1. Introduction

The increasing demand for accurate prediction of clouds and precipitation in numerical models at all scales has motivated the implementation of increasingly realistic cloud and precipitation microphysical parameterizations. These are typically bulk schemes, which are based on the assumption that the particle size distribution for each form of liquid and ice condensate takes a particular form. The simplifying assumptions about the form of the particle size distribution of ice and liquid condensate, which require a priori specification of parameter values, have an important effect on the details of cloud and precipitation development. It has been shown that, regardless of the details of the microphysical formulation, model output is nearly uniformly highly sensitive to small changes in these parameters (Fovell and Ogura 1989; Ferrier 1994; Walko et al. 1995; Ferrier et al. 1996; Meyers et al. 1997; Grabowski et al. 1999; Grabowski 2003). Traditional sensitivity experiments consist of defining a control simulation, perturbing a given parameter or set of parameters, rerunning the model, and comparing the results with the control. Several cloud-system-resolving model (CRM) sensitivity experiments have been conducted in this manner (Tao et al. 1995; Walko et al. 1995; Meyers et al. 1997; Grabowski et al. 1999; Wu et al. 1999; Petch and Gray 2001; Grabowski 2003; Saleeby and Cotton 2004; Gilmore et al. 2004; van den Heever and Cotton 2004), and have shown that CRM-simulated cloud fields are particularly sensitive to changes in the parameters that define particle size distributions.

A number of recent studies have used data assimilation techniques to study the effects of variation in model physics parameters (Aksoy et al. 2006; Tong and Xue 2008) or parameterization schemes (Meng and Zhang 2007). Instead of seeking to understand the nature of the uncertainty, the primary goal of these data assimilation experiments has been to account for it as a source of model error in the assimilation system and to produce a more accurate estimate of the state of the atmosphere. The principle that underlies sensitivity studies and data assimilation experiments is the fact that a functional relationship exists between parameters and model state. The extent to which this relationship can be quantified has been termed “parameter identifiability” in the ensemble data assimilation literature (Anderson 2001; Aksoy et al. 2006; Tong and Xue 2008). Regardless of whether the ultimate goal is to understand or to mitigate model uncertainty, two key pieces of information are needed: the functional relationship between parameters and how changes in parameter values map to changes in model output. Both pieces can be obtained by characterizing the space that results from a combination of 1) information about the set of physically realistic parameter values, 2) the associated output from a model of the system, and 3) the range of possible measurement values. This information can be quantified by assigning a probability distribution to each piece of information; the joint probability density function (PDF) that results from the conjunction of these probability distributions defines both the sensitivity and the data assimilation problem in the context of inverse problem theory (Vukicevic and Posselt 2008, hereafter VP08). The challenge is to construct the inverse problem in such a way that this joint PDF can be quantified. If this PDF is available, it can be used to examine which parameters have the most significant influence on the model, the relationships between parameters, and which model output variables are most sensitive to changes in the parameters. The resulting information can then be used to determine how to best perturb parameters in an ensemble forecasting or assimilation context.

Ensemble Kalman filter–type data assimilation methods address this problem by assuming each space is characterized by a Gaussian probability distribution and attempting to span (sample) the covariance of the analysis using an ensemble of model simulations. However, the assumption of Gaussian statistics can be problematic if the relationship between control variables and observations is nonmonotonic nonlinear (VP08). This is reflected in the fact that parameter identifiability has been shown to decrease when the numbers of parameters estimated in ensemble data assimilation increases. Aksoy et al. (2006) and Tong and Xue (2008) showed that, although a unique solution could be obtained when individual parameters were estimated, when multiple parameters were included in the assimilation the estimation of parameter values became much less reliable.

In contrast to methods that rely on the assumption of Gaussian statistics, Markov chain Monte Carlo (MCMC) algorithms have been shown to robustly characterize the solution space and to flexibly incorporate changes to the assumptions of the nature of uncertainty in observations and the model (Posselt et al. 2008; Tamminen 2004). In this study, we use an MCMC algorithm to characterize the nature of the uncertainty in simulations of convective clouds and precipitation associated with the specification of cloud microphysical parameters in a Lagrangian cloud model. We use a single-moment bulk microphysical scheme similar to those used in modern regional and global numerical prediction and climate models and examine a case of squall-line-type convection, which is known to have large sensitivity to changes in cloud microphysical parameters. The MCMC algorithm reveals a nonmonotonic relationship between parameters and model output that gives rise to a multimode parameter space. The examination of joint parameter–state PDFs indicates that this relationship changes with time over the course of the simulation. The addition of information in the form of additional observations or reduction of observation errors serves to eliminate many of the possible modes in the parameter space, but it cannot fully eliminate the occurrence of a nonunique solution, which is tied to nonuniqueness in the model formulation itself. We demonstrate that knowledge of the presence of nonmonotonic nonlinearity in the parameter–state relationships, along with an understanding of the physical nature of the system, can be used to adjust the inverse problem so that a one-to-one mapping between parameters and state can be achieved.

