1. Introduction
The present work seeks to calibrate the model parameters of the K-profile parameterization (KPP) model (Large et al. 1994, 1997) as implemented in the ocean model MIT general circulation model (MITgcm; Ferreira and Marshall 2006; Marshall et al. 1997a) of the tropical Pacific. The KPP model relies on a number of parameters whose default values are set based on a combination of theory, laboratory experiments, and atmospheric–oceanic boundary layer observations (Large et al. 1994, 1997). Our goal here is to quantify the uncertainties in these parameters where the ocean resolves the chaotic behavior of fluid dynamic models. The chaos in the response of the deterministic MITgcm model to the perturbation of these parameters leads to internal noise that in turn results in low signal-to-noise challenges, as will be discussed in more detail below.
An inverse modeling approach is adopted for the objective stated above in which a set of temperature, salinity, and horizontal current measurements are used to estimate the KPP parameters. Specifically, we employ a Bayesian approach to inverse problems that provides complete posterior statistics and not just a single value for the quantity of interest (Tarantola 2005). Traditionally, local model–data misfit of short-term turbulent mixing events is used to construct a cost function and then Bayesian inference is employed for the estimation of the uncertain parameters (Zedler et al. 2012; Sraj et al. 2014b). Here, instead, we use a test statistic for KPP parameter estimation that was introduced in Wagman et al. (2014) and in Zedler et al. (2016, manuscript submitted to Ocean Modell.). This statistic seeks to formulate a total cost function of different components representing the structures of turbulent mixing on both daily and multiyear time scales, takes data quality into account, and filters for the effects of parameter perturbations over those resulting from changes in the wind. At both time scales, the model and data are filtered before taking differences to capture the observations that are relevant to testing mixing physics parameters.
The end result of the Bayesian inference formulation is a multidimensional posterior (Malinverno 2002; Sraj et al. 2013; Olson et al. 2012) that can be directly sampled via a Markov chain Monte Carlo (MCMC) scheme. This, however, requires a prohibitive number of simulations of the forward model—one for every proposed set of parameters of the Markov chain. This practice renders Bayesian methods computationally prohibitive for large-scale models such as the MITgcm, where one model evaluation takes 22 h in computing time using 256 processors. To overcome this issue, we construct a surrogate model that approximates the forward model and can be used in the sampling MCMC. More precisely, we use the polynomial chaos (PC) method to construct the surrogate model from an ensemble of MITgcm model runs (Marzouk et al. 2007; Marzouk and Najm 2009). This approach further offers additional advantages such as computing model output sensitivities and additional statistics (Le Maître and Knio 2010).
The PC method has been extensively investigated in the literature, and its suitability for large-scale models has been recently demonstrated in various settings. Alexanderian et al. (2012) implemented a sparse spectral projection PC approach to propagate parametric uncertainties of three KPP parameters in addition to the wind drag coefficient during a hurricane event. The study demonstrated the possibility of building a representative surrogate model for a realistic ocean model; however, the inverse problem was not tackled. Winokur et al. (2013) followed up on Alexanderian et al.’s (2012) work and implemented an adaptive strategy to design sparse ensembles of oceanic simulations for the purpose of constructing a PC surrogate with even less computational effort. Sraj et al. (2013) extended the previous work and combined a spectral projection PC approach with Bayesian inference to estimate the parameterized wind drag coefficient using temperature data collected during a typhoon event. The same problem was also solved using a gradient-based search method (Sraj et al. 2014a). Mattern et al. (2012) have similarly exploited the virtues of such polynomial expansions for examining the response of ecosystem models to finite perturbations of their uncertain parameters. Tsunami (Sraj et al. 2014b; Ge and Cheung 2011) and subsurface flow modeling (Elsheikh et al. 2014) were also investigated using similar PC approaches.
