## 1. Introduction

The resolved scale of atmospheric models is limited by their finite temporal and spatial resolutions. For example, the horizontal resolution of operational state-of-the-art weather and climate global circulation models (GCMs) is on the order of ~10–100 km. As a result of the nonlinear nature of atmospheric dynamics and physics, unresolved processes significantly impact the resolved scale flow. Therefore, these unresolved processes need to be represented with simplified models, which are called parameterizations. In a typical atmospheric model, the parameterizations represent planetary boundary layer turbulence, moist convection, micro and macrophysical cloud processes, radiation and gravity wave drag, among other processes. All these parameterized processes are highly nonlinear and many are coupled with each other as well as with the resolved flow. Recent studies (e.g., Watanabe et al. 2012; Cesana et al. 2017, and references within) suggest that the uncertainty of subgrid-scale parameterizations are a major factor in uncertainty of simulated climate and also climate response to anthropogenic forcing. For example, Cesana et al. (2017) show that the ensemble spread of cloud and radiative response to an increase of the sea surface temperature from leading GCMs can be reproduced with a single GCM with different parameterizations. Therefore, improved parameterizations are likely to improve the skill and increase the confidence of climate and weather predictions. One way of parameterization improvement is to understand the sources of uncertainty in parameterizations and to determine which observations may be used to constrain them.

Turbulence, convection and cloud parameterizations are often developed and first validated for a few representative cases in a single-column-model (SCM) framework. For such validation, either observations or numerical results from high-resolution large-eddy simulations (LES) or cloud-resolving models (CRM) are used (e.g., Randall et al. 1996). Utilization of LES or CRM results is often more convenient because these high-resolution numerical results provide quantities that are difficult or impossible to measure accurately (e.g., vertical profiles of turbulent fluxes, subgrid-scale distributions of dynamic and thermodynamic variables, turbulent kinetic energy), and the uncertainty in initial conditions and forcing can be minimized. To help with development and validation of turbulence, convection and cloud parameterization, Smalley et al. (2019) designed a SCM observational test bed for parameterization development and validation using observations from the A-Train satellite constellations. They apply their framework to the Jet Propulsion Laboratory stochastic multiplume Eddy-Diffusivity/Mass-Flux (JPL-EDMF) parameterization (Suselj et al. 2012, 2013, 2014, 2019a,b, 2020, manuscript submitted to *J. Atmos. Sci.*, hereafter SLK) and study its response to various conditions and forcing that might be encountered in atmospheric models. The key idea of their approach is to extend the model validation from a few cases to over 1000, rigorously testing the model to a wide variety of environmental conditions. Their work focuses on a subtropical marine region characterized by transition of stratocumulus to shallow convection, which plays an important role in Earth’s energy budget. Many atmospheric models struggle to represent this cloud transition accurately (e.g., Teixeira 1999; Karlsson et al. 2008, 2010; Teixeira et al. 2011), presumably due to biases in parameterizations. Smalley et al. (2019) also use the SCM observational test bed to study the sensitivity of the JPL-EDMF results to representation of certain physical processes. The capability of this method is however fundamentally limited by (i) the available observations and (ii) the ad hoc selection of both the available observations and parameters to constrain.

The goal of this work is to develop methodology for identification of candidate observables (together with the required error characteristics and vertical resolution) that best constrain representation of most uncertain physical processes in the parameterizations of boundary layer, convection and cloud macrophysics in the SCM framework. In this work, the uncertainty of parameterized processes is characterized by the uncertainty of model parameters. This goal is achieved by performing three tasks: (i) rank the parameterized processes in terms of their impact on JPL-EDMF parameterization results (section 3b), (ii) identify which observables best constrain these influential processes (section 3c), and (iii) investigate the sensitivity of the JPL-EDMF results to the measurement error and vertical resolution of observables that constrain influential processes (section 3d). The objective is to contribute to the development of a future observational system that could be used to optimally constrain physical parameterizations in the Smalley et al. (2019) framework or in the fully three-dimensional model. This methodology is applied to the version of the JPL-EDMF parameterization described in SLK and summarized in section 2 and the appendixes of this work. Various versions of the JPL-EDMF parameterization have been implemented in different atmospheric models (Suselj et al. 2014; Angevine et al. 2018; Bhattacharya et al. 2018; Olson et al. 2019; Kurowski et al. 2019; Wu et al. 2020). We focus on two nonprecipitating cases representative of subtropical cumulus and stratocumulus convection that are based on measurements during the Rain in Cumulus over the Ocean field campaign (RICO case; van Zanten et al. 2011) and by the second research flight of the second Dynamics and Chemistry of Marine Stratocumulus Field Study (DYCOMS case; Ackerman et al. 2009), respectively. For the second and the third task an ensemble of LES results of the nonprecipitating versions of RICO case from van Zanten et al. (2011) and DYCOMS case from Ackerman et al. (2009) are used instead of observations.

The paper is organized as follows. In section 2 and in the appendixes an overview of the JPL-EDMF parameterization model is given. Section 3 provides a description of the studied cases, details on the methodology and the results of the aforementioned three tasks. A summary and conclusions are given in section 4.

## 2. Overview of JPL-EDMF parameterization model

The methodology developed in this work is applied to the nonprecipitating version of the stochastic multiplume JPL-EDMF parameterization model implemented in an SCM. A basic overview of the JPL-EDMF model and SCM are given below; for details see the appendixes and Suselj et al. (2013, 2019a,b, SLK).

The SCM is essentially a single grid column of the GCM and it solves prognostic equations for the grid-mean values of moist conserved thermodynamic variables, horizontal momentum, turbulent kinetic energy (TKE) and the saturation excess variance. For this, the SCM requires information about the subgrid-scale fluxes and the cloud properties (such as liquid water content and cloud fraction), which are computed by the JPL-EDMF model. The JPL-EDMF model splits the subgrid-scale domain into multiple convective updrafts and the remaining nonconvective environment. The quantities required by the SCM are computed somewhat differently for the two components, as described below.

If the surface buoyancy is positive, the JPL-EDMF parameterization model initializes multiple convective updrafts at the surface. There, each updraft is associated with its unique fractional area and values of thermodynamic and kinematic properties. Fractional area of the individual updraft is assumed to be constant from the surface until its termination height (which is the height where its vertical velocity ceases, and can be different for each of the updrafts). A set of coupled steady-state equations govern vertical evolution of updraft properties, which include moist conserved thermodynamic variables and velocity components. We assume a horizontally uniform distribution of these properties within each individual updraft. While ascending, the updrafts are subject to lateral entrainment of environmental air. The entrainment rate is unique for each updraft and is parameterized as a superposition of discrete stochastic entrainment events. Cloudiness and liquid water content within the updrafts are computed following saturation adjustment and the assumed uniform distribution of thermodynamic properties within each updraft.

The JPL-EDMF parameterization model assumes normal distribution of key thermodynamic properties in the nonconvective environment, which are used to compute partial cloudiness and cloud liquid water content. The environmental turbulent fluxes are computed following the eddy-diffusivity approximation and are a function of environmental TKE, which is diagnosed from the grid-mean TKE. The grid-mean cloud properties are computed as an area-weighted average of the updraft and environmental values. The turbulent fluxes follow the eddy-diffusivity/mass-flux approach, which combines contributions from the environment and updrafts and relies on the assumption of uniform distributions in individual updrafts.

We identified 16 parameters in the JPL-EDMF parameterization model that potentially impact the model results. These parameters, together with their default values, their ranges and short descriptions, are listed in Table 1, while the appendixes provide more details. We believe that for most of the parameters it is nearly impossible to objectively define a range of potential values. This is because, in addition to uncertainty arising from the simplified and incomplete description of reality captured by the JPL-EDMF model, our understanding of the underlying physical processes is also imperfect. For these reasons, we used our expert judgment to define the parameter ranges. The default values for some of the parameters [i.e., *c*(*w*, *q*_{t}), *c*(*w*, *θ*_{υ}), *α*_{w}, *a*_{s}] are based on measurements or theoretical considerations, while for the others they are simply near the middle of the range. Note that the default parameter values are different from the ones in SLK.

List of the JPL-EDMF model parameters varied in this work, together with the parameter range, default value, and short description. The upper part of the table shows parameters used for the updraft parameterization, and the lower part shows parameters used for the nonconvective environment.

## 3. Methodology and results

### a. Description of studied cases

We study the nonprecipitating versions of two cases that epitomize marine cumulus and stratocumulus convection. In this work, the results from LES models for these two cases are used as a reference data. The first case represents slowly deepening cumulus convection observed during the RICO campaign in the trades over the western Atlantic. The convective evolution for the RICO case is simulated for 24 h, as defined by the reference LES results. Here, we analyze the time averaged results between the 23rd and 24th simulation hour. For this case, the constant sea surface temperature, surface roughness, subsidence velocity and tendencies of thermodynamic variables from horizontal advection and radiation (for liquid-water potential temperature only) are prescribed. The setup as well as the reference results, which consist of 13 different LES model outputs simulated with 40 m horizontal and vertical grid spacing, are taken from the nonprecipitating version described by van Zanten et al. (2011). The second case represents a steady-state, nonprecipitating version of coastal marine stratocumulus and is based on measurements off the coast of California during the second research flight of the second phase of the DYCOMS field study. As this simulation reaches steady-state conditions relatively quickly, the case is simulated only for 6 h, and as for the RICO case, we analyze the results averaged over the last simulation hour. The setup and the LES reference data are both taken as nonprecipitating case from Ackerman et al. (2009). The DYCOMS reference data consist of results from 14 different LES models simulated with 50 m horizontal and variable vertical grid spacing. In this case, the surface fluxes, subsidence velocity and horizontal advective tendencies are prescribed. For both cases, the LES models are initialized and forced the same way and therefore the differences among their results reflect the differences in their dynamical solvers and subgrid parameterizations.