The remainder of this paper is organized as follows: Section 2 contains a description of the model framework and information on the control parameters and observations used in the sensitivity analysis. We outline the general inverse method in section 3 and describe how a Markov chain Monte Carlo algorithm can be used to sample the posterior joint probability distribution. We present the results of running the MCMC algorithm on the cloud model in section 4 and discuss caveats and implications. In section 5, we offer a summary of our results and our primary conclusions, present a brief discussion of the limitations of our study, and outline our anticipated future work.

2. Modeling framework

Uncertainty in cloud microphysical parameters affects simulated cloud and precipitation fields, which naturally feed back to the dynamics on multiple scales. Because of the complexity in the microphysics–dynamics relationship, a necessary first step in understanding the details of model error is to consider the effect of changes in model physics parameters on the model output in isolation from any feedback to the cloud-scale dynamics. Such an exercise requires a framework that accommodates realistic changes to the environment so that the cloud microphysical routine can respond as if it were embedded in a fully interactive model. To do this, we have constructed what we term a “Lagrangian column model,” which is designed to emulate the changes in environment experienced by an atmospheric column as it moves through a cloud system following the mean flow. The model consists of a single-column version of the National Aeronautics and Space Administration (NASA) Goddard Cumulus Ensemble (GCE) model (Tao and Simpson 1993; Tao et al. 2003). As in the fully three-dimensional version of the GCE, base-state vertical profiles of potential temperature and moisture are fixed and the model predicts perturbations on these base-state profiles. In our idealized framework, clouds are generated by forcing the model with time-varying vertical profiles of vertical motion and water vapor tendency. Advection is only allowed to operate on cloud liquid and ice condensate and only in the vertical direction. By specifying appropriate base-state potential temperature and water vapor and time-varying vertical profiles of vertical motion and water vapor forcing, the model can be used to simulate the flow through a range of different cloud systems.

Because organized deep convection produces the bulk of the warm-season precipitation globally, and has been shown to be highly sensitive to changes in cloud microphysical parameters, we use as our test case an idealized representation of squall-line-type convection. The added benefit to the examination of squall-line-type convection is that it contains two discrete cloud morphologies: convective, in which precipitation is primarily generated by the collision–coalescence (warm rain) process; and stratiform, in which the melting of snow and graupel play a key role. The simulation is constructed so that it is consistent with idealized (Moncrieff 1992) and observed (Houze 2004) squall lines. Profiles are assumed to be sinusoidal with the following functional forms for vertical motion:

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and for water vapor tendency

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Use of a sinusoidal function allows specification of full profiles of vertical motion and water vapor tendency using only three parameters: the maximum value w₀ and q₀, height of the maximum value z_0w and z_0q, and sine function half-width D_w and D_q—all of which are allowed to vary in time. The resulting profiles (Fig. 1) bear a close resemblance to the observed vertical distribution of latent heating and vertical motion in tropical squall lines (Houze 2004 and references therein). In our idealized squall-line simulation, the column enters the system at the leading edge of the cold pool, propagates through the convective precipitation region, and exits through the stratiform precipitation region. Base-state profiles of temperature and water vapor are representative of the tropical oceanic environment (Fig. 1a) and are generated by setting surface temperature and water vapor mixing ratio equal to values typical of the tropical Pacific warm-pool region, then decreasing temperature at a pseudo- (moist) adiabatic lapse rate. Water vapor mixing ratio q_υ is held constant with height until the atmosphere reaches liquid saturation, after which point q_υ decreases at a rate that maintains saturation with respect to liquid at temperatures above freezing and with respect to ice below. Time-evolving profiles of vertical motion and water vapor forcing (Figs. 1b,c, respectively) consistent with observed and simulated squall-line-type evolution (Moncreiff 1992; Fovell and Tan 1998; Houze 2004) are used to drive the simulation. The model is run with 60 vertical layers with constant 250-m vertical grid spacing and a 5-s time step, and the radiative transfer, surface flux, and microphysical parameterizations are all identical to those used in the NASA Goddard Cumulus Ensemble model (Tao and Simpson 1993; Tao et al. 2003; Lang et al. 2007). The time series of a number of model output variables are shown in Figs. 2a–e, and it can be seen that the model produces realistic time evolution of precipitation rate (PCP), liquid and ice water path (LWP and IWP, respectively) and top of the atmosphere shortwave (SW) and longwave (LW) radiative fluxes [outgoing SW radiation (OSR) and outgoing LW radiation (OLR)]. Time–height cross sections of mass mixing ratio for each condensate species are plotted in Figs. 2f–j; convective and stratiform regions of precipitation are clearly discernible.