What is common in the aforementioned PC applications is that the processes studied occurred over short time scales of a few days, so that the internal noise within the model was small. This enabled a successful PC expansion construction using traditional spectral projection (Sraj et al. 2013; Reagan et al. 2003; Alexanderian et al. 2012). In this study, however, the major hurdle of constructing a PC surrogate model was the internal noise present in the MITgcm model due to the perturbation of the chosen KPP parameters, which was amplified over time by nonlinear interactions. As will be explained in section 4b(1), a spectral projection technique failed to construct a PC expansion that faithfully represents the model. Instead, we resort to a compressed sensing technique, namely, the basis-pursuit-denoising (BPDN) method (Peng et al. 2014), to determine the PC expansion coefficients. This technique first seeks to estimate the noise inherent in the signal and then solve approximately an optimization problem assuming sparsity in the PC expansion to determine the nonzero PC coefficients. The BPDN method offers an additional advantage in that a smaller number of model runs compared to the spectral projection method is required to determine the PC coefficients, as shown in section 4c. BPDN was recently employed to build a proxy model for an integral oil–gas plume model (Wang et al. 2016) and for an ocean model with initial and wind forcing uncertainties (Li et al. 2015). In the former, the model output was noisy as a result of the iterative solver used in the double-plume calculation (Socolofsky et al. 2008) and BPDN proved to be successful in building a representative PC surrogate that excludes the model output noise. In the latter, no noise was present in the model output; however, the model was unable to produce realistic simulations for prespecified sets of parameters, as required by the spectral projection method. BPDN was therefore used as an alternative approach, as it does not have this requirement; a random sample of model parameters can be modeled instead (Doostan and Owhadi 2011).
We end the introduction section by asserting that the employed compressed sensing technique is not new but also is not a standard in large-scale applications. To our knowledge, BPDN has never been applied to a noisy, large-scale ocean model. Implementing this technique together with the choice of fitting a newly formulated test statistic (Wagman et al. 2014; Zedler et al. 2016, manuscript submitted to Ocean Modell.) for the purpose of inferring KPP model parameters represents the novelty of our current work.
The structure of the paper is as follows. Section 2 describes the MITgcm, presents the choice of the uncertain parameters, and describes the observation and cost function used in the estimation process. Section 3 introduces the Bayesian inference and its application to our specific problem. Section 4 discusses the PC method and presents several error studies to show the convergence of the constructed PC expansion. Section 5 presents the results of the inference of KPP parameters and section 6 summarizes our findings.
2. Model, uncertain parameters, and observations
a. MITgcm model
The MITgcm employed in this work is based on the primitive Navier–Stokes equations implemented in spherical coordinates with an implicit nonlinear free surface. The MITgcm implements the KPP turbulent mixing scheme (Adcroft 1995; Marshall et al. 1997a,b; Large et al. 1994) (see below) in a regional configuration based on that of Hoteit et al. (2008, 2010) for the simulation of oceanic flow. In particular, the domain chosen covers the region with latitudes from 26°S to 30°N and longitudes from 104°E to 70°W (Fig. 1). The time period of our model simulation is 2004–07. The initial and lateral boundary conditions are provided by the Ocean Comprehensible Atlas (OCCA) reanalysis that was developed for the 2004–07 time period using data assimilation in the MITgcm of available temperature and salinity ocean datasets (Forget 2010). The lateral boundary conditions for our model are implemented with a sponge layer (with a thickness of nine grid cells, and inner and outer boundary relaxation time scales of 20 and 1 days, respectively). For the lateral boundary conditions, the OCCA data assimilation product is interpolated at the model resolution of ⅓° and with a time step of 1 day. Therefore, at the initial time step, the boundary temperature and salinity conditions are approximately in equilibrium with the interior fields. Once our higher-resolution simulation starts, the velocity field quickly adjusts to the pressure gradient forces and establishes a realistic equatorial circulation. We note that our model runs on 256 processors and takes about 22 h for a single simulation. As described below in section 4, we needed to run the model 903 times, which required a total of about 5.5 million compute hours.
Model domain showing 2004–07 averaged sea surface temperature using default MITgcm KPP parameters and the TOGA TAO mooring array.
Citation: Monthly Weather Review 144, 12; 10.1175/MWR-D-15-0394.1
Model domain showing 2004–07 averaged sea surface temperature using default MITgcm KPP parameters and the TOGA TAO mooring array.
Citation: Monthly Weather Review 144, 12; 10.1175/MWR-D-15-0394.1
Model domain showing 2004–07 averaged sea surface temperature using default MITgcm KPP parameters and the TOGA TAO mooring array.
