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  • View in gallery

    Biases by time of day for 1-h predictions of shelter-height (a) temperature and (b) water vapor mixing ratio, and (c) anemometer-height wind speed.

  • View in gallery

    As in Fig. 1, but at the first model layer (~15 m AGL) and for wind speed.

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    Surface (a) sensible heat and (b) latent heat flux bias by time of day. A gap in the latent heat flux results from one or more missing observations at that particular time of day.

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    Experiment- and ensemble-mean observation increments for (a) 2-m T, (b) 2-m Qυ, and (c) 10-m wind speed.

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    First-layer (15 m AGL) analysis increments for (a) T, (b) Qυ, and (c) wind speed.

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    First-layer (15 m AGL) vs surface-layer diagnostic values for (a) T, (b) Qυ, and (c) wind speed at 1230 LDT from experiment DA-SLR-SW. All 100 ensemble members and 29 days are plotted.

  • View in gallery

    (a) Estimates of CZil for three estimate experiments (+, ◊, ). (b) A narrower date range to better see the details.

  • View in gallery

    Biases by time of day for 1-h predictions at the first model layer (~15 m AGL) of (a) temperature, (b) water vapor mixing ratio, and (c) wind speed.

  • View in gallery

    First-layer (15 m AGL) analysis increments for (a) T, (b) Qυ, and (c) wind speed.

  • View in gallery

    As in Fig 8, but for predictions of shelter-height and anemometer-height wind speed.

  • View in gallery

    Biases by time of day for 1-h predictions at the first model layer (~15 m AGL) of (a) temperature, (b) water vapor mixing ratio, and (c) wind speed.

  • View in gallery

    First-layer (15 m AGL) analysis increments for (a) T, (b) Qυ, and (c) wind speed.

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Ensemble Data Assimilation to Characterize Surface-Layer Errors in Numerical Weather Prediction Models

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  • 1 Naval Postgraduate School, Monterey, California
  • | 2 Cooperative Institute for Research in Environmental Sciences, University of Colorado, and NOAA/Earth System Research Laboratory, Boulder, Colorado
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Abstract

Experiments with the single-column implementation of the Weather Research and Forecasting Model provide a basis for deducing land–atmosphere coupling errors in the model. Coupling occurs both through heat and moisture fluxes through the land–atmosphere interface and roughness sublayer, and turbulent heat, moisture, and momentum fluxes through the atmospheric surface layer. This work primarily addresses the turbulent fluxes, which are parameterized following the Monin–Obukhov similarity theory applied to the atmospheric surface layer. By combining ensemble data assimilation and parameter estimation, the model error can be characterized. Ensemble data assimilation of 2-m temperature and water vapor mixing ratio, and 10-m wind components, forces the model to follow observations during a month-long simulation for a column over the well-instrumented Atmospheric Radiation Measurement (ARM) Central Facility near Lamont, Oklahoma. One-hour errors in predicted observations are systematically small but nonzero, and the systematic errors measure bias as a function of local time of day. Analysis increments for state elements nearby (15 m AGL) can be too small or have the wrong sign, indicating systematically biased covariances and model error. Experiments using the ensemble filter to objectively estimate a parameter controlling the thermal land–atmosphere coupling show that the parameter adapts to offset the model errors, but that the errors cannot be eliminated. Results suggest either structural errors or further parametric errors that may be difficult to estimate. Experiments omitting atypical observations such as soil and flux measurements lead to qualitatively similar deductions, showing the potential for assimilating common in situ observations as an inexpensive framework for deducing and isolating model errors.

Corresponding author address: Joshua Hacker, Naval Postgraduate School, Department of Meteorology, 589 Dyer Road, Monterey, CA 93943. E-mail: hacker@nps.edu

Abstract

Experiments with the single-column implementation of the Weather Research and Forecasting Model provide a basis for deducing land–atmosphere coupling errors in the model. Coupling occurs both through heat and moisture fluxes through the land–atmosphere interface and roughness sublayer, and turbulent heat, moisture, and momentum fluxes through the atmospheric surface layer. This work primarily addresses the turbulent fluxes, which are parameterized following the Monin–Obukhov similarity theory applied to the atmospheric surface layer. By combining ensemble data assimilation and parameter estimation, the model error can be characterized. Ensemble data assimilation of 2-m temperature and water vapor mixing ratio, and 10-m wind components, forces the model to follow observations during a month-long simulation for a column over the well-instrumented Atmospheric Radiation Measurement (ARM) Central Facility near Lamont, Oklahoma. One-hour errors in predicted observations are systematically small but nonzero, and the systematic errors measure bias as a function of local time of day. Analysis increments for state elements nearby (15 m AGL) can be too small or have the wrong sign, indicating systematically biased covariances and model error. Experiments using the ensemble filter to objectively estimate a parameter controlling the thermal land–atmosphere coupling show that the parameter adapts to offset the model errors, but that the errors cannot be eliminated. Results suggest either structural errors or further parametric errors that may be difficult to estimate. Experiments omitting atypical observations such as soil and flux measurements lead to qualitatively similar deductions, showing the potential for assimilating common in situ observations as an inexpensive framework for deducing and isolating model errors.

Corresponding author address: Joshua Hacker, Naval Postgraduate School, Department of Meteorology, 589 Dyer Road, Monterey, CA 93943. E-mail: hacker@nps.edu

1. Introduction

Coupling between the land and the atmosphere result from two processes: 1) molecular fluxes of sensible and latent heat through the interface and roughness sublayer, and 2) turbulent heat, moisture and momentum fluxes through the atmospheric surface layer. Turbulent fluxes at the top of the atmospheric surface layer provide lower boundary conditions for atmospheric models. As shown by many sensitivity studies (e.g., Pielke 2001; Cheng and Cotton 2004; Weckworth et al. 2004; Holt et al. 2006, and references therein) those fluxes can have profound impacts on weather simulations and forecasts. Interpretations of atmospheric sensitivity depend on faithful representation of the actual fluxes, and predictability studies relying on a model to faithfully reproduce atmospheric sensitivity (e.g., Hacker 2010) could produce misleading results if modeled coupling is poorly represented. Objective methods to identify and characterize land–atmospheric coupling errors appearing in either modeled process 1 or 2 are sparse.

The focus of this work is local coupling. One way to characterize local coupling is by the sensitivity of the fluxes to vertical gradients adjacent to the surface of the earth. When the coupling is strong, a small increase in the vertical gradient can produce a large increase in the fluxes to reduce the gradient. A weak vertical gradient can then be a sign of either strong coupling, or a lack of forcing from other system components that act to produce a strong gradient, making interpretation difficult. Vertical gradients and turbulent fluxes need to be evaluated together and in the context of other model components.