Even though the JPL-EDMF and LES models can represent microphysical cloud processes and thus can model precipitation, we study the nonprecipitating versions of these two cases. The reasons for this are that, first, for these two cases, microphysical processes do not significantly impact the thermodynamical structure of the atmosphere including turbulence, convection and clouds, which can be seen as the small difference of these atmospheric properties between the LES simulations with active and inactive microphysical processes (van Zanten et al. 2011; Ackerman et al. 2009). Therefore, to the first approximation microphysical processes can be studied separately for these two cases. Second, we have less confidence in the LES representation of microphysical processes because of the large spread in the surface rain rate from the LES ensembles for these two cases.

For each of the relevant LES outputs, we compute the median and interquartile range value from the ensemble, which we take to serve as the representative value and its uncertainty range, respectively. LES data are characterized with these two quantities instead of mean and the standard deviation, because the selected quantities are less susceptible to the outliers (e.g., Wilks 2011). For both cases, Fig. 1 shows the LES median values and interquartile ranges that are used for EDMF parameter estimation and for investigation of measurement requirement for parameter constraint (sections 3c and 3d, respectively). These outputs include the profiles of thermodynamic variables (water vapor and potential temperature), cloud properties (liquid water and cloud cover) and the turbulent fluxes of moist thermodynamic variables (turbulent fluxes of total-water mixing ratio and liquid-water potential temperature). Note that for visualization purpose we plot the profile of potential temperature (Figs. 1b,h) while temperature is used in this study. The temperature is obtained by multiplication of potential temperature with Exner function, for which we assume the same profile in all LES models. The LES values of integrated outputs used for parameter estimation are shown in the next section.

In the RICO case, the relatively well-mixed subcloud layer with the depth of approximately 500 m is capped by the cloud layer that extends to the height of approximately 3000 m (Figs. 1a–d). The uncertainty range of the water vapor and potential temperature profile is small throughout most of the subcloud and cloud layer (around 0.15 g kg^{−1} and 0.2 K, respectively) and it peaks in the upper cloud layer between around 2000 and 2500 m (Figs. 1a,b). This peak seems to reflect somewhat different cloud top heights and strength of detrainment of convective clouds in different LES models. Both cloud properties (i.e., the liquid water and the cloud fraction) are small throughout the cloud layer and are most uncertain in the upper cloud layer (Figs. 1c,d). Even though the absolute values of uncertainty ranges of cloud properties are small, they are both between 25% and 50% of their median values throughout most of the cloud layer. The uncertainty range of turbulent fluxes of moist thermodynamic variables is approximately 10% of the median values throughout the convective layer (Figs. 1e,f).

In the DYCOMS case, the boundary layer extends to the height of approximately 900 m and the upper part (above around 500 m) is dominated by cloudiness (Figs. 1g–i). The uncertainty range of water vapor and potential temperature within the boundary layer is around 0.05 g kg^{−1} and 0.2 K, respectively, and peaks at the top of the cloud layer (Figs. 1g,h). This peak reflects primarily the differences of the depth of the cloud layer among the LES models. The liquid water mixing ratio increases with the height from the cloud base, peaks at approximately 0.5 g kg^{−1} just below the cloud top and its uncertainty range is approximately 0.1 g kg^{−1} throughout most of the cloud layer (Fig. 1i). The cloud fraction reaches 100% between the heights of approximately 600 and 900 m, and is the most uncertain at the cloud base and at the cloud top (Fig. 1j). The uncertainty range of flux of total water mixing ratio is around 7 W m^{−2} throughout most of the cloud layer (Fig. 1k) and the uncertainty range of flux of liquid water potential temperature linearly increases with height and reaches maximum value of around 15 W m^{−2} in the lower cloud layer.

For both cases, the JPL-EDMF parameterization model in the SCM is run with a time step of 20 s and a vertical grid spacing of 20 m, but the results are aggregated on 40 m resolution for comparison to the LES outputs. In the SCM no explicit horizontal resolution is associated with these simulations. The depth of the SCM vertical domain is 4000 m, which is enough to encapsulate PBL and moist convective layers for these two cases. However, for the analysis we consider only the lowest 3500 m for RICO case and 1200 m for DYCOMS case. For brevity, in the rest of this work the JPL-EDMF parameterization model in the SCM will be referred as the JPL-EDMF model.

### b. Screening for influential parameters (MOAT analysis)

Using the probabilistic Morris (1991) one-at-the-time (MOAT) parameter screening method refined by Covey et al. (2013), Posselt et al. (2019) and Morales et al. (2019) we first identify JPL-EDMF parameters that are highly influential for selected model results. By “model results” here we mean a collection of *quantities of interest* (QIs) that characterize atmospheric conditions that the JPL-EDMF model is designed to represent, and influential parameters are those whose uncertainties most strongly impact QIs. The MOAT method is computationally efficient for highly nonlinear parameter impact and we use it to decrease the dimension of the JPL-EDMF model parameter space that warrants further investigation. Because it is well described in the above-cited references, here we only outline its key ideas and describe its application to our JPL-EDMF model.

As a first step, one needs to define a set of model parameters that are suspected to have an important impact on QIs, along with the range of possible values that each parameter can take. The parameter default values and possible ranges are listed in Table 1. If the number of parameters is denoted by *N*, the parameter space is represented by an *N*-dimensional hyper-rectangle. This parameter space is approximated with a discrete multidimensional parameter lattice with *L* equidistant points along each of the *N* dimensions. Here we take *L* = 20 but the results do not seem to be sensitive to the exact value. Instead of estimating parameter influence on QIs by evaluating the model for all *L*^{N} points defined by the parameter lattice, MOAT requires model evaluation only along certain paths on the parameter lattice. The first node of each path is a random starting point on the parameter lattice. The second node is a step of length Δ along the dimension spanned by the first parameter. The other nodes on the path are a step of length Δ along one of the dimensions from the previous node. The *n* + 1th node is a step Δ along the *n*th dimension from the previous node. The MOAT approach is called *one-at-the-time* because nodes along the path differ in the value of only one parameter, which simplifies evaluation of impact of parameter uncertainty on QIs. The increment Δ is chosen to be one-half of the width of the corresponding parameter range, so that the direction (positive or negative) of the step is uniquely determined if the parameter values are to stay within the prescribed range. This procedure is repeated for *M* number of paths, each with a different randomly chosen starting point. Even though Covey et al. (2013) showed that generally *M* = 20 is sufficient for the MOAT results to converge, we use *M* = 30. Therefore, the MOAT analysis requires *M* × (*N* + 1) model evaluations, which for moderate values of *M* and large values of *N* is many orders of magnitude smaller than model evaluation on all *L*^{N} nodes. For example, in our case *N* = 16 and therefore MOAT requires only 510 model evaluations, while the number of all nodes on the parameter lattice is approximately 6.5 × 10^{20}.

For each parameter *n*, the impact of its uncertainty (i.e., what is called the *elementary effect* in Covey et al. 2013) on the QI *d* along the *m*th MOAT path is expressed as: *δd*_{m,n} = *d*_{m,n+1} − *d*_{m,n}. Here *d*_{m,n} represents the JPL-EDMF simulated QI *d* with parameter values defined by the *n*th node along the *m*th MOAT path. The *δd*_{m,n} is the elementary effect of *n* on QI *d* because the *n*th and *n* + 1th node of the *m*th path differ only in the value of parameter *n*. The magnitude of the elementary effect, and all statistical measures derived from it, are expected to be inversely proportional to the predefined parameter range because the distance between the two nodes is half of the parameter range (Δ). The mean magnitude of the elementary effect of parameter *n* for QI *d* can be approximated following Covey et al. (2013) as

The standard deviation of an elementary effect is defined as

where its mean value is:

The standard deviation of the elementary effect, *d* to varying parameter *n*, and the response of QI *d* to the interaction between parameter *n* and the other model parameters. Separation of these two contributions is beyond the purpose of this work.

As in Posselt et al. (2019) we investigate three different criteria derived from values of

The goal of the MOAT analysis is simply to identify the parameters to which the model is most sensitive. Therefore, we only assess sensitivity of the JPL-EDMF QIs to model parameters and do not compare them against LES outputs. QIs are therefore not restricted to variables that can be diagnosed from the LES outputs (or observations). Instead, they are defined to characterize the most important aspects of dry boundary and moist convective layers and can be grouped into three broad categories that represent:

*Thermodynamic structure of the atmosphere*characterized with vertical gradients of moist conserved thermodynamic variables: liquid water potential temperature and total water mixing ratio (Δ*θ*_{l}and Δ*q*_{t}, respectively). For the two studied cases, the gradients are taken over different parts of the convective layers in order to capture the aspects that are most challenging for the JPL-EDMF model to represent accurately. We need to arbitrarily decide on these heights because we cannot take these gradients over the whole boundary and convective layer as they are constrained by initial conditions and forcing.