Citation: Monthly Weather Review 144, 12; 10.1175/MWR-D-15-0394.1
b. KPP model
In the ocean, turbulent mixing can ensue when there is net heat released to the atmosphere at the sea surface (i.e., at night), producing gravitationally unstable density inversions (convective mixing) and when there is sufficient vorticity-producing (on the x–z plane) vertical shear in the horizontal currents to overturn a nominally stratified water column (shear driven or Kelvin–Helmholz instability induced mixing). In general terms, the intensity of convective and shear-driven mixing depend on local water column properties and surface forcing conditions. Theoretically, the most vigorous turbulent mixing should occur when a weakly unstably stratified, strongly sheared flow is forced with strong winds and convection (i.e., at night). By contrast, the flow is most likely to be laminar when a strongly stably stratified, weakly sheared flow is forced with weak winds and large net heat going into the ocean (i.e., during the day). The KPP embodies these basic relationships, by making the intensity of mixing a function of the locally diagnosed properties of the water column that relate to the amenability to turbulent mixing (such as the bulk and gradient Richardson numbers), as well as the nonlocal surface wind stress and net heat flux forcing (through forcing parameters such as the friction velocity of wind and the Monin–Obukhov length). In the KPP, mixing is more intense under unstable convective surface forcing conditions. Readers interested in the details of the KPP are referred to Large et al. (1994) and Large and Gent (1999).
The KPP generates depth profiles of two quantities that are relevant for turbulent mixing, the eddy diffusivity/viscosity and a nonlocal term. The eddy diffusivity and viscosity at depth z can be thought of as a scale of the intensity of the turbulent mixing there, with larger values indicating more vigorous turbulence. The role of the nonlocal term is to enhance the turbulent fluxes of temperature and salinity (but not the horizontal velocity components) under convective forcing conditions.
There are nine parameters in the KPP that pertain to convective or shear-driven mixing. A list of the five uncertain parameters in the KPP model under investigation in this work is presented in Table 1 along with the default values used in MITgcm for each parameter. The critical bulk and gradient Richardson (
List of KPP parameters to be estimated using Bayesian inference along with their MITgcm default values and assumed uniform prior.
c. Observations and test statistic
The observational data for our experiment come from the TOGA TAO mooring array for the November 2003–November 2007 time period. The array consists of 77 moorings (Fig. 1) that are centered on the equator and that span the width of the tropical Pacific in the east–west direction, in the latitude range from 8°S to 8°N (McPhaden et al. 1998). The data used included measurements of temperature, salinity, and horizontal current components.
3. Bayesian inference
a. Formalism
List of scaling factors used to weight cost function components. Here uvel is east velocity, vvel is north velocity, temp is temperature, salt is salinity, and R is the correlation. In addition, 
b. Sampling method
The described Bayesian formulation requires sampling the resulting posterior [Eq. (10)] to estimate the KPP parameters. The MCMC methods are convenient and popular sampling strategies that require a large number of posterior evaluations. In our case, each posterior evaluation requires single MITgcm simulation for a given set of KPP parameters to compute E, which is computationally prohibitive. Thus, we seek to build a surrogate model for the quantity of interest (QoI) E, as described in the following section, and use the random-walk Metropolis MCMC algorithm (Metropolis et al. 1953) to accurately and efficiently sample the posterior. Since the scaling parameter S is included in our test statistic as a scalar correction to the data covariance matrix, it is also included as a hyperparameter to be estimated in addition to the model parameters p. We assume the priors for p and S are independent. This implies that for each MCMC step we can use Gibbs sampling to iteratively generate a value of p conditional on S and a value of S conditional on p as follows.
- We simulate p conditional on S, applying a sampling algorithm for p, but for just one iteration:
- We simulate S conditional on p; for the informative gamma distribution, we havewhich results in a gamma distribution of parameters
The two steps are repeated until convergence.
4. Accelerating Bayesian inference
To reduce the cost of sampling the posterior, we rely on a surrogate model of the QoI that requires a much smaller ensemble of model runs (Malinverno 2002; Marzouk and Najm 2009). Here, we rely on polynomial chaos expansions for representing the QoIs, which, in addition can efficiently provide statistical properties, such as the mean, variance, and sensitivities (Crestaux et al. 2009).