Local coupling described here differs from the slower, larger-scale coupling addressed by the Global Land–Atmosphere Coupling Experiment (GLACE; Koster et al. 2006; Guo et al. 2006). The GLACE effort focused on regional scales and time scales of several days in atmospheric general circulation models. Linking the variability among multiple models and in time to total boundary-condition forcing, they deduce areas around the globe where the modeled atmospheric water cycle is strongly tied to soil moisture. The resulting coupling is described as “local” because the analysis reveals that precipitation in regions of strong coupling can be related to the corresponding regional soil moisture and evaporation.

This work examines the potential of combining systematic forecast errors, systematic increments, and parameter estimation in a cycling data assimilation (DA) system to identify persistent error structures arising from model formulation. We focus on land–atmosphere exchanges through the modeled atmospheric surface layer by using a single-column model (SCM) to reduce the degrees of freedom, and data assimilation to constrain the local model state that is directly interacting with surface-layer fluxes. The Weather Research and Forecasting Model (WRF; Skamarock et al. 2008) SCM serves as the test bed for these experiments.

Conventional observations at shelter (2 m) and anemometer (10 m) height above ground level (AGL) can be assimilated, and effective assimilation assures that very short-range predictions of the variables to be assimilated are skillful. Elements in the state that are constrained via model-derived covariance estimates can then be evaluated for fidelity with independent observations. Here near-surface temperature, humidity (2 m above ground), and winds (10 m above ground) are assimilated by ensemble DA, while predictions of surface fluxes and the column above the assimilated observations are evaluated from independent observations. Systematic departures from the observations, even if small, help identify the source of model error. Experiments to estimate a parameter leading to known sensitivity in surface fluxes leads to further insight.

A goal of this work is to explore the potential for data assimilation to characterize coupling errors at locations containing widely deployed and relatively inexpensive surface-observation systems. Rather than be tied to intensive field programs or atypical observations that measure fluxes, soil state, and atmospheric response directly but perhaps less accurately, we hope to deduce the coupling errors. Here analysis starts with high-quality more complete observations available at the Southern Great Plains (SGP) Central Facility for the Atmospheric Radiation Measurement (ARM) experiment, then proceeds to limited observation sets.

Objective characterization of errors in atmospheric model formulation (model error or inadequacy) remains a challenging task. Traditional forecast verification, where model predictions are compared to either observations or analyses, are important but limited. Measured errors when nonlinearity is present include some inseparable combination of errors arising from chaotic error growth due to uncertain initial conditions and the model error.

Data assimilation offers a powerful addition to methodologies for understanding error sources. Very short-range forecasts are explicitly compared to observations before assimilating them, providing simple quadratic measures of forecast error. In an effective and well-tuned DA system, those errors are measured during the period of linear error growth, offering the possibility of disentangling model error and initial condition error. Effective DA goes further by incrementing the model state toward the observations, and the increments are consistent with both observation and forecast uncertainty. Analogous to estimating forecast bias, which results exclusively from model error when the observations are unbiased, the increments can be analyzed for systematic behavior that reveals space and time scales of model error.

Data assimilation can also provide objective estimates of model parameters (e.g., coefficients in physical parameterization schemes) that are correlated with observed variables. Parameter estimation can be used to quantify the linear component of model error resulting from mis-specification of parameters. In the limit of a set of parameters varying to account for all parametric error in a model, errors in model formulation remain. It remains a challenge to identify a term or set of terms in the equations responsible for the errors. But systematic errors remaining in parameter estimation experiments, or unphysical combinations of estimated parameters, offer the starting point for physical interpretation.

Ensemble-based DA is advantageous because the ensemble mean represents a pseudostate of the model with the randomlike (chaotic) errors filtered out. In the limit of an infinite ensemble size, the ensemble-mean error can be interpreted as a bias estimate conditioned on all observations and model states prior to the present time. That is, it is the bias relevant to current atmospheric dynamics, and may contain complicated spatial structures. Ensemble-mean error can be further averaged over some other conditional sample. Emphasis here is on systematic error as a function of time of day, and we average the ensemble-mean error over every 24-h period for a month to estimate the diurnal evolution of bias. Because the ensemble filters the error at a specific prediction time, fewer days are needed (than with a deterministic forecast system) to gain a robust estimate of the systematic error. Increments on the ensemble mean can be analyzed accordingly. Parameter estimates are fully time-dependent and can also evolve with the diurnal cycle.

Background errors and analysis increments are established and theoretically grounded sources of information for measuring bias in atmospheric models (e.g., Dee and Da Silva 1998; Martin et al. 2002, among others). Typical estimation studies emphasize the utility of error estimates for improving the data assimilation process, and make little attempt at linking the errors to a particular model component or process. Rodwell and Palmer (2007) examined short-range tendencies from multiple physical parameterization schemes to identify poor combinations and reject climate model configurations that are likely to be misleading. They lucidly make the case that the tendency analysis depends on assimilation, and on small increments indicating good predictions of the observations. The same requirements exist here, but we analyze the increments directly.

A series of recent papers has adopted and extended the graphical flux representation described in Betts (1992) to quantify local land–atmosphere coupling in terms of 2-m water vapor Qυ and temperature T (Santanello et al. 2009, 2011). By plotting the joint temporal evolution of 2-m T and Qυ, and assuming a mixed-layer model for the PBL, surface sensible and latent heat fluxes can be represented as vectors. The local energy budget can be approximately closed by plotting advection vectors on the same axes. That work compared several different combinations of planetary boundary layer (PBL) and land surface model (LSM) schemes to identify differences in modeled coupling, and also compared the models to observations.

Results from data assimilation experiments offer a complementary view of coupling errors. By assimilating frequently, the model remains close to the assimilated observations in the phase space defined by 2-m T and Qυ, and 10-m zonal and meridional wind components U and V. Covariances between the model-predicted (formally, diagnosed from predicted quantities) observations and the remaining model state constrain the latter to remain close to the atmosphere in the higher-dimensional phase space defined by a physical region in the model state. Accurate covariances keep the trajectories of the atmosphere and the model similarly close in all dimensions. Systematically erroneous covariances allow errors to grow in some dimensions disproportionally to others.

