RICO cumulus case. However, the 2Cat-Cld method reduces the number of sample points outside of cloud, degrading the simulation of rain evaporation. In contrast, both the 2Cat-CldPcp and 8Cat methods preferentially sample within the region of the sample space containing evaporation (out of cloud but within precipitation), leading to large improvements.

Table

RICO: percentage of sample points allocated to each category by each sampling method at the importance sampling level, time-averaged over the entire simulation. The more sample points placed in a particular category, the better the estimate of processes active in that category.

Figure

RICO: time-series plots of the four tendencies at the importance sampling level. The simulations in these plots use 32 sample points, and the plots show minutes 3321 to 4320 of the simulations. To improve readability, the LH-only method is not plotted. The evaporation tendencies are much more noisy in the 2Cat-Cld method than in the 2Cat-CldPcp or 8Cat methods.

To assess the performance of the sampling methods at levels away from the
importance sampling level, profile plots (over height levels) were generated
for simulations with 32 sample points. To reduce the role of a “lucky”
random seed in the comparison and thereby better distinguish the methods, an
ensemble of 12 simulations was used. Figure

RICO: mean profile plots of the four tendencies. The simulations in these plots use 32 sample points. For each configuration, an ensemble of 12 simulations is used, each with a different seed. Profiles are averaged over all 864 timesteps of the simulation and all 12 ensemble members. It is seen that all SILHS sampling methods are clearly convergent at all height levels.

We note that, in this paper, only the profile plots display an ensemble average. The time-series plots display a single simulation so that individual sample values can be seen. The plots displaying RMSE vs. the number of sample points are not strongly influenced by the choice of random seed.

The other simulated case is DYCOMS-II RF02, a drizzling
stratocumulus case. Figure

RICO: profile RMSE plots of the four tendencies. The simulations in
these plots use 32 sample points. For each configuration, an ensemble of 12
simulations is used, each with a different seed. RMSE values are averaged
over all 864 timesteps of the simulation and all 12 ensemble members. The
8Cat and 2Cat-CldPcp methods show improvement between 1000

DYCOMS-II RF02: the root-mean-square error (RMSE) of SILHS
simulations as a function of sample points for the DYCOMS-II RF02
stratocumulus case. The error is calculated at the importance sampling level
and is averaged over all timesteps of the simulation. The LH-only, 2Cat-Cld,
and 2Cat-CldPcp methods are expected to have roughly the same behavior in
a case like DYCOMS-II RF02 that has cloud fraction near 100

The similarity between the LH-only, 2Cat-Cld, and 2Cat-CldPcp
methods is expected. The 2Cat-Cld method, like the previous version
of SILHS, includes a condition that reverts to straight Latin
hypercube sampling in the event that cloud fraction exceeds
50

DYCOMS-II RF02. Percentage of sample points allocated to each category by each sampling method at the importance sampling level, time-averaged over the entire simulation. The more sample points placed in a particular category, the better the estimate of processes active in that category.

The reason for the improvement using the 8Cat method can be inferred from
Table

Figure

DYCOMS-II RF02: time-series plots of the four tendencies at the importance sampling level. The simulations in these plots use 32 sample points. The time range plotted includes minutes 161 to 360 of the simulation. The evaporation process is poorly sampled in all three sampling methods, but it is a relatively small term and makes a much smaller contribution to the sum of the three processes than autoconversion and accretion.

Number of sample points needed by each configuration of SILHS to
achieve a given RMSE in estimating the sum of the three processes, for the
RICO and DYCOMS-II RF02 cases. These numbers are estimated visually from
Figs.

Figure

DYCOMS-II RF02: mean profile plots of the four tendencies. The simulations in these plots use 32 sample points. For each configuration, an ensemble of 12 simulations is used, each with a different seed. Profiles are averaged over all 360 timesteps of the simulation and all 12 ensemble members. It is seen that all SILHS sampling methods are clearly convergent at all height levels. All of the lines overlap well, indicating that all three processes are sampled well by all three sampling methods.

DYCOMS-II RF02: profile RMSE plots of the four tendencies. The
simulations in these plots use 32 sample points. For each configuration, an
ensemble of 12 simulations is used, each with a different seed. RMSE values
are averaged over all 360 timesteps of the simulation and all 12 ensemble
members. All sampling methods show the largest RMSE at around 800

Table

An important consideration among Monte Carlo integration methods is their
computational cost. The cost of the new nCat method was tested against both
the prior SILHS importance sampling method and the cost of CLUBB. Eight SILHS
sample points were used in each simulation. Five RICO simulations were
performed, and Table

These costs may be compared with other costs in global climate simulations.
To this end,

We have developed a new (“nCat”) method to sample subgrid
variability in atmospheric models. The method divides the grid box
sample space into

The most flexible variant of the nCat method that we consider here breaks the grid box into eight categories, depending on whether a parcel contains cloud droplets or rain droplets, or is within the first mixture component of the PDF. This “8Cat” variant allows a fine degree of control over where the samples are placed.

Another variant has been created by lumping the eight separate categories into two: one that contains either cloud or precipitation, and one that contains neither cloud nor precipitation. This (“2Cat-CldPcp”) variant is useful when the user does not have an estimate of the optimal sampling fraction for each of the eight categories.

We have tested the 8Cat and 2Cat-CldPcp methods on a drizzling
cumulus case (RICO) and a drizzling stratocumulus case (DYCOMS-II
RF02). The improvement we find relies on two aspects of the
method. One aspect is an algorithm that limits the weight of
samples and thereby increases the number of samples in
“unimportant” but large-probability categories. This helps
prevent a user from becoming overzealous with importance sampling,
thereby leaving excessive noise in “unimportant” categories.
Another aspect is the choice of sampling variable to prescribe. We
prescribe

Cumulative run time of CLUBB and the different SILHS configurations over an 864-timestep RICO simulation. Each SILHS configuration uses eight sample points. The means and standard deviations of five simulations are shown in the table. All times are in seconds. The two nCat methods (2Cat-CldPcp and 8Cat) show no significant increase in computation time as compared to the original SILHS importance sampling method. All SILHS methods, with eight sample points, are more expensive than CLUBB.

The finer degree of control over the sampling in the nCat method allows us to improve sampling in evaporating (i.e., precipitating but non-cloudy) regions. This turns out to be a key to the improvement in the results. Evaporation of precipitation is an important process in the RICO case, but precipitation evaporates within only a small portion of a grid box, a portion that the nCat method can preferentially sample. Such fine-scale control of the sampling is not possible in less flexible methods, such as the former method in SILHS, 2Cat-Cld, which does not allow importance sampling on precipitation.

Quantitative improvements are realized by the 2Cat-CldPcp and
especially the 8Cat allocations. As compared to the 2Cat-Cld
method, the 8Cat allocation allows a reduction in the number of
sample points, given equal accuracy in the tendency of
autoconversion plus accretion plus rain evaporation. The reduction
is approximately a factor of 1.6 in DYCOMS-II RF02 and a factor of
8 in RICO (see Figs.

The CLUBB-SILHS code is freely available for non-commercial use
after registering for an account on the website

The modified integral that is estimated by using importance sampling is given
in Eq. (

We would like to find values of

The authors are grateful for financial support by the Office of Science, US Department of Energy, under grant DE-SC0008323 (Scientific Discoveries through Advanced Computing, SciDAC) and support by the National Science Foundation under grant AGS-0968640. This manuscript benefitted from the helpful comments of the editor, Klaus Gierens, and of two anonymous reviewers. Edited by: K. Gierens