As a result of the complexity of the MITgcm, constructing a surrogate for the different model outputs is not feasible. Instead, we construct a single surrogate for the test statistic E, which is the QoI in this case. This test statistic E is computed from the outputs of the model runs required for the construction of the surrogate, as explained below. This practice simplifies the PC calculation where only one surrogate model would be constructed (Pratola et al. 2013) that can be sampled directly in the posterior of Eq. (10) [or Eqs. (14) and (15)].
a. Polynomial chaos
Polynomial chaos is an efficient method of representing stochastic processes for the purpose of quantifying the uncertainties in a specific system (Le Maître and Knio 2010; Xiu and Tartakovsky 2004). This method is based on a probabilistic framework that represents the stochastic quantities of interest as truncated polynomial expansions (Ghanem and Spanos 1991). We briefly describe below the application of the PC method for the construction of a PC surrogate for the test statistic E.
b. Determination of PC coefficients
PC expansion coefficients 
1) Nonintrusive spectral projection
In our present work, a quadrature was built based on a Smolyak sparse nested grid (Petras 2000, 2001, 2003; Gerstner and Griebel 2003; Smolyak 1963) to reduce the number of expensive deterministic MITgcm runs. For a PC expansion of order r = 5 (total number of terms in the truncated PC expansion R + 1 = 252) and uncertain parameters m = 5, a total number of Q = 903 quadrature nodes were needed, corresponding to Smolyak level 5. A projection of the quadrature nodes is shown in Fig. 2 on a two-dimensional plane.
Projection of the Smolyak sparse quadrature nodes corresponding to level 5 on a 2D plane.
Citation: Monthly Weather Review 144, 12; 10.1175/MWR-D-15-0394.1
Projection of the Smolyak sparse quadrature nodes corresponding to level 5 on a 2D plane.
Citation: Monthly Weather Review 144, 12; 10.1175/MWR-D-15-0394.1
Projection of the Smolyak sparse quadrature nodes corresponding to level 5 on a 2D plane.
Citation: Monthly Weather Review 144, 12; 10.1175/MWR-D-15-0394.1
We therefore ran MITgcm 903 times as per the quadrature and calculated the test statistic from the different model outputs. The test statistic corresponding to the sparse quadrature is shown in Fig. 3 as a function of different parameter spaces, as indicated in each panel. The red dashed line in each panel represents the cases where the other parameters are set to the center of the corresponding priors (i.e., 
Test statistic E vs KPP parameters at the 903 sparse quadrature nodes. Each panel plots E against one of the uncertain parameters, as indicated. The red dashed line in each panel corresponds to the case when the other parameters are fixed to the midpoint of their uniform prior value (i.e., 
Citation: Monthly Weather Review 144, 12; 10.1175/MWR-D-15-0394.1
Test statistic E vs KPP parameters at the 903 sparse quadrature nodes. Each panel plots E against one of the uncertain parameters, as indicated. The red dashed line in each panel corresponds to the case when the other parameters are fixed to the midpoint of their uniform prior value (i.e.,
Citation: Monthly Weather Review 144, 12; 10.1175/MWR-D-15-0394.1
Test statistic E vs KPP parameters at the 903 sparse quadrature nodes. Each panel plots E against one of the uncertain parameters, as indicated. The red dashed line in each panel corresponds to the case when the other parameters are fixed to the midpoint of their uniform prior value (i.e.,
Citation: Monthly Weather Review 144, 12; 10.1175/MWR-D-15-0394.1
The PC expansion coefficients are computed from the output of the 903 quadrature runs using Eq. (19). Figure 4 plots the spectrum of the normalized PC coefficients, 
PC expansion normalized coefficients 
Citation: Monthly Weather Review 144, 12; 10.1175/MWR-D-15-0394.1
PC expansion normalized coefficients
Citation: Monthly Weather Review 144, 12; 10.1175/MWR-D-15-0394.1
PC expansion normalized coefficients
Citation: Monthly Weather Review 144, 12; 10.1175/MWR-D-15-0394.1
To further assess this convergence issue, we show in Fig. 5 the deterministic MITgcm realizations plotted on top of their PC expansion counterparts. The difference between the two sets of data confirms the inability of the NISP-estimated surrogate PC model to efficiently represent the QoI. In Fig. 6, we show the test statistic E, corresponding to 89 MITgcm model runs, where we vary 
Comparing test statistic E from MITgcm model runs superimposed with their PC surrogate counterparts constructed using NISP. The shown cases correspond to the sparse quadrature.