The next section describes the model, DA, and experiment configuration. Section 3 demonstrates the ability of DA to constrain the model to follow observations in the neighborhood of the observations, and shows that DA can overcome other model deficiencies. Section 4 interprets the systematic errors with the help of the Monin–Obukhov similarity theory (MOST) for the surface layer. Section 5 describes the parameter estimation experiments and interprets the remaining error. Section 6 examines the potential to extend these methods to locations with limited observations, and section 7 summarizes the key results and offers some contextual comments.

2. Ensemble data assimilation with an SCM

a. Experiment details

The primary goal of this work is to determine whether ensemble DA can be used to quickly diagnose errors in the surface-layer coupling between the model land and atmosphere without the need for atypical atmospheric observations that are usually only available at experimental locations or during special field campaigns. That goal demands experiments with and without intensive sampling. Availability of high-quality observations from many instruments makes the ARM Central Facility near Lamont, Oklahoma, a suitable experiment location. We examine the time period from 3 May to 14 July 2003, during which WRF 3D forecasts were also archived at the National Center for Atmospheric Research (NCAR) in support of the Bow Echo and Mesoscale Convective System (MCS) Experiment.

SCM initialization is at 1200 UTC [0700 local daylight time (LDT)] 3 May 2003, and the last assimilation is at 1200 UTC 1 June 2003. Observations are assimilated every 60 min on the hour using an ensemble adjustment Kalman filter (EAKF) as described in Anderson (2001, 2003) and implemented in the Data Assimilation Research Testbed (DART; Anderson et al. 2009) at NCAR. The EAKF is one approach to solving the statistical analysis equations that form the basis for data assimilation, and we present them here for completeness:
e1
e2
e3
An analysis xa results from combining a forecast xf with the analysis increment. An innovation vector, namely, the differences between observations yo and the model-predicted observations obtained by operating on the forecast with linear , is weighted and regressed onto the model state variables (elements of x) by . Weights for regressions are estimated from the forecast error covariance matrix f and the observation error variance (here diagonal). The analysis error covariance a results from scaling f by . Forecasts are samples from f in ensemble Kalman filters (Evensen 1994), and the forecast error covariance is estimated from the ensemble as the forecast covariance formed by taking the outer product of the ensemble of perturbations. The analysis ensemble is a sample from a. DART never forms these matrices, and the analysis equations are instead solved with a sequence of pairwise linear regressions between observation increments and analysis increments as in Anderson (2003).

Equations (1)(3) are derived by assuming a perfect model, resulting in unbiased xf and f. The underlying theory supports a linear evolution of model error (e.g., Dee and Da Silva 1998), and the forecast error covariances are modified accordingly. Covariance estimates for the model error are particularly difficult to obtain. Although that is not our goal here, the present work could be extended to augment f with model error covariances for lower boundary forcing.

All observations used in initial conditions, forcing, and assimilation are 30-min averages. Shelter height (2 m), anemometer height (10 m), and land surface (skin) verification data are also 30-min averages. This removes much of the small-scale, fast variability and is more faithful to the scale separation and averaging assumed by MOST. Verification at the first model layer (15 m) is accomplished with instantaneous rawinsonde observations.

b. Model and assimilation

WRF version 3.2 (Skamarock et al. 2008) is implemented within DART. A 3 × 3 horizontal grid stencil with doubly periodic lateral boundary conditions results in an SCM for the WRF (see the appendix). Evolving geostrophic wind components are assigned as the background wind in the WRF-SCM, so that the SCM only needs to provide the ageostrophic wind component, which will largely depend on the state of the modeled PBL. “Large scale” vertical velocities are imposed via an additional advective forcing term on the right-hand side of the WRF vertical-velocity equation, following Ghan et al. (1999). Geostrophic winds and the large-scale vertical velocities come from the North American Regional Reanalysis (NARR) 30-km product. No horizontal advection is imposed, and instead the data assimilation increments will account for the effects of advection on the observed atmosphere. These experiments use a vertically stretched grid of 81 levels where the lowest layer is approximately 15 m thick and the model top is at 16 km above ground.

PBL parameterization in these experiments is handled by the Mellor–Yamada–Janjić (MYJ) scheme, which is the Mellor and Yamada (1982) level-2.5 scheme augmented by Janjić (2001). Surface-layer parameterization follows MOST extended by Beljaars and Holtstag (1991) to the free-convection regime. The Noah land surface model (LSM; Chen and Dudhia 2001; Ek et al. 2003), which is a four-layer soil temperature and moisture model with canopy moisture and snow-cover parameterizations, predicts soil temperature and moisture. It also provides sensible and latent heat fluxes to the atmospheric model taking into account atmospheric state, net radiation at the surface, and land-use characteristics such as vegetation type and soil texture.

The Lin scheme (Lin et al. 1983; Chen and Sun 2002) parameterizes cloud microphysics. The Rapid Radiative Transfer Model (RRTM) from Mlawer et al. (1997), and the scheme outlined by Dudhia (1989) parameterize longwave and shortwave irradiance, respectively. Deep convective processes are omitted.

Although the SCM explicitly ignores most scale interactions, the integrated effects are contained in the NARR and imposed on the SCM through the geostrophic winds and vertical velocity advection. We show later that this simplified modeling approach, combined with ensemble DA of surface observations, results in small 1-h prediction errors. It follows that the increments provided by the DA system are small, allowing a safe diagnosis of error structures in the model.

The control (xc) initial conditions are constructed from the NARR valid at 1200 UTC 3 May, and control forcing is constructed from the NARR valid every 3 h during that period. No control run is needed for these experiments, but initial and forcing ensembles are implicitly centered on the control; they are constructed from random linear combinations of the NARR profiles as follows: 1) a 30-day NARR profile time series beginning at 1200 UTC is selected randomly from start dates between 1 May and 15 June 2003, inclusive, as a sample of a perturbed state x′; 2) a uniform random number α between 0 and 1 is drawn; and 3) an ensemble-member xn initial condition and time series of forcing profiles is generated with xn = αxc + (1 − α)x′. Repeating the process 100 times gives the 100-member ensemble. Choosing the forcing sample elements to begin at the same time of day ensures that diurnal variations are approximately preserved in the forcing.

c. Additional constraints

Observations are used to initialize the soil and also to partially constrain the surface energy balance in some of the experiments. By initializing the LSM soil temperature and moisture profiles with observations, soil states are initially in a realistic thermal balance with the observations that are immediately assimilated into the near-surface atmosphere. Observations coinciding with the SCM are given by the average of the east and west soil probes at the ARM Central Facility, which are 1 m apart. Soil temperature is measured by thermocouples, and soil moisture is estimated from the temperature and soil water potential measurements by calibrating against laboratory analysis. Further details can be found in Bond (2005). Assuming unbiased observations the relationship between the near-surface observations and the soil is also unbiased. Results will show that using soil observations in place of LSM-predicted soil states improves all model simulations, suggesting that disparate scaling between the LSM and the observations is not a problem here. Ensuing systematic deviations are attributable to model error.