Because this paper seeks to characterize model uncertainty associated with the specification of bulk microphysical parameters, we offer a brief description of the cloud microphysical parameterization here. The GCE model employs a single-moment bulk scheme, in which two classes of liquid (rain and nonprecipitating cloud) and three classes of ice (nonprecipitating pristine crystals, precipitating unrimed pristine crystals and aggregates, and graupel) are assumed to follow an exponential particle size distribution (Lin et al. 1983; Rutledge and Hobbs 1983; 1984; Tao et al. 2003; Lang et al. 2007) that, for cloud species x, takes the form

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Here, N_0x is the slope intercept and λ_x is the slope,

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where ρ_x is the condensate density, ρ is the air density, and q_x is the condensate mass mixing ratio. In addition to the specification of the slope intercept of the particle size distribution and the ice particle density (rain is assumed to have a constant density equal to liquid water at 0°C), the particle fall speed is parameterized using a power relationship,

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where the values assigned to the coefficient a_x and exponent b_x depend on the assumed ice crystal shape and are typically taken from in situ observations of precipitating ice (Locatelli and Hobbs 1974; Mitchell 1996; Heymsfield et al. 2002). Though this single-moment scheme is relatively simple compared with modern two- and three-moment and bin schemes, it is representative of the complexity of microphysical parameterizations that are currently used in regional prediction models and quite a bit more sophisticated than those used in GCMs, which typically use one species of liquid and ice as prognostic variables and diagnose the partition between suspended and precipitating condensate.

Our goals in this study are to use the model framework outlined earlier to explore the sensitivity of simulation output to changes in model physics assumptions and to characterize the degree of nonlinearity present in the parameter–state relationship. To this end, we examine the influence of changes to cloud microphysical parameters on simulated PCP, LWP, IWP, OSR, and OLR, as they are directly related to cloud microphysical processes relevant to the study of cloud radiative feedbacks in the climate system and are available as retrievals from current satellite remote sensing platforms. These variables also serve as observations in our inverse modeling algorithm, and we note that because they are output directly from the model, we require no observation operator. Avoiding the use of a complicated (and likely nonlinear) observation operator (as would be necessary for radar reflectivity observations, for example) allows for a cleaner evaluation of the relationship between changes to parameters and the effect on model output. We note the pros and cons of our choice of observations later in section 5.

Thorough examination of parameter–state relationships necessitates perturbation of all parameters simultaneously; it is the relationships between parameters and their joint effect on model output that is of interest. Because this is a computationally demanding exercise, it is useful to briefly explore the model sensitivity in a more simplistic way before we begin to ensure that the model is appropriately sensitive to changes in cloud microphysical parameters. To this end, we run two experiments in which we examine the sensitivity of precipitation rate, LWP, IWP, OLR, and OSR at 120 min of simulation time to systematic changes in the values of the graupel fall speed coefficient and the graupel density (Fig. 3). These plots are analogous to Figs. 1b and 2b in VP08, and they effectively map the functional relationship between the chosen microphysical parameter and the model output. It can be seen that the model is sensitive to changes in each of these parameters; model output varies widely across the range of parameter values. The graupel fall speed coefficient exhibits nonmonotonic nonlinearity; multiple values of this parameter produce identical values of each of the model output variables. The graupel density exhibits a nonlinear relationship with each of the output variables, however, the nature of the relationship is monotonic; each value of graupel density maps to a single value of each of the model output variables. The primary conclusions that arise from an examination of Figs. 2 and 3 are that the model 1) produces a realistic simulation that is sensitive to changes in microphysical parameters, 2) exhibits nonlinear relationships between parameters and state, and 3) includes two distinct cloud microphysical regimes.