Citation: Monthly Weather Review 144, 12; 10.1175/MWR-D-15-0394.1
Comparing test statistic E from MITgcm model runs superimposed with their PC surrogate counterparts constructed using NISP. The shown cases correspond to the sparse quadrature.
Citation: Monthly Weather Review 144, 12; 10.1175/MWR-D-15-0394.1
Comparing test statistic E from MITgcm model runs superimposed with their PC surrogate counterparts constructed using NISP. The shown cases correspond to the sparse quadrature.
Citation: Monthly Weather Review 144, 12; 10.1175/MWR-D-15-0394.1
Test statistic E from MITgcm model runs when varying 
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Test statistic E from MITgcm model runs when varying
Citation: Monthly Weather Review 144, 12; 10.1175/MWR-D-15-0394.1
Test statistic E from MITgcm model runs when varying
Citation: Monthly Weather Review 144, 12; 10.1175/MWR-D-15-0394.1
As a conclusion, the construction of a converging PC expansion using the NISP method is not successful and the surrogate model is not representative of the QoI of MITgcm; therefore, it cannot be used for further analysis.
2) Basis-pursuit denoising
This 
Here, we applied the BPDN to determine the PC coefficients for the surrogate model of the test statistic. Instead of using a Monte Carlo sampling method to generate realizations, as in Peng et al. (2014), we take advantage of the previously simulated 903 MITgcm realizations and use them to solve the optimization problem. The resulting PC normalized coefficients spectrum 
PC expansion normalized coefficients 
Citation: Monthly Weather Review 144, 12; 10.1175/MWR-D-15-0394.1
PC expansion normalized coefficients
Citation: Monthly Weather Review 144, 12; 10.1175/MWR-D-15-0394.1
PC expansion normalized coefficients
Citation: Monthly Weather Review 144, 12; 10.1175/MWR-D-15-0394.1
To further check the effect of increasing the PC order r on the surrogate accuracy, the PC expansion is sampled to find the probability distribution function (PDF) of the test statistic E using different polynomial orders. The PDFs are shown in Fig. 8, where we observe that as the PC order is increased, the PDFs get closer to each other, indicating that the higher-order modes carry little energy and that a sufficiently large basis has consequently been selected for the BPDN reconstruction. In addition, in Fig. 9 we also show the cumulative distribution functions (CDFs) constructed using the full-model runs versus those obtained from the PC surrogates determined using BPDN with different bases. Figure 9 also indicates that the CDFs obtained using higher-order bases agree with each other and with the CDFs obtained from the full-model runs directly. We note that constructing CDFs using the independent model runs is not feasible, as those model runs have different weights and are not uniformly sampled; a much larger set of uniformly sampled runs would be required to extract the CDFs directly from the outputs, but in the present setting the corresponding computational cost is prohibitive.
PDFs of test statistic E with increasing order of PC constructed using a BPDN-estimated PC surrogate model.
Citation: Monthly Weather Review 144, 12; 10.1175/MWR-D-15-0394.1
PDFs of test statistic E with increasing order of PC constructed using a BPDN-estimated PC surrogate model.
Citation: Monthly Weather Review 144, 12; 10.1175/MWR-D-15-0394.1
PDFs of test statistic E with increasing order of PC constructed using a BPDN-estimated PC surrogate model.
Citation: Monthly Weather Review 144, 12; 10.1175/MWR-D-15-0394.1
CDFs of test statistic E with increasing order of PC constructed using a BPDN-estimated PC surrogate model.
Citation: Monthly Weather Review 144, 12; 10.1175/MWR-D-15-0394.1
CDFs of test statistic E with increasing order of PC constructed using a BPDN-estimated PC surrogate model.
Citation: Monthly Weather Review 144, 12; 10.1175/MWR-D-15-0394.1
CDFs of test statistic E with increasing order of PC constructed using a BPDN-estimated PC surrogate model.
Citation: Monthly Weather Review 144, 12; 10.1175/MWR-D-15-0394.1
Comparing test statistic E from MITgcm model runs with their PC surrogate counterparts (left) superimposed and (right) scatterplot. The shown cases correspond to the sparse quadrature and PC is constructed using BPDN. The NRE is also indicated.