Sensible and latent heat fluxes are given by MOST, and are functions of the aerodynamic surface and first model layer values of temperature, moisture, and velocity. Values for T0 and Q0 (aerodynamic surface) are inherited from the energy balance solution in the LSM during the previous model time step. Wind, T, and Qυ values on the first model level are given from the dynamic solution of the previous step. Sensible and latent heat fluxes define exchange coefficients, which force the LSM.

Given sensible and latent heat fluxes at the surface, ground heat flux is diagnosed in the LSM. Then the surface energy balance is used to solve for a radiative skin temperature . We adopt the values for emissivity ɛ = 0.985, and albedo range of α = 0.18–0.23, agreeing with the WRF specifications for the U.S. Geological Survey vegetation classification of mixed dry and irrigated cropland and pasture. Note that the LSM provides a radiative surface temperature, which is inconsistently used by the surface-layer scheme to represent the aerodynamic surface temperature. When the PBL is unstable, which is typical during the day in these experiments, we can expect that Ts > T0, and vice versa when the PBL is stable. The first-order effect of the inconsistency between the LSM and the surface-layer scheme should produce an excess of upward sensible heat flux during the day and a deficit at night. One method of addressing the inconsistent usage of radiative temperature for aerodynamic temperature is given by Zilitinkevich (1995), and used here as in the standard WRF-Noah configuration. A parameter modulates surface heat fluxes by determining the ratio of momentum to heat roughness lengths; as described later that parameter is useful to test whether errors in the surface-layer scheme arise from mis-specified parameters.

d. Experiment design

We design a set of experiments to provide a basis for deductive reasoning to identify model error associated with land–atmosphere coupling through the atmospheric surface layer. We first make use of the extensive and high-quality observations available at the ARM Central Facility, but show later that a subset of experiments can lead us to the same central conclusions. Table 1 provides a key to experiment naming.

Table 1.

Key to experiment naming.

Table 1.

A reference simulation is denoted FREE because it lacks any data assimilation, and is the ensemble derived from forcing the WRF-SCM with geostrophic winds and vertical velocities from the NARR. An ensemble cycling while assimilating the hourly near-surface observations is called DA. We show later that after the first few cycles the assimilation weakly forces the SCM while producing accurate hour-long predictions of the lower atmosphere. That result allows interpretation of inter-relationships between model variables.

When incrementing a model state in data assimilation, we have the option of incrementing the soil temperature and moisture (i.e., the soil state). Assuming the covariance between the soil state and the near-surface atmospheric-model state is sampled robustly in the ensemble, we expect that incrementing the soil in a perfect model will improve predictions. It is not immediately clear whether incrementing the soil in an imperfect model will be helpful, and the result will depend on the severity of any imperfections. In the DA run we do not include increments to the soil, and thus include a third simulation. DA-SLI updates the model soil using the covariance between the predicted values of the observations and predicted soil state.

Use of soil-state observations throughout an experiment permits explicit evaluation of the role of LSM error. Observed soil temperature and moisture replace model soil moisture and temperature hourly (denoted SLR). Precipitation directly modifies the LSM state via the soil moisture in SLR. For FREE-SLR, this ensures that the SCM soil state does not drift too far from reality, and should provide more accurate lower boundary forcing for the atmospheric column. For DA-SLR the relationship between the atmospheric model and LSM can be understood with help from the DA-SLI experiments. In the limit of a perfect atmospheric prediction, an imperfect atmosphere-soil covariance would produce an imperfect soil state. When replacing the soil with observations, and assuming unbiased observation, the soil-atmosphere covariance is ignored by the data assimilation. Degraded atmospheric predictions, compared to SLI, then implicate biased covariances.

Finally, the surface energy balance can be better constrained by imposing observed 30-min average downwelling shortwave irradiance at the surface (denoted SW). Between the hourly forcing update times, irradiance is linearly interpolated. The effects of clouds are therefore included in SW experiments, even if the SCM does not contain any cloud water. Longwave irradiance may be inconsistent, but we focus on the daytime here, when longwave irradiance is small compared to shortwave irradiance.

In all DA experiments we evaluate the ensemble prior to assimilation (a 1-h forecast). Spurious transients that may be introduced by the assimilation process do not contaminate any results (although we found no evidence of transients). Systematic departures of the background ensemble from the observations indicate a systematic model error. The analysis increment acts to reduce those errors each time observations are assimilated.

Observations are 30-min-averaged 2-m temperature T2 and water vapor mixing ratio Q2, and 10-m wind components (U10, V10). The 2-m observations are shelter height and the 10-m observations are anemometer height. To avoid overfitting the observations, we impose observation error variance estimates of 1 K2 and 10−6 g2 g−2 for T2 and Q2, respectively, and 2 m2 s−2 for the wind components. The results will demonstrate these values are large enough to prevent overfitting, but small enough to produce a robust systematic increment toward the observations.

Forcing the Noah LSM requires data at a single height above ground, taken to be the first model layer when coupled with the atmospheric model. The DA constrains the first model layer via the covariances between observation heights and the first model layer, which are established by the surface-layer scheme. This differs from forcing the LSM directly from the observations, where a wind profile must be assumed to reduce the wind speed from 10 to 2 m above ground. The approach here is consistent with the goal of quantifying errors in the land–atmosphere coupling as determined by the profiles through the surface layer.

3. Observation-space and neighboring state-space errors

Data assimilation results in skillful predictions of the observations to be assimilated, and DA constrains the model state in the neighborhood of the assimilated observations via the model-estimated covariances. The atmospheric model elements controlling land–atmospheric coupling are thereby isolated and can be analyzed unambiguously. This section provides supporting results. We start by taking advantage of the atypical observations available at the ARM site, including downwelling solar irradiance, soil temperature, and soil moisture. By strongly constraining the soil and downwelling radiation, which are large contributors to the surface energy balance, imperfections in the land surface model are eliminated from the analysis. Later in section 6 we remove these constraints and use only conventional observations.

Ensemble-mean prediction error approaching observation error shows that model evolution reasonably approximates the evolution of the atmosphere. Further averaging of 24-h periods during the experiment gives a robust estimate of mean error, which we interpret as systematic error.