3. Inverse problem framework

Model uncertainty can be quantified using the framework of inverse problem theory, as postulated in Mosegaard and Tarantola (2002). The most general solution to an inverse problem consists of the conjunction of the various information spaces that define the state of knowledge of the system (Rodgers 2000; Mosegaard and Tarantola 2002; Tarantola 2005; VP08). The character and magnitude of the uncertainties in observations and forward model translate directly to the extent and shape of their corresponding information spaces, and the uncertainty in the resulting estimate is similarly characterized by the extent and shape of the solution information space. If uncertainties in measurements, model, and prior estimates are represented by probability distributions, then the solution is defined as the joint posterior distribution of the observations and control parameters. The optimal parameter estimate is, by definition, the location in the joint distribution P(x, y) with the greatest probability density, and the error characteristics for each control parameter can be examined by considering each parameter’s marginal probability distribution. Similarly, the relationship between parameters can be examined by considering their joint distribution, whereas the functional relationship between parameters and state is expressed in the joint distribution of parameters and model output state. The marginal probability distribution for each parameter, subset of parameters, or parameter observation pair is computed by integrating the joint PDF over all parameters except the parameter[(s) and observation(s)] of interest. The ensemble Kalman filter and variational data assimilation methods are specific types of solution to the general inverse problem (VP081513-Vukicevic1">VP08), but the fundamental strength of the information-based framework lies in its generality. It treats contributions from the model, model error, observations, and prior state as probability distributions with no assumed form and characterizes the solution as the joint posterior probability distribution that results from conjunction of the component PDFs.

Following VP08, the definition of conditional probability can be used to write the joint PDF of model parameters x and observations y in the familiar Bayes theorem form

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where P(x|y) is the posterior probability density to be solved for, P(x) is the prior probability density of the set of parameters x, P(y|x) is the probability density that describes the likelihood that the parameters are the true parameters (given the uncertainty in the model and observations), and P(y) is the prior probability density of the observations. The optimal parameter estimate is, by definition, the maximum likelihood point in the conditional distribution P(x|y), and the error characteristics for the set of control parameters can be examined by considering the properties (e.g., moments) of the conditional PDF. Note that we have made no assumptions about the form of the probability distributions in (6), hence the complete information on the parameter interaction and parameter–state functional relationship is present in the solution to this formulation of the generalized inverse problem.

It remains, then, to find a way to compute the solution to (6). Most modern data assimilation techniques and retrieval algorithms are based on the assumption that probabilities are Gaussian. The derivation of the solution to (6) for the case of Gaussian PDFs has been well documented (Rodgers 2000; Kalnay 2003; Evensen 2006; Lewis et al. 2006; VP081513-Vukicevic1">VP08; among many others) and will not be repeated here. Instead, we address the question of how to compute each PDF in the case where PDFs are not assumed to be Gaussian, and, in fact, may not have a unique mode. Perhaps the most direct method of solution is to resolve each of the parameters of interest into discrete bins across their ranges of natural variability, assign a form and magnitude for the measurement uncertainty, and convolve the resulting PDFs via numerical computation. Such an exercise formed the basis for the algorithm described in VP08; however, the computational expense of doing so increases as N^M, where N is the desired resolution of the parameter space and M is the number of parameters allowed to vary. For joint probability distributions that contain a moderate amount of mass in each dimension, the theory of empty spaces (Tarantola 2005) dictates that the amount of empty space in the solution space (e.g., the percentage of the space for which values of the parameters are associated with very low probability) tends to increase exponentially with each additional dimension. It follows that an effective method of characterizing the probability density function is one that avoids empty (low probability) regions of the space in favor of regions with relatively high probability density, where high-density regions are identified via a predefined measure of fit between simulation output and observations (e.g., cost function). Markov chain Monte Carlo algorithms comprise a class of such methods. Markov chain Monte Carlo methods address the problem of empty parameter spaces by sampling the probability distribution in such a way as to revisit those regions with high probability while avoiding regions with low probability (Tamminen and Kyrola 2001, hereafter TK01; Tarantola 2005; Posselt et al. 2008). Specifically, the MCMC algorithm is made up of a random walk that consists of multiple successive iterations (see flowchart in Fig. 4). In each iteration, test values x̃ of all parameters are randomly drawn from a “proposal” distribution {x̃ ∼ q[x̃; x⁽ⁱ⁾]} that depends on the current estimate x⁽ⁱ⁾. Proposed parameter values are then used in the forward model, and the resulting simulated measurements [ỹ = f (x̃)] and prior [p(x̃)] are compared with observations via a cost function derived from the specified form of P(y|x). The proposed parameter values are accepted as a sample in the posterior distribution with the probability