Citation: Monthly Weather Review 144, 12; 10.1175/MWR-D-15-0394.1
Comparing test statistic E from MITgcm model runs with their PC surrogate counterparts (left) superimposed and (right) scatterplot. The shown cases correspond to the sparse quadrature and PC is constructed using BPDN. The NRE is also indicated.
Citation: Monthly Weather Review 144, 12; 10.1175/MWR-D-15-0394.1
Comparing test statistic E from MITgcm model runs with their PC surrogate counterparts (left) superimposed and (right) scatterplot. The shown cases correspond to the sparse quadrature and PC is constructed using BPDN. The NRE is also indicated.
Citation: Monthly Weather Review 144, 12; 10.1175/MWR-D-15-0394.1
For further validation of our surrogate model, an independent set of 954 MITgcm runs was conducted simultaneously by varying the same set of KPP uncertain parameters in an identical model setup and using the same test statistic. The sample is shown in Fig. 11 as a 2D projection of the 
Projection of the 954 nodes corresponding to an independent random sample on the 
Citation: Monthly Weather Review 144, 12; 10.1175/MWR-D-15-0394.1
Projection of the 954 nodes corresponding to an independent random sample on the
Citation: Monthly Weather Review 144, 12; 10.1175/MWR-D-15-0394.1
Projection of the 954 nodes corresponding to an independent random sample on the
Citation: Monthly Weather Review 144, 12; 10.1175/MWR-D-15-0394.1
Test statistic E generated using the MITgcm model vs KPP parameters for the 954 independent random samples. Each panel plots E against one of the uncertain parameters, as indicated.
Citation: Monthly Weather Review 144, 12; 10.1175/MWR-D-15-0394.1
Test statistic E generated using the MITgcm model vs KPP parameters for the 954 independent random samples. Each panel plots E against one of the uncertain parameters, as indicated.
Citation: Monthly Weather Review 144, 12; 10.1175/MWR-D-15-0394.1
Test statistic E generated using the MITgcm model vs KPP parameters for the 954 independent random samples. Each panel plots E against one of the uncertain parameters, as indicated.
Citation: Monthly Weather Review 144, 12; 10.1175/MWR-D-15-0394.1
The PC expansion is calculated for the different sample points and is shown in Fig. 13 (left) versus the MITgcm realization. The PC expansion appears to reproduce the mean of the deterministic model. Figure 13 (right) shows the same but in a scatterplot where MITgcm realizations are plotted against their PC counterparts. The NRE is calculated and found to be ~4.2%, which is also quite acceptable, indicating that BPDN is successful in producing a surrogate that is able to accurately simulate the QoI.
Comparing test statistic E from MITgcm model runs with their PC surrogate counterparts (left) superimposed and (right) scatterplot. The shown cases correspond to the 954 independent random samples and PC is constructed using BPDN. The NRE is also indicated.
Citation: Monthly Weather Review 144, 12; 10.1175/MWR-D-15-0394.1
Comparing test statistic E from MITgcm model runs with their PC surrogate counterparts (left) superimposed and (right) scatterplot. The shown cases correspond to the 954 independent random samples and PC is constructed using BPDN. The NRE is also indicated.
Citation: Monthly Weather Review 144, 12; 10.1175/MWR-D-15-0394.1
Comparing test statistic E from MITgcm model runs with their PC surrogate counterparts (left) superimposed and (right) scatterplot. The shown cases correspond to the 954 independent random samples and PC is constructed using BPDN. The NRE is also indicated.
Citation: Monthly Weather Review 144, 12; 10.1175/MWR-D-15-0394.1
We end this section by stating that in the present setting the QoIs are seen to exhibit appreciable variation even when infinitesimal changes in model parameters are considered, as indicated in Fig. 6. The resulting nonsmoothness manifests itself as high-amplitude high-frequency noise superimposed on a milder variability. The compressed sensing was robust with this noise and was effective in capturing the underlying variability. However, the presence of the noise effectively precludes an application of a straightforward convergence analysis.
c. PC sensitivity to ensemble size
Normally, a number N of random samples is required to compute the PC coefficients using BPDN; this number is much less than the number of the unknowns, that is, the size of the PC expansion (R + 1 = 252) such that N ≪ R + 1. In the above results, we instead used an ensemble of Q = 903 sparse quadrature MITgcm model runs corresponding to Smolyak level 5 (Fig. 2) to compute the PC coefficients using BPDN due to their availability (upon using NISP method) where in this case N = Q ≫ R + 1. Here, we explore the possibility of utilizing a small number of MITgcm realizations to build a faithful PC expansion surrogate. To create a smaller ensemble and avoid running new expensive MITgcm simulations, we consider lower levels of refinement of the Smolyak quadrature shown in Fig. 14 in the canonical vector space for levels 1, 2, 3, and 4 with total numbers of nodes of 11, 51, 151, and 391, respectively. These levels are nested and therefore the nodes in each level are a subset of the higher levels. Thus, a small number of model runs N can be extracted from the original Q = 903 runs, as per the Smolyak levels, and used to construct different PC models using BPDN.