Given a 1-h prediction from the ith ensemble member, and a verifying observation yo, the systematic prediction error at time-of-day t is . Operator 〈·〉 denotes the mean over i ensemble members, operator denotes the mean over the experiment, and both are evaluated at t. This is a mean error for a specific local time of day estimated from the ensemble and one-month sample. Mean error quantifies systematic departures from the evolution of the atmosphere at the observed locations and in the observed variables. For ease of discussion, we will refer to the mean error as “bias.”

Observations correspond to values diagnosed within the surface-layer scheme that adhere to surface-layer similarity theory. In a data assimilation context, these are predicted observations, and are the elements of vector xf. Formally, the DA system is optimized on the error in predicting observations, making them the natural place to start evaluation.

Bias-reduction effects of using the ensemble filter (DA experiments) are clear when comparing to the baseline experiments (FREE experiments in Fig. 1), and DA results appear relatively insensitive to additional configuration choices. This result is broadly consistent with the results in Rostkier-Edelstein and Hacker (2010), who found that forecast skill of the assimilated observations, with a simpler SCM, was primarily determined by whether or not DA was used. In the absence of soil observations, DA removes most T2 and Q2 biases, and reduces wind speed bias by a factor of at least 2.

Fig. 1.
Fig. 1.

Biases by time of day for 1-h predictions of shelter-height (a) temperature and (b) water vapor mixing ratio, and (c) anemometer-height wind speed.

Citation: Monthly Weather Review 141, 6; 10.1175/MWR-D-12-00280.1

Assuming that scaling between soil observations and the LSM model is not an important issue here, a perfect atmospheric model (including the coupling schemes) with SLR-SW would lead to the most skillful predictions within either the DA (if functioning well) or FREE groups. Here soil-state replacement (circles) reduces T2 and Q2 bias magnitude compared to FREE by itself, suggesting a lack of scaling issues. Including observed downwelling irradiance as well results in T2 biases with greater magnitude than FREE. That is, FREE-SLR-SW is more biased than either FREE or FREE-SLR, although it is possible that observations of downwelling irradiance are biased in some way. Wind speed biases are slightly improved compared to FREE, but show little difference between FREE-SLR-SW and FREE-SLR. When either SLR or SW are included, biases in T2 and 10-m wind speeds are in the same direction (cool and slow) as those reported by Zhang and Zheng (2004), providing evidence that the SCM is behaving qualitatively similar to three-dimensional implementations in the most relevant model components. The cool bias, most notable in the FREE simulations, is also consistent with weak PBL entrainment in the MYJ scheme (Hu et al. 2010). The dry bias opposes expectations from a lack of entrainment, and may result because no moisture is advected into the column.

Biases among the DA group rank similarly to those in the FREE group. Soil observations (SLR) improve both FREE and DA experiments during daytime, while adding shortwave irradiance (SLR-SW) increases the bias magnitudes. Biased temperatures much of the day in FREE-SLR-SW compared to FREE-SLR, and DA-SLR-SW (triangles) compared to other DA experiments, provide an initial indication that the modeled coupling between the land and the atmosphere is deficient.

The land surface and the first atmospheric model layer control fluxes through the surface, and balloon-borne soundings can be used to quantify biases in predicting the first model layer. Similar to anemometer-height winds, the lowest SCM grid point (approximately 15 m AGL) predictions are systematically slow (Fig. 2c). Although results are mixed, DA removes much of the bias. Temperature and moisture prediction errors depend on the particular experiment. First, smaller 2-m T biases when assimilating observations (Fig. 1a) can be coincident with greater bias magnitudes at 15 m AGL (Fig. 2). For example, DA-SLR-SW is least biased in 15-m T, but shows slightly greater bias in 2-m T (Fig. 1a). Also, DA-SLR is least biased in 2-m T during the afternoon, coinciding with the only warm bias at 15 m AGL. Finally, DA successfully moistens the profile at 2 m to offset the dry bias (Fig. 1b), but consequently overmoistens the first model layer.

Fig. 2.
Fig. 2.

As in Fig. 1, but at the first model layer (~15 m AGL) and for wind speed.

Citation: Monthly Weather Review 141, 6; 10.1175/MWR-D-12-00280.1

The T and Qυ profiles between the surface and layer 1, which determine surface fluxes, cannot simultaneously satisfy the observations at 2 m and 15 m AGL, immediately indicating poor model covariances between surface-layer profiles and the first model layer. Use of soil observations (SLR) also suggest that the profile cannot simultaneously satisfy the upper and lower boundaries imposed by the soil state and the atmosphere at 15 m AGL. The latter interpretation is somewhat obscured by the energy balance solution giving the skin temperature in the LSM. The LSM gives the radiative temperature at the interface that agrees with the diagnosed atmospheric surface-layer flux and flux into the soil. Assuming that the energy balance solution is accurate relative to its input, it is not clear whether the atmospheric or soil model is the problem, and we can at this point only say that the coupling between them is deficient.

Surface flux biases are mostly within 50 W m−2 during the night, still quite large, but even when soil observations are used directly the model shows a surplus of upward sensible heat flux and deficit of upward latent heat flux during the day (Fig. 3). Combined SLR-SW improves the Bowen ratio (sensible to latent heat flux) predictions. FREE-SLR-SW Bowen ratios are less biased than those in FREE-SLR; DA-SLR-SW Bowen ratios are less biased than either DA-SLR or DA-SLI-SW. In general, DA improves Bowen ratios compared to FREE. Although DA-SLI-SW biases at 2 m were smaller in magnitude than DA-SLR-SW (Fig. 1), the Bowen ratio is slightly more biased.

Fig. 3.
Fig. 3.

Surface (a) sensible heat and (b) latent heat flux bias by time of day. A gap in the latent heat flux results from one or more missing observations at that particular time of day.

Citation: Monthly Weather Review 141, 6; 10.1175/MWR-D-12-00280.1

Although contradictory results complicate the emerging interpretation, the contradictions suggest coupling deficiencies. Atmospheric predictions appear to benefit from greater flux errors, which may result because the diagnosed fluxes are compensating for other errors. Results for DA-SLR-SW suggest that the model is adjusting the surface-layer profiles rapidly to agree with prescribed soil temperature. Combined use of soil observations and atmospheric data assimilation produces bias among the smallest in 15-m T (Fig. 2a), but is generally not the least biased 2-m T or sensible heat fluxes.

Systematic observation increments show whether the assimilation system is working correctly to oppose the biases. Data assimilation cannot formally eliminate bias (cf. Dee and Da Silva 1998), and systematic observation and analysis increments must be less than the bias when observation uncertainty is nonzero. But DA should produce increments that systematically oppose the bias. The increments in Fig. 4 oppose the biases in Fig. 1, showing that the DA is working as expected.