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where the acceptance ratio ρ is defined as

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If the prior parameter probability distribution is constant (e.g., uniform), and the proposal distribution is symmetric {e.g., q[x̃; x⁽ⁱ⁾] = q[x⁽ⁱ⁾; x̃]}, then the acceptance ratio reduces to

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which can easily be computed from the specified form of the observation uncertainty PDF applied to the current and proposed parameter values. Note that this also eliminates the need for an a priori term in the cost function.

The value of the acceptance ratio determines whether the proposed set of parameters is saved as a sample of the posterior distribution. If the new parameters generate a better fit to observations (ρ > 1), then the proposed set of parameters is saved as the next sample in the distribution [x⁽ⁱ⁺¹⁾ = x̃]. If not (ρ < 1), then a test value is drawn from a uniform (0, 1) distribution. If this value is less than ρ, then the proposed parameter set is also accepted as a sample in the distribution. Otherwise, the proposed set of parameters is rejected, and new proposed parameter values are drawn based on the original set. The accept/reject procedure ensures that parameter sets that provide a better fit to the observations are immediately accepted, those that provide a similar fit are considered, and those that lead to simulated observations that are very different from the measurements are rejected. In the process, regions of the parameter space with relatively high probability are preferentially sampled, whereas regions with low probability are avoided, and a sample of the posterior distribution is produced using far fewer iterations than direct computation of the PDF. A more complete description of the theoretical underpinnings of the MCMC algorithm can be found in Mosegaard and Tarantola (2002) and in Tarantola (2005), whereas tools for assessing convergence of MCMC to sampling a stationary distribution are discussed in Gelman et al. (1996, 2004).

It should be noted that, in the process of sampling the joint parameter–observation information space, MCMC identically returns a solution to Bayes theorem (6). The computation of the cost function (measure of fit between model state and observations) for the set of MCMC iterations is identical to the computation of P(y|x), and the parameter values generated over multiple MCMC iterations is a sample of P(x|y). The application to parameter sensitivity follows from the fact that, in sampling the joint parameter PDF, MCMC effectively returns a nonlinear estimate of the parameter–state sensitivity (Jacobian) matrix. However, in contrast to linear estimates of sensitivity, the PDF output from MCMC contains a description of the full functional relationship between parameters and model output given the true model parameters. As such, the sample of the PDF returned by the MCMC algorithm can be used to characterize the relationship between parameters and model state, and it allows the sources of error that dominate the uncertainty in the result to be robustly quantified. Testing different error assumptions, including specification of not only different error magnitudes but also different error probability distributions, is straightforward, as is the introduction of new observations and the evaluation of new physical models.

Compared with methods that are constrained to particular forms of the PDFs in (6), the design of an MCMC algorithm is highly flexible, and the choice of prior estimate of control parameters P(x) and the magnitude and form of the observation uncertainty P(y|x) can be adapted to suit the specific inverse problem. Our implementation of the MCMC algorithm is documented in Posselt et al. (2008) and is based on the work of TK01), who employed a Gaussian proposal distribution centered on the current parameter estimate. However, in contrast to the technique used in TK01, we do not use the covariance between parameters to generate parameter perturbations. In the case of a unimodal joint parameter distribution, generating parameter perturbations from a sample of the parameter covariance can improve the efficiency of the MCMC algorithm. However, if the parameter joint PDF is multimodal (e.g., the functional relationship between parameters and observations is nonmonotonic), parameter–parameter covariance can change sign for different combinations of parameter values. Because parameter covariance in the TK01 algorithm is updated infrequently, nonmonotonicity can lead to parameter perturbations that are based on an improper covariance estimate and to an inability to effectively sample the parameter space. Consequently, we have chosen to apply uncorrelated Gaussian perturbations to our set of control parameters. It should also be noted that, if perturbations to parameters are too large, the algorithm will not sample the PDF with sufficient resolution; however, if perturbations are too small, the algorithm will be inefficient and may not visit all regions of the space. For this reason, the variance of the proposal distribution is adaptively changed during a burn-in period to give the desired sampling rate. Samples of the posterior PDF obtained during burn in are not included in the final analysis. For a discussion on optimal sampling rates, the reader is referred to Gelman et al. (1996) and Haario et al. (1999).