The 2D Smolyak sparse quadrature nodes corresponding to levels 1–4 in the canonical vector space.
Citation: Monthly Weather Review 144, 12; 10.1175/MWR-D-15-0394.1
The 2D Smolyak sparse quadrature nodes corresponding to levels 1–4 in the canonical vector space.
Citation: Monthly Weather Review 144, 12; 10.1175/MWR-D-15-0394.1
The 2D Smolyak sparse quadrature nodes corresponding to levels 1–4 in the canonical vector space.
Citation: Monthly Weather Review 144, 12; 10.1175/MWR-D-15-0394.1
In Fig. 15, we show the NRE of the original 903 Smolyak runs computed using PC models built employing quadrature nodes from the different Smolyak levels. We observe that the error increases as N decreases; however, the increase is not significant. In fact, even with level 2 (51 model runs) the NRE is around 5%. We note that in all the results below we used the PC expansion model constructed using the level-5 Smolyak runs with BPDN.
NRE results found using different numbers of nodes corresponding to different Smolyak levels of refinement.
Citation: Monthly Weather Review 144, 12; 10.1175/MWR-D-15-0394.1
NRE results found using different numbers of nodes corresponding to different Smolyak levels of refinement.
Citation: Monthly Weather Review 144, 12; 10.1175/MWR-D-15-0394.1
NRE results found using different numbers of nodes corresponding to different Smolyak levels of refinement.
Citation: Monthly Weather Review 144, 12; 10.1175/MWR-D-15-0394.1
d. Statistical moments and sensitivity analysis
The sensitivities of each of the uncertain KPP parameters are calculated and summarized in Table 3. It is notable that the test statistic is most sensitive to 
Reporting mean and standard deviation of test statistics in addition to the total sensitivity index of each uncertain parameter.
e. Response surfaces
Another advantage of the PC expansion representation is the possibility of sampling the surrogate model with almost no computational cost, as it only requires polynomial evaluations. This allows us to construct a response surface for the E function of the different uncertain KPP parameters shown in Fig. 16. Each curve shows an E function of a single uncertain parameter 
Response curves of test statistic E as a function of different parameters (in canonical space). For each curve, the other four parameters are set to 
Citation: Monthly Weather Review 144, 12; 10.1175/MWR-D-15-0394.1
Response curves of test statistic E as a function of different parameters (in canonical space). For each curve, the other four parameters are set to
Citation: Monthly Weather Review 144, 12; 10.1175/MWR-D-15-0394.1
Response curves of test statistic E as a function of different parameters (in canonical space). For each curve, the other four parameters are set to
Citation: Monthly Weather Review 144, 12; 10.1175/MWR-D-15-0394.1
Response surfaces of test statistic E as a function of (top) 
Citation: Monthly Weather Review 144, 12; 10.1175/MWR-D-15-0394.1
Response surfaces of test statistic E as a function of (top)
Citation: Monthly Weather Review 144, 12; 10.1175/MWR-D-15-0394.1
Response surfaces of test statistic E as a function of (top)
Citation: Monthly Weather Review 144, 12; 10.1175/MWR-D-15-0394.1
5. KPP inference
In this section, we report on the results of applying Bayesian inference to update the PDFs of the KPP parameters. To this end, we sample from the posterior distributions [Eqs. (14) and (15)] using a random-walk MCMC algorithm (Metropolis et al. 1953) that requires a large number of MITgcm simulations to compute E for each sampled set of KPP parameters. To avoid such computational cost, we used the constructed PC surrogate discussed in section 4 to compute E predictions instead and use them in Eqs. (14) and (15). We note that we ran the MCMC sampler 106 times to attain convergence, as discussed below.