Fig. 4.
Fig. 4.

Experiment- and ensemble-mean observation increments for (a) 2-m T, (b) 2-m Qυ, and (c) 10-m wind speed.

Citation: Monthly Weather Review 141, 6; 10.1175/MWR-D-12-00280.1

Analysis increments from the data assimilation more clearly demonstrate problems with model profiles in the lower atmosphere. In a perfect model, the increments would also systematically have a sign opposing the bias. Although 15-m wind analysis increments systematically oppose the slow bias, increments can also systematically cool and moisten the model at 15 m AGL (Fig. 5). Systematic analysis increments with the same sign as the bias maintain the bias. At 1230 LDT, DA-SLI-SW and DA-SLR-SW maintain both temperature and moisture biases at 15 m AGL. Although DA-SLR increments oppose the bias at 1230 LDT, it does maintain the moist bias. At 1830 LDT, DA with all configurations increment to oppose the moist bias. DA-SLR and DA-SLI-SW increment to oppose the 15-m T bias, but DA-SLR-SW maintains the cool bias.

Fig. 5.
Fig. 5.

First-layer (15 m AGL) analysis increments for (a) T, (b) Qυ, and (c) wind speed.

Citation: Monthly Weather Review 141, 6; 10.1175/MWR-D-12-00280.1

Biased analysis increments can result from either violation of the linearity assumptions underlying the ensemble filter, or model biases. The first possibility is easily eliminated by noting that the predicted shelter and anemometer height observations vary strongly, and linearly, with the first-layer predictions. Figure 6 shows first-layer model variables against predicted observations for every ensemble member and day in DA-SLR-SW, valid at 1230 LDT. Vertical moisture and wind speed correlations, and covariances, remain strong and largely invariant from day to day. Temperature shows more variability. Isolated groups of points show that on any individual day T15 can vary weakly with T2 and the scatter indicates periods of weaker correlation, but signs of nonlinearity are absent. Anderson (2003) showed that linearity need only be satisfied locally in phase space, and Fig. 6 shows that the linear assumption is generally not violated.

Fig. 6.
Fig. 6.

First-layer (15 m AGL) vs surface-layer diagnostic values for (a) T, (b) Qυ, and (c) wind speed at 1230 LDT from experiment DA-SLR-SW. All 100 ensemble members and 29 days are plotted.

Citation: Monthly Weather Review 141, 6; 10.1175/MWR-D-12-00280.1

Small biases in observation space, combined with persistent and small increments, permit deeper analysis of the model behavior. Various DA experiments show slightly different errors, and choosing one for further evaluation would require a judgment on which variable is most important. Avoiding that question, we choose to interpret DA-SLR-SW so that we can eliminate first-order errors in the land surface model as the primary source of surface-layer coupling errors.

4. Interpretation

Increments can be interpreted in the context of MOST. In ensemble data assimilation an analysis increment is proportional to the covariance between the ensemble of predicted observations (yf) and predictions of state variable (xf). From the statistical analysis Eq. (1) we can write the ensemble-mean increment from observation yo:
e4
where σxy = cov(xf, yf).
MOST applied to the surface layer assumes constant vertical turbulent fluxes. It predicts nondimensional vertical wind and temperature gradients that follow universal functions of the Monin–Obukhov scale height L:
e5
e6
where here L is defined as
e7
Here κ = 0.4 is the Von Kármán constant, is the mean surface-layer potential temperature, and g is the acceleration due to gravity. Scaling variables are defined from the Reynolds-averaged vertical eddy fluxes of momentum and heat :
e8
e9
where the primes denote turbulent quantities.

The unstable surface-layer scheme used in these experiments largely follows Louis (1979), who integrated the flux–profile relationships as in Panofsky (1963), Paulson (1970), and Barker and Baxter (1975). For the present analysis we ignore the modification of the velocity scale u* proposed by Beljaars and Holtstag (1991).

The MOST flux–profile relationships integrated from the roughness height for momentum z0m to any height z in the surface layer:
e10
e11
where ψM and ψH are functions of z/L. They are meant to compactly represent the integrals of φM and φH, respectively, and modify the profile to account for stability. The ratio of roughness lengths R = z0m/z0t, and is used so the aerodynamic surface potential temperature can be substituted for the radiometric “skin” potential temperature θs. Analogous to zero wind at the roughness height z0m, θs is defined at z0t. The relationship applies at any z in the constant-flux layer, and the WRF implementation assumes that the entire lowest model layer is within it. Thus, the fluxes result from integration from z0 to the height of the lowest model layer.

Equations (8)(11) can be used to interpret the model biases and increments. During statically unstable daytime conditions θs > θ15, cold biases for T15 indicate that |θ15θs| is too large, and θ* is too large from Eq. (11). Then from Eq. (9), is also too large, consistent with overprediction of sensible heat fluxes shown in Fig. 3. Equation (10) shows that underprediction of 15-m wind speeds means that u* and, therefore, the momentum flux is too small. Too little momentum flux with too much sensible heat flux also consistent the inverse relationship between u* and θ* seen in Eq. (9).

Clearly for experiment DA-SLR-SW, and to different degrees for other experiments, the model needs to persistently weaken the temperature gradient (reduce upward sensible heat fluxes) and strengthen the momentum gradient (increase downward momentum flux). From Eq. (4) we can easily see that σxy has an incorrect sign for all DA experiments at 1230 LST. Given that the linearity assumption in the DA is valid, we next test for parametric errors leading to biased covariances.

5. Parameter estimation and residuals

Model deficiency can be broadly classified as one of either: 1) incorrect formulation of model equations (structure), or 2) incorrect specification of a parameter appearing (most often) in physical parameterization schemes. Experiments here examine the potential for the latter in the surface-layer scheme. A hypothesis that the structure of the surface-layer parameterization is reasonable, but that one or more coefficients may be incorrectly specified, is testable to some extent. To ensure a reasonable (but surely not exhaustive) test, the focus here is on a single parameter known to produce sensitivity in forecasts by exerting strong control over surface fluxes.

Parametric error can be identified by dynamically estimating one or more parameters, and determining whether the model is improved as measured by reduced biases. State augmentation in the DA process provides a simple and theoretically grounded method to estimate parameters. Rather than estimating just the state, an augmented state z(x, p) is formed by appending a vector of parameters p. The augmented state z replaces every instance of x in Eqs. (1)(3) and the covariance estimates are similarly augmented. Parameters can be updated in the assimilation as long as the parameters are detectably correlated with predicted observations. State augmentation to estimate parameters within physical parameterization schemes has drawn recent attention in the mesoscale modeling community (e.g., Aksoy et al. 2006; Tong and Xue 2008).