We choose as control variables 10 cloud microphysical parameters known to have a significant influence on simulated clouds, precipitation, and radiative fluxes. The set of chosen parameters, along with their units and range of variability, are detailed in Table 1. We assign a bounded uniform PDF as the prior parameter distribution and set the parameter ranges to correspond to observations of ice particle fall speeds (Locatelli and Hobbs 1974; Mitchell 1996) and liquid and ice particle size distributions (Tokay and Short 1996; Roy et al. 2005; Heymsfield et al. 2002). In specifying a bounded uniform as the parameter prior, we make the prior distribution nearly noninformative, though there is some potential dependence on the specified parameter range. We tested a variety of ranges for parameter values and found that the PDF returned by MCMC is insensitive to the exact range of parameter values used in the algorithm, as long as it spans the range variability observed in nature.

In each experiment, a truth state is generated by running the model with parameter values consistent with what is currently used in cloud–resolving and mesoscale models (Table 1). Observations are drawn from this truth state, and observation errors are assumed to have a Gaussian form, with variance consistent with the rigorous quantification of uncertainty performed on the analogous retrieved quantities from the Tropical Rainfall Measuring Mission (TRMM; L’Ecuyer and Stephens 2002, 2003). It should be noted that the form of the observation error is not constrained to be Gaussian (Posselt et al. 2008); we use this form in our experiments to be consistent with the error estimates associated with TRMM retrievals. Observation uncertainty standard deviations are listed in Table 2, and the truth state from which observations are drawn is plotted in Figs. 2a–e.

Convergence of the MCMC algorithm to sampling a stationary distribution is assessed by comparing distributions generated by drawing random sets of parameters from the full sample. According to the results presented by Haario et al. (1999), between 20 000 and 40 000 iterations should be sufficient to sample a 10-dimensional parameter space if the target (posterior) distribution is multivariate Gaussian. In reality, as was indicated in the preliminary results presented in Fig. 3, the underlying PDF is non-Gaussian, and we find that approximately 100 000 iterations are necessary to sample the underlying PDF. To ensure a robust sample, we ran MCMC out to 4 × 10⁶ iterations. Convergence to sampling a stationary distribution was ensured by drawing random samples of 100 000 parameter sets from the full MCMC sample and computing the integrated absolute difference between histograms. We found an average of less than 1% difference between any two given random samples, indicating robust sampling of the underlying (true) PDF. To illustrate convergence of the MCMC algorithm to sampling a stationary distribution, we have plotted the joint PDF of two of the model parameters for successive numbers of iterations (Fig. 5). As early as 5 000 iterations, the shape of the underlying PDF is beginning to emerge, and by 50 000 iterations, the characteristics of this particular joint PDF appear to be stationary. Recall that 10 parameters are varied in this example; direct numerical computation of the PDF (assuming a resolution of 10 bins per parameter) would require 10¹⁰ simulations; MCMC returns a robust sample of the PDF using fewer than 10⁶ simulations.

4. Probabilistic assessment of model physics uncertainty

5. Summary and conclusions

In this paper, we introduced a Lagrangian column framework for evaluating model physics parameterizations and described how nonlinear inverse methods can be used to examine the details of model physics uncertainty. We then used this framework to explore the characteristics of uncertainty due to specification of parameters in a single-moment bulk cloud microphysical parameterization. The MCMC inverse method returned a robust sample of the joint PDF of the parameters and model output variables, from which we were able to obtain information on the sensitivity of model output to changes in parameters. This information includes details of the relationship between parameters, which parameters had the greatest effect on model output, and the nature of the parameter–state functional relationship. The major findings of this study include the following:

• All of the microphysical parameters had some effect on the solution, but the effects differed depending on the model state variable examined. Parameters that define the fall velocity of snow and graupel, along with the slope intercept of the rain size distribution and the cloud-to-rain autoconversion mass mixing ratio threshold, had the greatest effect on simulated precipitation rate and cloud liquid and ice water paths. Parameters that define the graupel particle size distribution had the greatest effect on OLR, whereas parameters that define the snow particle size distribution had the greatest effect on OSR.