Figure 2 shows the three MOAT criteria computed for each QI separately and averaged across all QIs for RICO (top row) and DYCOMS (bottom row). The simple averages across QIs are used only as a general guidance as they depend on selected QIs, and are sensible because all of these MOAT criteria are dimensionless. For both cases, almost all QIs are most sensitive to the parameter controlling the magnitude of lateral entrainment for convective updrafts [i.e., parameter *ϕ* in Eq. (B7)]. This is because for most QIs the rank of *R*_{ϕ} values in Fig. 2 are both close to one. For the RICO case, the exceptions are integrated TKE (∫*e dz*), convective mass-flux (mf) and convective liquid water path (cLWP). The integrated TKE is most sensitive to parameters controlling eddy-diffusivity length scale and in particular to the ones that define its dependence on vertical stability [i.e., parameters *N*_{0} and *α*_{τ} in Eq. (C3)], and to the parameter controlling dissipation rate of the TKE [parameter *a*_{diss} in Eq. (C6)]. The mf and cLWP are both most sensitive to updraft parameters, in particular to the parameter in the updraft vertical velocity equation [parameters *w*_{b} in Eq. (B4)] and less sensitive to parameters defining the magnitude and the intermittency of updraft entrainment rate [parameters *ϕ* and *s*_{f} from Eqs. (B7) and (B6)]. For the DYCOMS case the only notable exception is the integrated TKE, which is most sensitive to the parameter *a*_{diss}.

The MOAT criteria: (left) rank of *x* axes) and average across all QIs (last columns) for model parameters indicated by *y* axes. Results are for the (top) RICO case and (bottom) DYCOMS case.

Citation: Monthly Weather Review 148, 10; 10.1175/MWR-D-20-0114.1

The MOAT criteria: (left) rank of *x* axes) and average across all QIs (last columns) for model parameters indicated by *y* axes. Results are for the (top) RICO case and (bottom) DYCOMS case.

Citation: Monthly Weather Review 148, 10; 10.1175/MWR-D-20-0114.1

The MOAT criteria: (left) rank of *x* axes) and average across all QIs (last columns) for model parameters indicated by *y* axes. Results are for the (top) RICO case and (bottom) DYCOMS case.

Citation: Monthly Weather Review 148, 10; 10.1175/MWR-D-20-0114.1

The top-left two panels in Fig. 2 show that for the RICO case, in addition to the majority of QIs being highly sensitive to parameter *ϕ*, most of the QIs are also somewhat sensitive to parameters controlling intermittency of updraft entrainment rate (*s*_{f}), updraft vertical velocity equation (*w*_{b}), and dissipation rate of the TKE (*a*_{diss}). Except for the integrated TKE, none of the QIs for the RICO case are significantly sensitive to any of the other twelve JPL-EDMF parameters. The bottom-left two panels in Fig. 2 show that for the DYCOMS case most of the QIs are fairly sensitive to parameter *w*_{b} and to almost all of the parameters controlling the mixing length for the eddy-diffusivity part of the parameterization (and in particular to the two parameters *N*_{0} and *a*_{diff}), and to the dissipation rate of TKE (*a*_{diss}).

Based on the discussion above, we chose parameters *ϕ*, *w*_{b}, *a*_{diss}, and *s*_{f} to be most influential for the RICO case, and parameters *ϕ*, *w*_{b}, *a*_{diss}, *a*_{diff}, and *N*_{0} to be most influential for the DYCOMS case. In fact, as shown by the second column of Fig. 2 some of those influential parameters explain a relatively small fraction of the variability of QIs, or are responsible for significant variability of just one or two QIs. However, a decrease of sixteen parameters to four or five influential ones is sufficient for further analysis. As shown below, the noninfluential parameters have very limited impact on the QIs.

The fact that most JPL-EDMF QIs are highly sensitive to parameter *ϕ* agrees with our previous studies (Suselj et al. 2013; Smalley et al. 2019), which show that the JPL-EDMF results are highly sensitive to the entrainment rate parameterization. The entrainment rate parameterization has proven to be a difficult problem for virtually all mass-flux models (e.g., Angevine 2005; de Rooy et al. 2013) and it has a strong impact on many aspects of convection and circulation in GCMs (e.g., Wang et al. 2007; Oueslati and Bellon 2013). We might have expected that the JPL-EDMF results would not be sensitive to parameter *w*_{b}. This is because De Roode et al. (2012) shows that the updraft vertical velocity should be reasonably well modeled as long as the two updraft vertical velocity parameters (i.e., *w*_{a} and *w*_{b}) are related following Eq. (B5), which is the case for our experiments. The fact that our results disagree to some degree with the analysis of De Roode et al. (2012) is probably because in their study they only diagnostically estimate the two vertical velocity parameters, while here the vertical velocity formulation is coupled to model dynamics. We should also stress that we found much stronger sensitivity to parameter *w*_{b} for QIs in the stratocumulus (DYCOMS) case than in the cumulus (RICO) case, and the analysis of De Roode et al. (2012) is focused on cumulus-dominated cases. Overall, our results agree with Smalley et al. (2019) who argue that for shallow convective cases the JPL-EDMF results are most sensitive to the lateral entrainment rate parameterization, while for stratocumulus-dominated cases the parameterization of eddy-diffusivity length scale is important as well. Perhaps it might seem surprising that JPL-EDMF results are fairly insensitive to surface updraft fractional area (i.e., parameter *a*_{u}) as the convective mixing is essentially proportional to the updraft area. This result is however consistent with Suselj et al. (2019b) where we showed that the low sensitivity of convection to surface updraft area is a result of a balance between convection and large-scale forcing.

For both cases, the values of effect ratio, *ϕ* are consistently higher than the effect ratios of other influential parameters, with the exception of QIs that characterize moist physical processes in the RICO case. This result seems to indicate that of the influential parameters, *ϕ* seems to have the most consistently strong impact on the majority of QIs. The lowest values of effect ratios are found for the noninfluential parameters corresponding to the QIs that represent moist physical processes and TKE in the RICO case. These results point to possible residual impact of corresponding noninfluential parameters on these QIs. The effect ratios of influential QIs that characterize thermodynamic structure of atmosphere and moist processes are generally larger in the RICO compared to the DYCOMS case, which points to more linear response of the QIs to parameter change for RICO case. This is also confirmed by the results in section 3c, where the values of DYCOMS influential parameters constrained by reference LES data show large independence and nonlinearities.

Figures 3 and 4 show PDFs of QIs obtained from all nodes along the 30 MOAT paths for the RICO and DYCOMS cases, respectively. To confirm that QIs are fairly insensitive to noninfluential parameters, we recomputed the MOAT analysis varying only influential parameters while keeping the noninfluential ones fixed at their default values (given in Table 1). These two figures also show histograms of QIs from this modified MOAT analysis, which we refer to as PDFs with influential parameters, as opposed to the original analysis with full parameters. The PDF with influential parameters was obtained from a smaller number of JPL-EDMF simulations because the number of nodes per MOAT path was five for RICO and six for DYCOMS instead of 17 for the full parameter set. Figures 3 and 4 also show the values from the JPL-EDMF with the default parameter values, and where possible also the uncertainty range from the LES outputs. As described in section 3a the LES uncertainty ranges represents the interquartile ranges from the LES ensemble. Exceptions are convective LWP and mass-flux for RICO case, for which the LES range represents the ensemble means of cloud core and cloud sampling. This sampling has been used in previous studies to constrain convective properties for shallow convection case (e.g., Siebesma et al. 2003; Suselj et al. 2013), but because it is not designed for stratocumulus cases, we do not show these LES results for DYCOMS case.

RICO case. PDFs of QIs from JPL-EDMF model evaluations on all nodes along 30 MOAT paths for full parameter set (blue bars) and from influential parameter set (black lines). The red lines show the uncertainty range from the LES (i.e., for mf and cLWP the ensemble mean of cloud core and cloud sampling and for the rest of the QIs the interquartile range from the LES ensemble). The green lines are the values from JPL-EDMF model with default parameters.

Citation: Monthly Weather Review 148, 10; 10.1175/MWR-D-20-0114.1

RICO case. PDFs of QIs from JPL-EDMF model evaluations on all nodes along 30 MOAT paths for full parameter set (blue bars) and from influential parameter set (black lines). The red lines show the uncertainty range from the LES (i.e., for mf and cLWP the ensemble mean of cloud core and cloud sampling and for the rest of the QIs the interquartile range from the LES ensemble). The green lines are the values from JPL-EDMF model with default parameters.

Citation: Monthly Weather Review 148, 10; 10.1175/MWR-D-20-0114.1

RICO case. PDFs of QIs from JPL-EDMF model evaluations on all nodes along 30 MOAT paths for full parameter set (blue bars) and from influential parameter set (black lines). The red lines show the uncertainty range from the LES (i.e., for mf and cLWP the ensemble mean of cloud core and cloud sampling and for the rest of the QIs the interquartile range from the LES ensemble). The green lines are the values from JPL-EDMF model with default parameters.

Citation: Monthly Weather Review 148, 10; 10.1175/MWR-D-20-0114.1

DYCOMS case. PDFs of QIs from JPL-EDMF model evaluations on all nodes along 30 MOAT paths for full parameter set (blue bars) and from influential parameter set (black lines). The red lines show the uncertainty range from the LES (i.e., the interquartile range from the LES ensemble). The green lines are the values from JPL-EDMF model with default parameters.

Citation: Monthly Weather Review 148, 10; 10.1175/MWR-D-20-0114.1

DYCOMS case. PDFs of QIs from JPL-EDMF model evaluations on all nodes along 30 MOAT paths for full parameter set (blue bars) and from influential parameter set (black lines). The red lines show the uncertainty range from the LES (i.e., the interquartile range from the LES ensemble). The green lines are the values from JPL-EDMF model with default parameters.