We often do not know the true value of the parameter, and parameters can adjust in response to errors that are intuitively or physically unrelated to the parameter itself. Evaluation here is simply to determine whether the errors are reduced when the parameter is allowed to vary. Reduced errors indicate that the evolving parameter is successfully accounting one or more errors in the model formulation. Remaining errors qualitatively alike those when the parameter is fixed indicate that the parameter is not able to address the leading source of error.

Each ensemble member has a unique value of a given parameter, and temporal variation results from updating the parameter distributions with the data assimilation. Initial parameter values are chosen by drawing a random number from a uniform distribution. Each assimilation updates the distribution. Although many parameters are expected to vary in time and space, dynamics governing parameter evolution are rarely known. The simplest possible model is persistence model so that for time k, which is suitable here. When the DA updates the distribution, the variance of the parameters gets smaller, and over several assimilation cycles will be too small to be affected by the observations. To prevent this, the parameter estimation uses the same adaptive inflation commonly used for model state variables (Anderson 2009).

A parameter controlling the bulk aerodynamic roughness lengths for momentum (z0m) and heat (z0t), thereby modulating the ratio of heat flux to momentum flux in the surface layer, is a good candidate for estimation because forecasts are known to be sensitive to it (Chen et al. 1997; Trier et al. 2011). Zilitinkevich (1995) suggested parameterizing the ratio as a function of the Reynolds number such that
e12
where κ = 0.4 is the Von Karman constant and ν ≈ 1.5 × 10−5 m2 s−1 is the kinematic viscosity of air near sea level. Although the physical meaning of z0m is clear, thermal roughness is not directly definable without a suitable definition for the temperature at z0t. A suitable definition consistent with Monin–Obukhov surface-layer parameterization, and also readily observable, is obtained by specifying z0t to equal the height at which the temperature is equal to the radiometric temperature (Zeng and Dickinson 1998).

The constant CZil strongly modulates the thermal coupling between the land and the atmosphere. Greater CZil results in smaller z0t and smaller sensible heat fluxes. Smaller CZil results in greater z0t and greater sensible heat fluxes. The model default value is CZil = 0.1 following Chen et al. (1997). LeMone et al. (2008) estimated a range of 0.1–0.5 from aircraft observations over the Southern Great Plains, and also found that CZil = 0.5 produced offline estimated fluxes in better agreement to observed fluxes. They reasoned that the optimal value may vary by season from smaller in the winter (shorter vegetation) to larger in the summer (deeper canopy). Trier et al. (2004) found CZil = 1.0 provided the best PBL simulations in a mesoscale model.

Values of CZil < 0 are arguably not possible because it would result in a negative roughness. Values of CZil > 1 would result in z0tz0m. If sample elements end up outside of the range [0, 1] following an assimilation, the whole distribution is contracted, without modifying the mean (estimated value), until the first and last ranked values differ by exactly 0.99. Then if needed, the distribution is shifted so that all values lie within [0, 1]. A shift is rarely necessary.

Estimated CZil varies in time and depends on the model configuration (Fig. 7), but all indicate that the parameter is adapting to weaken gradients through the surface layer. Compared to the default CZil = 0.1 used in all prior experiments, a greater CZil increases R by Eq. (12) and decreases |θ15θs| by Eq. (11). Sensible heat flux is smaller. Momentum fluxes are insensitive to CZil.

Fig. 7.
Fig. 7.

(a) Estimates of CZil for three estimate experiments (+, ◊, ). (b) A narrower date range to better see the details.

Citation: Monthly Weather Review 141, 6; 10.1175/MWR-D-12-00280.1

Allowing CZil to vary according to the assimilated observations improves biases and analysis increments, consistent with the errors and increments shown in sections 34. Temperature biases at 15 m AGL during the entire day are smaller by a Kelvin or more (Fig. 8a). Results for water vapor mixing ratio and wind speed are somewhat mixed, but all show small response to CZil variations (Figs. 9b,c). Increments are correspondingly smaller for temperature, while mixed for the other variables (Fig. 9). Biases and increments at shelter and anemometer height are all smaller (not shown). Daytime radiometric surface temperature (corresponding to θs) is also improved from too cool by approximately 3 K to a bias less than a few tenths of a degree (also not shown).

Fig. 8.
Fig. 8.

Biases by time of day for 1-h predictions at the first model layer (~15 m AGL) of (a) temperature, (b) water vapor mixing ratio, and (c) wind speed.

Citation: Monthly Weather Review 141, 6; 10.1175/MWR-D-12-00280.1

Fig. 9.
Fig. 9.

First-layer (15 m AGL) analysis increments for (a) T, (b) Qυ, and (c) wind speed.

Citation: Monthly Weather Review 141, 6; 10.1175/MWR-D-12-00280.1

Results show that some of the error in surface-layer gradients can be reduced by allowing CZil to adapt, but that large biases at 15 m AGL can persist. Model covariances are biased, and the thermal and momentum profiles through the surface layer are incorrect. Accurate shelter- and anemometer-height observations and radiometric skin temperature, combined with near-perfect soil state and downwelling shortwave irradiance, can coexist with the larger biases at 15 m AGL.

6. Generalization to more readily available observations

An ability to reach qualitatively similar conclusions from more readily available observations could extend the utility of DA for deducing model errors to a much greater range of soil conditions, synoptic conditions, and microclimates. The experiments and analysis in sections 35 benefited from atypical and high-quality observations of radiometric skin temperature, downwelling shortwave irradiance, and soil temperature and moisture. Although instruments measuring those quantities are available at research locations such as the ARM Central Facility or during field experiments, they are not generally deployed at typical observing locations. A set of simpler experiments quantify the potential, and assume that atypical observations are unavailable. We find that evidence for deductive reasoning is present, but details change.

Experiments DA, DA-SLI, and DA-SLI-CZil use the shortwave irradiance at the surface results from the Dudhia (1989) radiative transfer scheme rather than observed downwelling irradiance (no SWRAD). Soil state observations are also not used (no SLR). The ensemble filter assimilates shelter- and anemometer-height observations as before, and collocated rawinsondes are used to verify the profiles.