The outputs of the MCMC algorithm are the sample chains for the different KPP parameters in addition to the scaling parameter S. We show in Fig. 18 the trace plots of these parameters (chains versus MCMC iteration number) that clearly demonstrate well-mixed chains. This indicates that the distribution of the chains is expected to remain unchanged with further sampling and to converge to a stationary distribution. To further test for MCMC convergence, we computed the running mean for the different KPP parameters and plotted the results in Fig. 19. The plot shows that the running means have reached a steady state, indicating a good level of convergence among the chains. We also computed the empirical autocorrelation at lag s 
Chains of parameters using MCMC samples. The MITgcm default values are indicated as horizontal lines.
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Chains of parameters using MCMC samples. The MITgcm default values are indicated as horizontal lines.
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Chains of parameters using MCMC samples. The MITgcm default values are indicated as horizontal lines.
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Running mean of the different chains.
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Running mean of the different chains.
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Running mean of the different chains.
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Autocorrelation at lag s of the different MCMC chains.
Citation: Monthly Weather Review 144, 12; 10.1175/MWR-D-15-0394.1
Autocorrelation at lag s of the different MCMC chains.
Citation: Monthly Weather Review 144, 12; 10.1175/MWR-D-15-0394.1
Autocorrelation at lag s of the different MCMC chains.
Citation: Monthly Weather Review 144, 12; 10.1175/MWR-D-15-0394.1
In Fig. 18, we also show the MITgcm default KPP parameters (from Table 1) as horizontal lines in each panel for comparison with the range of the chains. The chains of both 
To confirm our conclusions from the chains, we estimated the marginalized posterior distributions from the MCMC chains of each parameter, discarding the first 2 × 105 burn-in iterations, using kernel density estimation (KDE) (Parzen 1962; Silverman 1986). We show these marginalized posterior PDFs in Fig. 21. Regarding 
PDFs of parameters using KDE. The shaded regions correspond to the 95% intervals of high posterior probability.
Citation: Monthly Weather Review 144, 12; 10.1175/MWR-D-15-0394.1
PDFs of parameters using KDE. The shaded regions correspond to the 95% intervals of high posterior probability.
Citation: Monthly Weather Review 144, 12; 10.1175/MWR-D-15-0394.1
PDFs of parameters using KDE. The shaded regions correspond to the 95% intervals of high posterior probability.
Citation: Monthly Weather Review 144, 12; 10.1175/MWR-D-15-0394.1
The posterior distributions are consistent with results of previous efforts to calibrate the KPP model (Large et al. 1994; Large and Gent 1999). They show that the default critical bulk and gradient Richardson numbers and the nonlocal convective parameter 
6. Summary and conclusions
In this work, we have presented a polynomial chaos–based method for the inference of five KPP parameters. Below, we remark on the PC construction process and the KPP inference results.
The method relied on building a surrogate model for a newly developed test statistic instead of the model output. The advantage was to avoid building surrogates for several model outputs and for different time scales. The PC construction was implemented using two techniques. First a nonintrusive spectral projection method was used that required a quadrature of level 5 nodes (Q = 903 model runs). The computed PC expansion suffered from convergence issues as a result of the presence of internal noise in the predictions. This resulted in overfitting of the data with increasing levels of PC refinement. The resulting PC model was thus unsuitable and discarded. The second technique used was BPDN, which tolerates noise in the model and assumes sparsity in the PC expansion, requiring a smaller number of model runs. This technique proved to be successful, as it excludes the noise from the model when finding the PC coefficient that leads to a faithful PC surrogate. Several error metrics were computed to check the validity of the PC model.
As follow-up research, we are seeking to infer all nine KPP parameters in a calibration where the wind is treated as an adjustable parameter. The additional number of parameters dramatically increases the number of required expensive model runs. Thus, a different approach is needed where we will investigate an adaptive approach to PC (Winokur et al. 2013).
Acknowledgments
This research made use of the resources of the Supercomputing Laboratory and computer clusters at King Abdullah University of Science and Technology (KAUST) in Thuwal, Saudi Arabia. IS, SZ, CJ, and IH are supported in part by KAUST Award CRG-1-560 2012-HOT-007; SZ and OK are supported in part by the Office of Advance Scientific Computing Research, U.S. Department of Energy, under Award DE-SC0008789.
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