Citation: Monthly Weather Review 148, 10; 10.1175/MWR-D-20-0114.1

DYCOMS case. PDFs of QIs from JPL-EDMF model evaluations on all nodes along 30 MOAT paths for full parameter set (blue bars) and from influential parameter set (black lines). The red lines show the uncertainty range from the LES (i.e., the interquartile range from the LES ensemble). The green lines are the values from JPL-EDMF model with default parameters.

Citation: Monthly Weather Review 148, 10; 10.1175/MWR-D-20-0114.1

The two most important results for both cases are that first, for most QIs the two MOAT PDFs with full and influential parameter sets are close to each other. This again confirms the unimportant role of the noninfluential parameters for the QIs. Second, the histograms of QIs from the JPL-EDMF show either a unimodal peak with the LES values close to this peak, or a nearly uniform PDF over some range with the LES values within this range. This second result indicates that the JPL-EDMF can represent the values of QIs found in the LES. The few exceptions are discussed below. The QIs with default JPL-EDMF values do not necessarily fall within the LES range. This is because most of the default JPL-EDMF parameter values are in the middle of the parameter range instead of being optimized for the two cases. The PDFs of QIs with the full parameter set are generally wider than the PDFs for the influential parameter set. This is because varying the noninfluential parameters somewhat increases the range of QIs. There are however a few exceptions (e.g., in Figs. 3a,b), which are due to inherently coarse and stochastic sampling of the parameter space with the MOAT analysis.

For both cases, the peak values of vertically integrated TKE (∫*e dz*; Figs. 3e and 4e) from both MOAT PDFs are outside the LES range and are generally underestimated compared to the LES results. This is consistent with Suselj et al. (2019b), who show that the JPL-EDMF tends to underestimate the LES TKE. However, in addition to turbulent motions, the LES TKE also includes a gravity wave contribution which is particularly noticeable in the conditionally unstable and stable layer above convective clouds. These gravity waves do not contribute to subgrid-scale mixing of thermodynamic variables and are not represented by the JPL-EDMF model. Therefore, it is unclear whether the LES vertically integrated TKE is a suitable reference for the JPL-EDMF TKE. The results from the JPL-EDMF with the default values are not necessarily close to the LES results, as the default values are not chosen to represent the two cases particularly well.

For RICO the other notable exceptions are as follows. The peaks of CC from both MOAT PDFs are at smaller values than the LES indicates (Fig. 3g); however, the LES values are well within the range of these PDFs. The influential MOAT PDF of vertically integrated TKE shifts to higher values compared to the full parameter MOAT PDF (Fig. 3e). This seems to reflect the fact that the TKE is somewhat sensitive to parameters *p*_{τ} and *α*_{τ} (see Fig. 2), which are not taken to be influential. The PDFs of turbulent fluxes from influential parameters do not reach small values found in the full parameter PDF set (Figs. 3c,d), while the lower values of *z*_{top} are more likely compared to the full set (Fig. 3j). The difference between these two parameter sets indicates that noninfluential parameters exert some small effect on those QIs. For DYCOMS, one notable feature is that the LWP from both MOAT PDFs shows two peaks, one close to zero and the other just below the LES range (Fig. 4h). The first peak corresponds to zero or small CC, and the second one agrees well with the LES range. The PDFs from Δ*q*_{t} and integrated turbulent flux of moisture (Figs. 4b,d) show a wide quasi-uniform PDF corresponding to different strengths of PBL mixing. The corresponding liquid water potential temperature PDFs are more peaked (Fig. 4c), showing the difference between water and heat mixing. Nevertheless, Figs. 3 and 4 show that the JPL-EDMF model parameters can be optimized to simulate QIs as obtained by the LES ensemble, and that the noninfluential parameters have only minor effect on most of the QIs.

### c. EDMF parameter estimation

Our next goal is to understand which observables best constrain the influential parameters. For this, the influential parameter space is again discretized with a multidimensional lattice and we perform JPL-EDMF simulations for parameter combinations on all points of this lattice. This is feasible to do only because (i) the dimensionality of the influential parameter space is much smaller than the dimensionality of the full parameter space, (ii) we discretize the parameter space with a relatively small number of points along each of the dimensions, and (iii) the JPL-EDMF is run in the SCM instead of the three-dimensional model. For the RICO and DYCOMS cases, we consider the four or five influential parameters, respectively, identified by the MOAT analysis. Because the MOAT analysis shows that the JPL-EDMF results are generally most sensitive to parameters *ϕ* and *w*_{b}, their ranges are discretized with ten or nine equidistant bins, respectively, and the ranges of other parameters are discretized with seven equidistant bins. In these simulations, the noninfluential parameters are kept at their default values given by Table 1. Comparing each set of JPL-EDMF simulation results with constraining observables derived from LES output (as a proxy for measurements), we compute the probability that these JPL-EDMF simulation results are consistent with the proxy measurements. This probability, which is computed via Bayes’s theorem assuming uniform priors, is then assigned to the corresponding point in the multidimensional parameter space.

Given observation of constraining observable *y*, the posterior distribution of a point characterized by influential parameters written in vector **n** is (Posselt 2016)

where *y* and its measurement error, respectively, and *y* with influential parameters given by **n**. Here, we use the median of the LES ensemble as a proxy for the measured values of constraining observable, and the one-half of the uncertainty range (i.e., interquartile range) of the LES ensemble as a proxy for measurement error. As described below we consider two types of observables. For profile observables, all data are obtained on vertical levels indicated with index *k* (*k* = 1, …, *K*), and for vertically integrated observables *K* = 1. Equation (5) follows the assumption of a normal measurement error distribution. The denominator represents the sum over all points and is a normalization factor that assures that the sum of probabilities over all points equals unity. To avoid division by zero in Eq. (5), we set the minimum value of *K* vertical levels. The results are not sensitive to that particular value (not shown).

For this work we consider two types of constraining observables: (i) the profile observables consist of vertical profiles of water vapor, temperature, liquid water mixing ratio, cloud fraction and turbulent fluxes of moist conserved variables; and (ii) the vertically integrated observables consist of liquid water path (LWP), cloud cover (CC), depth of the cloud layer (*z*_{top}), and vertically integrated turbulent fluxes. The representative values of profiles observables together with the error estimates are shown in Fig. 1, and the error estimates of the vertically integrated observables are shown in Figs. 3 and 4, for RICO and DYCOMS, respectively.

If the posterior distribution *p*^{n,y} constrained by observable *y* strongly deviates from the (uniform) prior, then this observable provides a strong constraint on the JPL-EDMF parameter values. One way of quantifying the information content of this PDF is with the Shannon (1948) information-entropy, which can be written as

where the sum is evaluated over all points on the influential parameter lattice. This information-entropy essentially measures the amount of uncertainty of the parameter values (e.g., Majda and Wang 2006), so that smallest *S*^{y} values indicate that observable *y* highly constrains parameter values. For illustration, we consider two examples. Parameter values are most uncertain (least well defined) if the *p*^{n} is a uniform PDF. If the number of points on the influential parameter lattice is *n*_{d} then *S* = log*n*_{d}. On the other hand, if the parameter values are fully determined (i.e., the *p*^{n} values are one for one of the points and equal zero for the rest of the *n*_{d} − 1 points), then *S* = 0.

For the two simulation cases, Fig. 5 shows Shannon information entropy of the posterior distribution of parameter values constrained by all considered observables. Because among other factors, the shape of the parameter PDF and therefore *S* values depend on the uncertainty of constraining observables and because this uncertainty is unique to the LES ensemble, comparison of *S* values for different constraining observables is used as a general guidance. In the section 3d we study the sensitivity of the EDMF results to the uncertainty of selected constraining observables. For both studied cases, all profile observables better constrain the JPL-EDMF parameters compared to the corresponding integrated observables (i.e., the values of *S* are consistently lower for the former than the latter). This is not surprising as the amount of information in profile observables is larger. The only exceptions are the cloud cover and cloud fraction for the RICO case, but neither of them strongly constrain parameter values. Unlike in the RICO case, the cloud fraction profile and cloud cover provide a very strong constraint on parameter values in the DYCOMS case. The different roles of these two observables for the two cases seem to be related to the differences in physical processes behind cloud formation, and also to the uncertainty of the cloud fraction from the reference LES. For the RICO case, the cloud formation is dominated by shallow convection (i.e., convective updrafts in the JPL-EDMF model) and the corresponding LES results are associated with large relative uncertainty (see section 3a). For the DYCOMS case, the cloud fraction is primarily a result of the stratiform cloudiness (i.e., condensation in the nonconvective environment) and not directly associated with convection, and is associated with relatively smaller LES uncertainty. Because the measure of observational uncertainty is taken from the spread in the LES results and is not related to a real observational system, we recomputed the *S* while multiplying the uncertainty by a factor of 0.5 and 2 (not shown). As expected the *S* values tend to decrease for all constraining observables when the uncertainty decreases and vice versa. However, the main conclusions discussed above are not sensitive to these modifications.

Shannon information entropy of posterior multidimensional PDF constrained by observables indicated by *x* axis. (left) RICO case and (right) DYCOMS case. Blue bars are for profile observables and red bars for the integrated observables. Horizontal black line indicates the value of the prior.