• Marginal PDFs tended to mask multimodality in the solution, and in some cases, a wrong estimate of parameter values could be obtained if the mode of the parameter marginal PDF was used.

• Joint PDFs of parameter pairs proved to be effective for revealing nonuniqueness (nonmonotonic nonlinearity) in the functional relationship between parameters that define the crystal shape and particle size distribution and simulated precipitation rate, LWP, IWP, and radiative fluxes.

• Joint PDFs of parameters with model state variables returned useful information on the details of the sensitivity of model output to changes in parameters, and they indicated that the functional relationship between parameters and model output changes depending on cloud morphology.

• Reductions in observation uncertainty served to increase the definition of the discrete solutions (as represented in multiple modes in the parameter space) and to exacerbate the problem of nonuniqueness and reduced parameter identifiability.

• Knowledge of the physical nature of the system can be used to greatly reduce, or even eliminate, the potential for a multimode solution. In this case, endemic nonmonotonic nonlinearity was addressed by imposing a simple requirement that graupel particles fall faster than snow. The flexibility of the MCMC framework makes implementation of this type of constraint trivial.

Multimodality in the joint parameter PDF has implications for the decrease in parameter identifiability noted in previous studies; with increasing dimension in the parameter space, the potential for a multimode solution increases. However, it is interesting to note that the realization of multimodality in the posterior PDF depends on which parameters are included in the sample. Because of nonlinearity in parameter–parameter and parameter–state relationships, parameter identifiability does not decrease smoothly as the number of parameters included as control variables in the assimilation increase. Instead, there is likely to be a sudden decrease in parameter identifiability as the potential for nonuniqueness in the parameter–state relationship is realized.

Nonmonotonic nonlinearity in the model–state functional relationship (nonuniqueness) and associated multimode parameter joint PDF reveals potential problems for data assimilation techniques that rely on the assumption of Gaussian posterior probability distributions. As is well known, this assumption leads to mean estimates that may deviate significantly from the maximum likelihood parameter values, depending on the nature of nonlinearity in the model. Although it may be possible for ensemble-type assimilation systems to return a robust estimate of the maximum a posteriori solution through successive addition of information from observations (VP08), a key goal of ensemble prediction is to more fully span the true range of variability in the natural system. As such, eliminating viable parameter combinations simply for the sake of returning a robust estimate may unrealistically restrict the robustness of ensemble prediction systems. Our results suggest that ensemble prediction systems should represent uncertainties inherent in physical parameterizations, even if parameter estimation is not included in the data assimilation system. Ensemble forecast studies that include mixed physics ensembles demonstrate the potential benefit of this approach (Stensrud et al. 2000). Determining the optimal approach for representing uncertainties in model physics that have a nonmonotonic nonlinear relationship to observations is one of the outstanding challenges in modern data assimilation and ensemble prediction research.

The fact that parameter–state relationships can change between convective and stratiform times within the same simulation implies that the relationships between changes in parameters and changes in model output are sure to differ for different cloud system types. These are also likely to change if the microphysics is allowed to interact with the dynamics in the model. Because cloud microphysical processes in nature directly feed back to cloud-scale (and larger) dynamics, we plan to run MCMC-based studies of parameter sensitivity using 2D and 3D versions of the GCE model. In addition, we note that, because the PDFs produced by the inversion are conditional and depend on the chosen model and observations, use of observations more directly related to the details of the particle size distribution (e.g., radar reflectivity) will change the nature of the solution space and may serve to render a unimodal PDF. Conversely, observations that are less directly related to cloud microphysical characteristics (e.g., wind and temperature) will likely result in an even greater degree of nonuniqueness in the parameter–observation relationship. We leave the detailed investigation of how increased (or decreased) observation information content associated with the use of different observing systems contributes to parameter identifiability for future work.

In the near future, we plan to apply the framework introduced in this paper to other cloud system types (e.g., shallow convection, airmass convection, stratus, and frontal clouds) and to explore the utility of the nonlinear inverse framework for understanding uncertainty in GCM parameterizations. We also note that the MCMC algorithm operates over the length of the simulation, requiring parameter values to be held constant. VP08 showed that the nonuniqueness in the model solution could be mitigated through sequential addition of observation information. Hence, we plan to apply the nonlinear analysis technique developed in VP08 to a subset of the microphysical parameters examined here for the purpose of examining whether the sequential update of parameter values may serve to reduce the solution to a single-parameter set.