Citation: Monthly Weather Review 148, 10; 10.1175/MWR-D-20-0114.1

Shannon information entropy of posterior multidimensional PDF constrained by observables indicated by *x* axis. (left) RICO case and (right) DYCOMS case. Blue bars are for profile observables and red bars for the integrated observables. Horizontal black line indicates the value of the prior.

Citation: Monthly Weather Review 148, 10; 10.1175/MWR-D-20-0114.1

Shannon information entropy of posterior multidimensional PDF constrained by observables indicated by *x* axis. (left) RICO case and (right) DYCOMS case. Blue bars are for profile observables and red bars for the integrated observables. Horizontal black line indicates the value of the prior.

Citation: Monthly Weather Review 148, 10; 10.1175/MWR-D-20-0114.1

In the following analysis, we consider only the five observables that most tightly constrain parameter values (i.e., the ones corresponding to the lowest *S* values, estimated separately for the RICO and DYCOMS cases), which for both cases include the profiles of water vapor, liquid water and temperature. In addition to these, for RICO the turbulent flux of the liquid-water potential temperature and the depth of the cloud layer are considered, and for DYCOMS the cloud fraction and the cloud cover.

Next, we examine the shape of the posterior PDFs of the influential JPL-EDMF parameter values constrained by the five above-described observables. Because it is hard to visualize the four- or five-dimensional PDFs, we plot the marginal one-dimensional (1D) and two-dimensional (2D) PDFs. For both cases, the 2D PDFs are plotted only for observables corresponding to the lowest values of information entropy. The marginal PDFs are integrated multidimensional PDF values along the dimensions spanned by all the parameters not shown. On the 1D PDF plot, we also show the most likely parameter values, which correspond to the maximum probabilities in the multidimensional PDFs.

For the RICO case, Fig. 6 shows the marginal 1D PDFs constrained by the five observables, and Fig. 7 shows marginal 2D PDFs constrained by the profile of water vapor. Measurements of all shown observables strongly constrain values of parameter *ϕ* (Fig. 6a). In particular, the water vapor profile best determines the value of this parameter (which can be seen in the highest peak of the corresponding parameter PDF) which is consistent with lowest value of information entropy for the water vapor profile. All observables indicate that the most likely value of parameter *ϕ* is between 7 and 10, which is a range close to the default value. For most of the observables, the maximum value of parameter *ϕ* from the marginal 1D PDF is close to the maximum value from the 4D PDF. Note that for *z*_{top} the most likely value of parameter *ϕ* from the 1D PDF is lower than for most of other parameters and there is more than a single most likely value from the 4D PDF. This is because of multiple maxima in the 4D PDF indicating that for different parameter combinations, the depth of the cloud layer can be the same.

RICO case. Marginal 1D PDF of influential parameter values constrained by observations of profiles of water vapor (blue lines), liquid water (red lines), temperature (green lines), turbulent flux of liquid water potential temperature (orange line), and depth of the cloud layer (brown line). The vertical dashed lines show the default parameter values. The colored filled circles represent the maximum values at which the probability from the four-dimensional PDF is at its maxima. The horizontal dashed lines represent 1.5 and 2 times the prior.

Citation: Monthly Weather Review 148, 10; 10.1175/MWR-D-20-0114.1

RICO case. Marginal 1D PDF of influential parameter values constrained by observations of profiles of water vapor (blue lines), liquid water (red lines), temperature (green lines), turbulent flux of liquid water potential temperature (orange line), and depth of the cloud layer (brown line). The vertical dashed lines show the default parameter values. The colored filled circles represent the maximum values at which the probability from the four-dimensional PDF is at its maxima. The horizontal dashed lines represent 1.5 and 2 times the prior.

Citation: Monthly Weather Review 148, 10; 10.1175/MWR-D-20-0114.1

RICO case. Marginal 1D PDF of influential parameter values constrained by observations of profiles of water vapor (blue lines), liquid water (red lines), temperature (green lines), turbulent flux of liquid water potential temperature (orange line), and depth of the cloud layer (brown line). The vertical dashed lines show the default parameter values. The colored filled circles represent the maximum values at which the probability from the four-dimensional PDF is at its maxima. The horizontal dashed lines represent 1.5 and 2 times the prior.

Citation: Monthly Weather Review 148, 10; 10.1175/MWR-D-20-0114.1

RICO case. Marginal 2D PDF (in percent units) of all influential parameter combinations constrained by observations of water vapor profile. The black star indicates the default parameter values.

Citation: Monthly Weather Review 148, 10; 10.1175/MWR-D-20-0114.1

RICO case. Marginal 2D PDF (in percent units) of all influential parameter combinations constrained by observations of water vapor profile. The black star indicates the default parameter values.

Citation: Monthly Weather Review 148, 10; 10.1175/MWR-D-20-0114.1

RICO case. Marginal 2D PDF (in percent units) of all influential parameter combinations constrained by observations of water vapor profile. The black star indicates the default parameter values.

Citation: Monthly Weather Review 148, 10; 10.1175/MWR-D-20-0114.1

Investigating the other marginal 1D PDFs, it appears that none of the observables strongly constrain the other two parameters that in addition to parameter *ϕ* directly control the updraft properties (i.e., parameters *w*_{b} and *s*_{f}; Figs. 6b and 6d). Most observables indicate that the values of *w*_{b} are in the lower range and the values of *s*_{f} are in the higher range of the parameter values and relatively far from the default values. We argue that marginal 1D PDFs of these two parameters are not very informative for the other two updraft parameters. This is because most of the JPL-EDMF results are more highly sensitive to the parameter *ϕ*, and because the optimal values of the other two parameters are not independent of the value of that parameter as is revealed by the marginal 2D PDFs (Fig. 7). Figures 7a and 7b show that there is more than one optimal parameter combination of *ϕ*, *w*_{b}, and *s*_{f} and that this optimal combination lies in a certain two-dimensional plane in the three-dimensional parameter space (i.e., on the ridge where the values are high). In particular, 2D PDF shows a negative correlation between optimal values of parameters *ϕ* and *w*_{b} and positive correlation between parameters *ϕ* and *s*_{f}. Overall, these results indicate that parameter estimation should either consider all of these three updraft parameters simultaneously, or that if the parameters are estimated successively the values of the latter ones will be highly dependent on the previous ones, which might camouflage the physical meaning of the values of these parameters.

The other 2D parameter PDFs (i.e., Figures 7d–f) show that the most likely value of parameter *a*_{diss} is independent of those of the three other influential parameters. This parameter does not directly impact the updraft properties, but rather the mixing in the eddy-diffusivity part of the model, which seems to be the reason for this behavior. Therefore, the corresponding marginal 1D PDF is more informative (Fig. 6c) and it seems that this parameter can be estimated independently. While the temperature and cloud layer depth only weakly constrain this parameter value, other observables indicate that the value of *a*_{diss} should be in the higher part of the range.

For the DYCOMS case, Fig. 8 shows the marginal 1D PDFs of influential parameters constrained by five observables, and Fig. 9 the marginal 2D PDFs constrained by cloud fraction profile. It appears that marginal 1D PDFs barely provide any useful information about the parameter values, and that most likely values can be dependent on constraining observables. For the parameter *ϕ*, even though most of the 1D PDFs peak around the highest values from its range, the peaks from the five-dimensional PDFs are at the lower range for most of the parameters (Fig. 8a). The values of parameters *w*_{b} and *a*_{diff} seem to be poorly constrained by any observables (Figs. 8b,e). This is consistent with the MOAT analysis, which shows that out of the five influential parameters, most of the JPL-EDMF simulation results are least sensitive to these two parameters. Parameter *a*_{diss} seems to be constrained only by the cloud fraction profile or cloud cover (Fig. 8c). The parameter *N*_{0} seems to be relatively well constrained by all of the observables, but its optimal value is highly dependent on constraining observables (Fig. 8d). The fact that this parameter is fairly well constrained by observables is consistent with the MOAT analysis, which shows that the most of the JPL-EDMF results are fairly sensitive to this parameter.

DYCOMS case. Marginal 1D PDF of influential parameter values constrained by observations of profiles of water vapor (blue lines), liquid water (red lines), temperature (green lines), cloud fraction (solid black line), and cloud cover (dashed black line). The vertical dashed lines show the default parameter values. The colored filled circles represent the maximum values at which the probability from the five-dimensional PDF is at its maxima. The horizontal dashed lines represent 1.5 and 2 times the prior.

Citation: Monthly Weather Review 148, 10; 10.1175/MWR-D-20-0114.1

DYCOMS case. Marginal 1D PDF of influential parameter values constrained by observations of profiles of water vapor (blue lines), liquid water (red lines), temperature (green lines), cloud fraction (solid black line), and cloud cover (dashed black line). The vertical dashed lines show the default parameter values. The colored filled circles represent the maximum values at which the probability from the five-dimensional PDF is at its maxima. The horizontal dashed lines represent 1.5 and 2 times the prior.

Citation: Monthly Weather Review 148, 10; 10.1175/MWR-D-20-0114.1

DYCOMS case. Marginal 1D PDF of influential parameter values constrained by observations of profiles of water vapor (blue lines), liquid water (red lines), temperature (green lines), cloud fraction (solid black line), and cloud cover (dashed black line). The vertical dashed lines show the default parameter values. The colored filled circles represent the maximum values at which the probability from the five-dimensional PDF is at its maxima. The horizontal dashed lines represent 1.5 and 2 times the prior.

Citation: Monthly Weather Review 148, 10; 10.1175/MWR-D-20-0114.1

DYCOMS case. Marginal 2D PDF (in percent units) of all influential parameter combinations constrained by observations of cloud fraction profile. The black star indicates the default parameter values.

Citation: Monthly Weather Review 148, 10; 10.1175/MWR-D-20-0114.1

DYCOMS case. Marginal 2D PDF (in percent units) of all influential parameter combinations constrained by observations of cloud fraction profile. The black star indicates the default parameter values.

Citation: Monthly Weather Review 148, 10; 10.1175/MWR-D-20-0114.1

DYCOMS case. Marginal 2D PDF (in percent units) of all influential parameter combinations constrained by observations of cloud fraction profile. The black star indicates the default parameter values.

Citation: Monthly Weather Review 148, 10; 10.1175/MWR-D-20-0114.1

The reason that most of the 1D PDFs are not very informative for the DYCOMS case is the complexity and nonlinearity of the multidimensional PDFs. Some of those features are better seen in the 2D parameter PDFs. Figure 9 shows that, unlike for the RICO case, there is a strong correlation between the updraft parameters and the eddy-diffusivity parameters, as shown for example by the positive correlation between parameters *ϕ* and *a*_{diss} (Fig. 9b). The 2D PDFs show very nonlinear behavior with multiple maxima, as seen for example in the 2D PDFs of *ϕ* and *a*_{diff} or *N*_{0} and *a*_{diff} (Figs. 9d,j). The high nonlinearity of the parameters in 2D histograms for the DYCOMS case seems to be consistent with the high effect ratio

For the DYCOMS case, different constraining observables might indicate quite different values of most likely influential parameters. But even with widely different parameter values, the JPL-EDMF results can be quite similar. Figure 10 shows an example: the results of two JPL-EDMF simulations in which the influential parameters were taken as the most likely values constrained by cloud fraction profile (with the following parameter values: *ϕ* = 7.3, *w*_{b} = 1.9, *a*_{diss} = 0.5, *N*_{0} = 4 × 10^{−3}, and *a*_{diff} = 1.3) and liquid water mixing ratio (parameter values: *ϕ* = 14, *w*_{b} = 2.3, *a*_{diss} = 2.25, *N*_{0} = 5.5 × 10^{−3}, and *a*_{diff} = 2.6). As in the rest of this study, the noninfluential parameters were kept at their default values. Even though the values of parameters were very different for the two simulations, both of these two simulations represent the vertical profiles of thermodynamic variables and the profiles of condensed water and cloud fraction reasonably well. For example, the CF equals one for both cases and LWP is 0.10 kg m^{−2} when the parameters are constrained by cloud fraction profile and 0.12 kg m^{−2} when constrained by liquid water profile. Both of these results fall within the range of likely values from LES. By design, each of the two solutions closely follows the values of constraining observables.

Profile of (top) moist conserved thermodynamic variables (liquid water potential temperature and total water mixing ratio) and (bottom) cloud properties (condensed water and cloud fraction) for DYCOMS. The blue and the red lines are for the values of influential parameters taken from the maximum PDF from liquid water and cloud fraction.

Citation: Monthly Weather Review 148, 10; 10.1175/MWR-D-20-0114.1

Profile of (top) moist conserved thermodynamic variables (liquid water potential temperature and total water mixing ratio) and (bottom) cloud properties (condensed water and cloud fraction) for DYCOMS. The blue and the red lines are for the values of influential parameters taken from the maximum PDF from liquid water and cloud fraction.

Citation: Monthly Weather Review 148, 10; 10.1175/MWR-D-20-0114.1

Profile of (top) moist conserved thermodynamic variables (liquid water potential temperature and total water mixing ratio) and (bottom) cloud properties (condensed water and cloud fraction) for DYCOMS. The blue and the red lines are for the values of influential parameters taken from the maximum PDF from liquid water and cloud fraction.

Citation: Monthly Weather Review 148, 10; 10.1175/MWR-D-20-0114.1

### d. Measurement requirement for JPL-EDMF parameter constraint

Next we aim to understand how sensitive the JPL-EDMF modeled quantities are to the characteristics (defined as vertical resolution and measurement uncertainty) of observables that constrain influential parameters in this model. These results can help determine minimum requirements for future observational systems. In section 3c we showed that the values of many influential parameters are interdependent and that different parameter combinations can yield similar JPL-EDMF results. Therefore, studying dependence of estimated parameters on the characteristics of constraining observables would not be highly informative.

To achieve our stated goal, we first estimate the posterior PDF of influential JPL-EDMF parameters constrained by modified proxy observables and then consider only the most likely parameter values (i.e., the ones corresponding to peaks in the multidimensional PDF). Next, we investigate how the JPL-EDMF models quantities obtained with these most likely parameter values compare to the reference LES results and how sensitive they are to characteristics of the constraining observables. The modeled quantities include the vertical profiles of moist conserved thermodynamic variables, liquid water and cloud fraction. As before, in these JPL-EDMF simulations the noninfluential parameters are kept at their default values.

In this work, the proxy constraining observables (which are taken to be the median values from the LES ensemble) are modified in two ways: their vertical resolution is degraded, and after that, random noise representing measurement uncertainty is introduced. When degrading vertical resolution, we simply average data from a certain number of consecutive vertical levels and consider only the levels that do not contain repeated data, as this approach mimics the behavior of the observational system. The introduced random noise follows a normal distribution with zero mean and prescribed variance, and it is vertically uncorrelated. After introduction of the random noise, we limit constraining observables to stay within the physically meaningful range (i.e., nonnegative water mixing ratio, positive temperature and cloud fraction between zero and one). Even though this limitation can modify the characteristics of introduced random noise, we use the square root of its prescribed variance as

For certain characteristics of constraining observables, we define the vertically mean square error for JPL-EDMF modeled quantities (*x*), constrained by observable *y* as

where *x* (for which the ensemble mean from the LES results is used), and *x* with influential parameters written in vector **n**_{y} and where these parameters are constrained by observable *y*. As before the index *k* represents the vertical levels. To compare the results for different modeled quantities and to obtain robust results (with respect to the introduced noise), we define the normalized error of model quantity *x* as

where *x* over all values of influential parameters, and the *E*_{x}. For each such estimation, a different realization of random noise is introduced into the constraining variables, and a new set of most likely posterior parameter values is computed from which the new values of quantity *x* are obtained. The *nE*_{x} therefore measures the expected value of normalized errors of modeled quantity *x* and is normalized so that the value of zero represents the best possible simulation of quantity *x* and a value of one corresponds to the expected error of *x* if the values of influential parameters were chosen at random (but limited to the prescribed range).

For the RICO case, Fig. 11 shows the normalized error (*y* = *q*_{υ} or *T*), and Fig. 12 shows the normalized error of model quantities for DYCOMS case constrained by profiles of cloud fraction and water vapor (i.e., *y* = CC or *q*_{υ}). The *q*_{υ} for RICO and CC for DYCOMS, see Fig. 5), and the other one is relatively easy to observe and at the same time provides meaningful constraint on the posterior PDF.

RICO case. Normalized error of modeled quantities [*x* is modeled quantity and includes profiles of temperature (purple line), water vapor (yellow line), liquid water (green line), and cloud fraction (black line) profiles constrained by the (left) water vapor profile and (right) temperature. The *x* axes), for the 40 m (full lines with diamonds), and 200 m (dashed lines) vertical resolution of constraining observables.

Citation: Monthly Weather Review 148, 10; 10.1175/MWR-D-20-0114.1

RICO case. Normalized error of modeled quantities [*x* is modeled quantity and includes profiles of temperature (purple line), water vapor (yellow line), liquid water (green line), and cloud fraction (black line) profiles constrained by the (left) water vapor profile and (right) temperature. The *x* axes), for the 40 m (full lines with diamonds), and 200 m (dashed lines) vertical resolution of constraining observables.

Citation: Monthly Weather Review 148, 10; 10.1175/MWR-D-20-0114.1

RICO case. Normalized error of modeled quantities [*x* is modeled quantity and includes profiles of temperature (purple line), water vapor (yellow line), liquid water (green line), and cloud fraction (black line) profiles constrained by the (left) water vapor profile and (right) temperature. The *x* axes), for the 40 m (full lines with diamonds), and 200 m (dashed lines) vertical resolution of constraining observables.

Citation: Monthly Weather Review 148, 10; 10.1175/MWR-D-20-0114.1

DYCOMS case. Normalized error of modeled quantities [*x* is modeled quantity and includes profiles of temperature (purple line), water vapor (yellow line), liquid water (green line), and cloud fraction (black line) profiles constrained by the (left) water vapor profile and (right) cloud fraction. The *x* axes), for the 40 m (full lines with diamonds), and 200 m (dashed lines) vertical resolution of constraining observables.

Citation: Monthly Weather Review 148, 10; 10.1175/MWR-D-20-0114.1

DYCOMS case. Normalized error of modeled quantities [*x* is modeled quantity and includes profiles of temperature (purple line), water vapor (yellow line), liquid water (green line), and cloud fraction (black line) profiles constrained by the (left) water vapor profile and (right) cloud fraction. The *x* axes), for the 40 m (full lines with diamonds), and 200 m (dashed lines) vertical resolution of constraining observables.

Citation: Monthly Weather Review 148, 10; 10.1175/MWR-D-20-0114.1

DYCOMS case. Normalized error of modeled quantities [*x* is modeled quantity and includes profiles of temperature (purple line), water vapor (yellow line), liquid water (green line), and cloud fraction (black line) profiles constrained by the (left) water vapor profile and (right) cloud fraction. The *x* axes), for the 40 m (full lines with diamonds), and 200 m (dashed lines) vertical resolution of constraining observables.

Citation: Monthly Weather Review 148, 10; 10.1175/MWR-D-20-0114.1

As might be expected, the normalized error of all modeled quantities, *y*. For small uncertainties of observables, the *y* = *x*). This is expected, because the model parameters are optimized for the modeled quantity.

For the RICO case, the values of normalized errors

For the RICO case, both the cloud fraction and liquid water profiles are poorly constrained by both the temperature and water vapor measurements. However, as discussed before, their values are small and LES results are associated with high uncertainty. For the DYCOMS case, the cloud fraction best constrains the four thermodynamic modeled quantities if its vertical resolution is fine, even though the error increases to 0.5. For this case, the measurements of water vapor provide much less constraint.

These results can help in the design of observational systems to constrain parameters in the JPL-EDMF model or models generally. In particular, one can investigate the requirements of both the maximum measurement uncertainty and minimum vertical resolution of such an observational system. To do this, expert judgment is needed to first decide on what simulated quantity is important and what is the acceptable normalized error of the modeled quantity. Let us assume that the accepted normalized error is 0.5. For the RICO case, if the temperature and water vapor profiles are the most important modeled quantities, then observations of water vapor with an uncertainty of 4 g kg^{−1} and vertical resolution of 40 m are acceptable, or uncertainty of 2.5 g kg^{−1} can be tolerated if the vertical resolution is 200 m. Alternatively, if temperature is measured, its uncertainty can be approximately 4 K if the vertical resolution of such measurements is 40 m or 2 K if the vertical resolution is 200 m (Fig. 11). For the DYCOMS case, if the most important modeled quantities are cloud fraction and liquid water profiles, an observational system with measurements of cloud fraction with 40 m vertical resolution and measurement uncertainty of 0.5, or with a vertical resolution of 200 m and the measurement uncertainty of 0.35, would be acceptable. If the water vapor profile is measured at 20 m vertical resolution then the acceptable measurement uncertainty is 1 g kg^{−1}, or if the vertical resolution is 200 m, then the acceptable measurement uncertainty is 0.25 g kg^{−1} (Fig. 12).

## 4. Summary and discussion

In this work we developed a quantitative approach for identification of observables that best constrain parameterization of physical processes in atmospheric models. We investigated requirements for vertical resolution and measurement error of these observables. This new approach was applied to JPL-EDMF parameterization for two nonprecipitating boundary layer marine cases, one representing stratocumulus (DYCOMS) and the other shallow convection (RICO) and includes three tasks. First, we ranked the uncertainty in the parameterization of processes by their influence on the JPL-EDMF model results while accounting only for the uncertainties of the model parameters. Next, we identified the observables that best constrain these influential model parameters. Finally, we investigated the sensitivity of the JPL-EDMF results to the measurement error and vertical resolution of the constraining observables.

For the shallow convective case, the JPL-EDMF model results are most sensitive to parameters in the formulation of convective updraft properties, and in particular to the ones in the vertical velocity equation and in the lateral entrainment rate formulation. For this case, the JPL-EDMF model results are also somewhat sensitive to the parameter in the dissipation rate of the TKE, which impacts the eddy-diffusivity mixing. We found that measurements of profiles of temperature, water vapor, liquid water, and liquid water potential temperature fluxes can constrain these influential parameters well. There is a strong relation between the optimal values of the influential updraft parameters, but not between the updraft parameters and the parameter that impacts eddy-diffusivity mixing. This result indicates that the updraft parameters should be estimated simultaneously. For the stratocumulus case, the JPL-EDMF model is also highly sensitive to parameters in the formulation of the eddy-diffusivity mixing length. For this case, in addition to measurements of water vapor, temperature and liquid water, the profile of cloud fraction and cloud cover constrains these influential parameters. Contrary to what is found for the shallow convective case, the values of the updraft and the eddy-diffusivity parameters are highly correlated. The difference between the JPL-EDMF model sensitivity in the shallow convective and stratocumulus cases agrees with Smalley et al. (2019). The results show interdependence of the required vertical resolution and error characteristics of the observational system. If the observations are associated with larger error, their vertical resolution has to be finer and vice versa. However, results indicate that observations need to resolve temperature inversion at the top of the cloud layer in order to provide a meaningful constrain for model parameters.

Satellite observations currently offer some of the potential observables that we found to be influential. In particular, cloud properties such as cloud fraction and cloud cover are available. The cloud cover can be derived from combined radar/lidar cloud masks such as the *CloudSat* Geoprof-lidar product (Mace and Zhang 2014). Furthermore, measurements of these observables are considered to be fairly accurate relative to the other considered observables. *CloudSat* radar reflectivity profiles constrained by MODIS optical depth can also be used to derive profiles of liquid water mixing ratio (Leinonen et al. 2016). These measurements are limited by the vertical sampling resolution of the radar (240 m), the minimum sensitivity of the radar (~ −28 dB*Z*), and the effects of surface clutter masking the lowest 1 km of the atmosphere. Even though LWP was not found to be a particularly useful observable for constraining parameters in either of the studied case, numerous LWP datasets exist including those from solar reflectance measurements and passive microwave radiometers (Stephens and Kummerow 2007). However, there are numerous large sources of uncertainty in each method including most prominently precipitation influence on the microwave LWP (Lebsock and Su 2014) and three-dimensional radiative transfer artifacts on the solar reflectance method.

Temperature and water vapor profiles, which both provide good constraint on the JPL-EDMF parameters for both the cases, from either the infrared or microwave sounding or occultations of radio waves are not sufficiently accurate nor do they have the vertical resolution required to constrain model parameters. However, the recent Earth Science Decadal Survey (NASEM 2018) recommended the incubation of technologies to measure temperature and water vapor within the planetary boundary layer. In particular, the Decadal Survey recommends measurements of temperature and water vapor with a vertical resolution of 200 m. Given this expected resolution for a possible future mission, our results would suggest a requirement of 2 K in temperature and 2.5 g kg^{−1} in water vapor to constrain key model parameters for cumulus case, and even better for stratocumulus case. We suggest that the approach outlined here provides a potentially useful tool to provide quantitative requirements for the PBL instrument incubation and the associated preformulation activities.

In this work, we studied the parameter uncertainties for the two test cases independently. Because the JPL-EDMF parameterization is particularly designed to represent stratocumulus and cumulus dominated boundary layers as well as the transition between those two regimes (Suselj et al. 2013; SLK), the future work will investigate parameter estimation across multiple regimes. For this we will turn to the observational framework designed by Smalley et al. (2019), which will allow for simultaneous analysis of ~1000 different cases. This framework, however, introduces new complications. For example, for the current study we assumed that the only source of the JPL-EDMF model uncertainty was physical parameterizations. This is because the initial conditions and forcing were defined in exactly the same way as in the reference LES simulations. On the other hand, using the observational framework of Smalley et al. (2019) we will also need to account for uncertainties arising from the initial conditions and forcing following approach of Posselt et al. (2019).

The results shown here indicate that the profile observables of thermodynamic properties and cloud fraction can successfully constrain the most uncertain processes in the JPL-EDMF parameterization via Bayesian inference. As these observables do not directly evaluate the physical processes (e.g., entrainment rate for convective plumes), these results show the value of indirect inference. Therefore, it is possible to constrain the physical processes without directly measuring them as proposed by other works (e.g., Luo et al. 2014). This is an advantage of our approach because direct measurements are often difficult to obtain and can rely heavily on assumptions, and are sometimes applicable only to certain types of models. On the other hand, the correlation between the JPL-EDMF model parameters indicates some underlying physics, which is difficult to decipher with a measurement of single observable for a single case.

The methodology developed here is in this work applied to the JPL-EDMF parameterization model. However, some of those results might be relevant for other parameterization models, as the influential parameters of the JPL-EDMF model represent processes that are uncertain in many other parameterization models. For example, most convective parameterizations represent convective plumes (e.g., Tiedtke 1989; Rio and Hourdin 2008; Neggers 2015; Sakradzija et al. 2016; Angevine et al. 2018) and parameterize lateral entrainment, and some include representation of a plume’s vertical velocity with an equation similar to the JPL-EDMF model. Similarly to what has been found for the JPL-EDMF model, the entrainment rate parameterization is known to be highly uncertain in other models (e.g., de Rooy et al. 2013). For TKE-based parameterizations, the uncertainty of the mixing length formulations has been a long-standing issues, and many alternative formulations have been proposed (e.g., Nakanishi and Niino 2004; Bogenschutz and Krueger 2013; Olson et al. 2019). Therefore, if the uncertain processes in the JPL-EDMF can be observationally constrained, these formulations might be used in other parameterization models.

## Acknowledgments

The research was carried out at the Jet Propulsion Laboratory, California Institute of Technology, under a contract with the National Aeronautics and Space Administration (80NM0018D0004). Parts of this research were supported by the NASA MAP Program and the JPL Research and Technology Development fund. We thank three anonymous reviewers for their helpful comments.

## Data availability statement

Data available upon request.

## APPENDIX A

### JPL-EDMF Parameterization

The JPL EDMF parameterization is implemented in single-column model (SCM), which solves prognostic equations for the grid-mean values of moist conserved variables, horizontal components of the velocity vector, turbulent kinetic energy (TKE), and the variance of saturation excess. The moist conserved variables include liquid-water potential temperature, *θ*_{l} = *θ* − *L*_{υ}/(*c*_{p}*π*)*q*_{l}, and total-water mixing ratio *q*_{t} = *q*_{l} + *q*_{υ} (here the symbol *θ* represents potential temperature, *q*_{l} and *q*_{υ} liquid and water-vapor mixing ratios, respectively; *π* is the Exner function and *c*_{p} and *L*_{υ} are constants that represent the specific heat of air at constant pressure and the latent heat of water, respectively). In the JPL-EDMF parameterization, the subgrid-scale motions are represented with a combination of multiple convective plumes/updrafts, each associated with a uniform distribution of model variables, and locally driven turbulence in a nonconvective environment for which we assume a joint normal distribution of prognostic variables. Condensation is computed following these subgrid-scale distribution assumptions in the nonconvective environment and convective updrafts separately. Properties of convective updrafts are modeled with the mass-flux model, whereas mixing and condensation in the nonconvective environment are represented with the eddy-diffusivity and subgrid-scale condensation models, respectively.

The key outputs from the JPL-EDMF parameterization are the vertical subgrid-scale turbulent fluxes (which characterize vertical subgrid-scale mixing of prognostic variables) and moist properties (such as cloud fraction and condensed water). The vertical subgrid-scale fluxes enter prognostic equations in the SCM model and are written as in Suselj et al. (2019a,b):

where *φ* and *φ* represents any of the prognostic moist conserved variables or horizontal wind component. Symbols *a*_{e}, *w*_{e}, and *φ*_{e} are the horizontal area, vertical velocity, and mean value of variable *φ* in the nonconvective environment, respectively, and *a*_{i}, *w*_{i} and *φ*_{i} are the same variables for the *i*th convective updraft where *i* = 1, …, *I* labels convective updrafts.

The grid-mean properties of moist variables (e.g., cloud fraction and liquid water content) are computed as a fractional-area weighted-average from the updrafts and environment:

The appendixes B and C describe the mass-flux and eddy-diffusivity models, respectively.

## APPENDIX B

### Mass-Flux Model

All convective updrafts are initialized in the surface layer, and assumed to have a constant fractional area until they terminate (at the height where their vertical velocity reaches zero, which can be different for each of the updrafts). During their ascent, updrafts interact with the environment through lateral entrainment. A set of coupled ordinary differential equations governs the vertical evolution of updraft properties. Condensation, and thus liquid water and cloud formation within an updraft, occurs if its total water mixing ratio exceeds its saturated water content.

First we describe the parameterization of updraft boundary conditions in the surface layer. We assume that the updrafts represent the right tail of the near-surface vertical velocity distribution, which is approximated with a normal probability density function (PDF) with zero mean and standard deviation *σ*_{w}, which in the model is proportional to the Deardorff (1970) convective velocity scale

where *α*_{w} is a model parameter. Therefore, in the surface layer, all modeled updrafts collectively represent the vertical velocity PDF that exceeds some prescribed threshold *w*_{min}, and for numerical reasons we truncate this PDF at *w*_{max} = 3*σ*_{w}. Because the total surface layer fractional updraft area is an integral of the velocity PDF from *w*_{min} to *w*_{max}, the *w*_{min} can be written as a function of total surface updraft area (*a*_{u}; see Eq. A15 in Suselj et al. 2019b). The velocity range represented by updrafts is discretized into *I* equidistant bins, where the mean velocity of each bin is represented by one of the updrafts. We assume further that the surface-layer vertical velocity PDF is a part of a joint normal PDF of vertical velocity, virtual potential temperature and total water mixing ratio. The standard deviations of virtual potential temperature and total water mixing ratios are assumed to be proportional to convective temperature and water vapor mixing scales with proportionality factors of *c*(*w*, *q*_{t}) and between the vertical velocity and virtual-potential-temperature *c*(*w*, *θ*_{υ}) are prescribed. The near-surface virtual potential temperature and total water mixing ratio for each of the updrafts are computed as an integral of the thermodynamic variables over the vertical velocity part of the PDF.

The vertical evolution of the moist-conserved thermodynamic variables and horizontal wind speed components in the updraft is modeled with a steady-state equation:

where *φ* = {*θ*_{l}, *q*_{t}, *u*, *υ*}, *ϵ*_{i} represents the lateral entrainment for the *i*th updraft (described below), and *φ* for the *i*th updraft (which is assumed zero for moist conserved variables) and follows (Suselj et al. 2019b):

As in Suselj et al. (2019b), these source terms decrease effective entrainment rate for horizontal momentum by a factor of 3. The updraft vertical velocity is governed by

In Eq. (B4), *θ*_{υ} is virtual potential temperature and *g* the gravitational constant), and *w*_{a} and *w*_{b} are constants. The terms on the rhs of Eq. (B4) include parameterization of pressure perturbation and subplume variability. Based on LES simulation of archetypal shallow convection cases, De Roode et al. (2012) show that the coefficients *w*_{a} and *w*_{b} are related with the following equation:

which helps reduce the number of parameters. In this work we consider parameter *w*_{b} while the parameter *w*_{a} is constrained by Eq. (B5).

In the JPL-EDMF model the lateral entrainment is assumed to be a result of a superposition of discrete stochastic entrainment events. For the updraft that travels the height of *δz* the probability of an entrainment event is described with a binomial distribution and is expressed as *δz*/(*s*_{f}*L*_{ϵ}) (where *s*_{f}*L*_{ϵ} represents the average height over which one entrainment event occurs). Each entrainment event is assumed to entrain a constant fraction of air into the updraft denoted with *s*_{f}*ϵ*_{0}. For an updraft that travels the depth of a model layer Δ*z* the entrainment rate is parameterized as in Suselj et al. (2019a):

where the symbol *x*, as the Poisson distribution is a superposition of binomial distributions. The factor *s*_{f}, which characterizes intermittency of entrainment (a larger *s*_{f} means the entrainment is more intermittent) is not present in parameterizations described in our previous works. Note that the entrainment rate averaged over a sufficiently large number of updrafts is independent of this factor because

As in Suselj et al. (2019b), the length *L*_{ϵ} is parameterized as

with *ϕ* a constant that essentially controls the magnitude of the mean entrainment rate, *z*_{top} the depth of the moist convective layer, *p*_{ϵ} = 0.85, and *z*_{0} = 1 m. We keep the value of *p*_{ϵ} constant because the *z*_{top} is fairly constant for each of the studied cases.

## APPENDIX C

### Eddy-Diffusivity Model

Turbulent fluxes in the nonconvective environment are parameterized with the eddy-diffusivity approach and are functions of prognostic turbulent-kinetic-energy (TKE):

where *l*_{0} is the mixing length scale, *α*_{h;m} is shorthand notation for stability functions (*α*_{h} is used for *φ* = {*θ*_{l}, *q*_{t}} and *α*_{h} is used for *φ* = {*u*, *υ*}), which depend only on Richardson number as detailed in Suselj et al. (2019b), and *e*_{e} is the TKE in the nonconvective environment (obtained from the total TKE via the JPL-EDMF decomposition). The key uncertainty in this part of the parameterization comes from specification of mixing length *l*_{0}, which in the JPL-EDMF model is a combination of the surface length scale (which is a product of von Kármán constant *k* = 0.4 and the height from the surface *z*) and the free tropospheric length scale (which equals

where *z*_{sf} = 0.1*z*_{pbl} is the depth of the surface layer and *z*_{pbl} is the depth of the planetary boundary layer (PBL). The formulation in Eq. (C2) that combines the two length scales has also been proposed by Teixeira and Cheinet (2004). The time scale *τ* is a combination of the time scale of the neutral atmosphere (*τ*_{0}) with a correction for stability when the Brunt–Väisälä frequency (*N*) exceeds a threshold value of *N*_{0} and is formulated as

where *τ* is defined so that it decreases with increasing stability, and therefore the magnitude of mixing in a nonconvective environment also decreases with stability. Similar Brunt–Väisälä frequency dependence of mixing length has been proposed by Deardorff (1976).

The time scale for the neutral atmosphere (*τ*_{0}) is defined as the ratio of the depth of the dry layer (*z*_{dry}) and near-surface velocity scale (which is a combination of the Deardorff surface convective velocity scale

One important part of the parameterization is dissipation of TKE in its prognostic equation (see Suselj et al. 2019b, for details). The TKE dissipation rate is modeled as *ϵ*_{e} = −0.16*e*^{3/2}/*l*_{ϵ}, where *l*_{ϵ} is the dissipative length scale that is a combination of the surface length scale (*kz*) and free tropospheric dissipative length scale *l*_{ϵ,0}:

where

In addition to turbulent mixing, the eddy-diffusivity part of the model includes parameterization of condensation for the nonconvective environment, which is based on a PDF approach similar to the one described by Cheinet and Teixeira (2003). The key assumption is that in this part of the model domain, the moist conserved variables follow a joint normal distribution. Given the mean saturation excess [defined as *s* = *q*_{t} − *q*_{s} where *q*_{s}(*T*, *p*) is a saturation mixing ratio] and its variance, analytical formulations of partial cloudiness and condensed water in the environment can be derived. For this, the mean saturation excess is computed directly from prognostic moist conserved variables, and a prognostic equation for its variance is solved. A key uncertainty in this prognostic equation is the parameterization of the dissipation rate, which is modeled similarly to the dissipation of TKE: *ϵ*_{s,s} are variance of saturation excess and its dissipation rate, and *a*_{s} is a parameter).

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