We find a clear signal of modeled land–atmosphere coupling error. Predicted-observation temperature biases (Fig. 10a) show some notable differences between the experiments. Warm biases from DA-SLI-CZil are greater than the others during the night and are smaller during the midafternoon. Compared to Fig. 1a the biases are warm during the afternoon and evening, instead of cool when observed shortwave irradiance was included. The coolest bias is at 0700–0800 LDT here, and 0900–1000 LDT when shortwave irradiance is included (Fig. 1a). Using observed shortwave irradiance appears to reduce biases during the morning transition, but increase bias while the PBL deepens through the morning. Water vapor mixing ratio biases show little difference between experiments except between approximately 2200–0200 LDT when DA-SLI-CZil shows greater moist bias than all other experiments. Including shortwave irradiance led to dry biases during this period (Fig. 1b). Wind speed errors are comparable to each other, and similar to earlier experiments.

Fig. 10.
Fig. 10.

As in Fig 8, but for predictions of shelter-height and anemometer-height wind speed.

Citation: Monthly Weather Review 141, 6; 10.1175/MWR-D-12-00280.1

Small differences between predicted-observation errors, whether or not soil observations and shortwave irradiance observation are given to the model, broadly demonstrate the control that the ensemble filter exerts on the state. Because these experiments are intended to isolate a small number of model components compared to a fully 3D model implementation, the design largely ignores 3D dynamics. In a 3D experiment, model errors influencing predictions nearby in a horizontal direction could change this result because dynamics could propagate those errors into the region influencing the predicted observations.

Cool biases in DA and DA-SLI at 15 m AGL and corresponding warm biases at shelter height would suggest that the model systematically overpredicts |θθs| in a convective PBL. But without the benefit of soil or radiometric skin temperature observations, we cannot directly verify it. As discussed earlier, it does suggest that CZil may be biased low. Biases among all the experiments at 15 m AGL are the same sign. Biases amongst the experiments at shelter–anemometer vary, and the vertical gradients lack a clear and consistent error except that they are too strong. Incrementing the soil in DA-SLI improves afternoon warm bias at 2 m AGL and the cool bias at 15 m AGL.

Allowing CZil to vary according to the observations is effective at reducing the cool bias at 15 m AGL, particularly during the afternoon, but the model still overpredicts the vertical gradient. At 1230 LDT, the 2-m T bias is approximately −0.3 K (cool), and the 15-m T bias is around −3 K (Fig. 11). Greater cool bias at 15 than 2 m means the gradient is stronger than observed, and the same result is evident at 1630 LDT. Parameter estimation further suggests that the coupling should be weaker. A time-mean estimate of CZil = 0.23 from DA-SLI-CZil is consistent with the reduced coupling, which is enabled by the variable CZil.

Fig. 11.
Fig. 11.

Biases by time of day for 1-h predictions at the first model layer (~15 m AGL) of (a) temperature, (b) water vapor mixing ratio, and (c) wind speed.

Citation: Monthly Weather Review 141, 6; 10.1175/MWR-D-12-00280.1

From remaining warm 2-m T biases during the afternoon coexisting with greater CZil values, we deduce that θs is still biased warm. Therefore the soil temperature is biased warm, and the DA cannot warm the first model layer to reduce the vertical temperature gradient because it lacks flexibility to modify the shape of the thermal profile in Eq. (11).

Analysis increments again offer direct evidence of model deficiencies. Increments at shelter and anemometer height systematically offset the biases (not shown) in all experiments, but increments at 15-m T often fail to offset the biases there (Fig. 12). Afternoon analysis increments cool the 15-m T instead of warming it, indicating biased representation of the covariance between 2-m and 15-m T. The vertical temperature gradient error cannot be eliminated.

Fig. 12.
Fig. 12.

First-layer (15 m AGL) analysis increments for (a) T, (b) Qυ, and (c) wind speed.

Citation: Monthly Weather Review 141, 6; 10.1175/MWR-D-12-00280.1

Further moistening by analysis increments at 1630 and 1230 LDT, helping to maintain the moist biases at 15 m AGL, lends further evidence. At these times the 2-m water vapor biases are small (Fig. 10b.) Moist biases at 15 m AGL at 1230 LDT indicate a vertical gradient biased weak, consistent with the biases and negative latent heat flux biases in the experiments that benefitted from additional observations (Fig. 3). At 0630 LDT, the latent heat flux is typically small and negative. In this case, the moist bias at 15 m AGL is consistent with the small negative latent heat flux bias in the earlier experiments.

Some ambiguity remains in interpreting the parameter estimation experiments because it is always possible that CZil is responding to error sources distinct from the surface layer per se. One alternate explanation is given by the cool, moist biases from lack of entrainment noted in Hu et al. (2010). The cool, moist biases could force strong surface-layer gradients and therefore strong fluxes, leading to CZil adapting to weaken then. Estimated CZil is weakening the gradients as would be expected, but other results suggest that biases from lack of entrainment are not dominant. Referring to Figs. 4 and 5, for example, sometimes opposing signs of the 2- and 15-m increments show that a 2-m warming and drying does not always lead to a 15-m warming and drying. Hu et al. (2010) showed consistent cool and moist biases through the surface layer and PBL when the PBL is convectively unstable, which if dominant would always give increments of the same sign at 2 and 15 m.

Sampling error arises in covariance estimates derived from the ensemble. The expected effect is overestimated covariances, leading to analysis increments (e.g., at 15 m AGL) that are too large. We see here that they instead can have the incorrect sign. Given the strong covariance between the predicted observation and the first model layer (Fig. 6), increments in the wrong direction are not likely to result from sampling error.

The results in this section demonstrate the potential for identifying model errors without the need for atypical observations. Focus here is on land–atmosphere coupling errors through the surface layer, where an instrument shelter, anemometer, and collocated rawinsonde system gathering conventional observations appears sufficient. To be certain of the generality, this analysis should be repeated in other locations characterized by different land surface properties and microclimates, and during different seasons. It is clear that more detailed analysis is possible from combining data assimilation and unconventional observations, but here the most important characteristic is to constrain the model state around the process of interest with effective assimilation, and attempt parameter estimation with the most relevant parameters.

7. Summary

A series of experiments with the WRF-SCM demonstrates the combination of data assimilation and parameter estimation for elucidating errors in land–atmosphere coupling through the atmospheric surface layer. Via ensemble data assimilation of 2-m T and Qυ, and 10-m wind components (U, V), the model follows the observations during a month-long simulation for a column over the ARM Central Facility near Lamont, Oklahoma. One-hour errors in predicted observations are systematically small but nonzero, and the systematic errors measure bias as a function of local time of day. Analysis increments for state elements nearby (15 m AGL) can be too small or have the wrong sign, indicating systematically biased covariances and model error. Experiments using the ensemble DA to objectively estimate a parameter controlling the thermal land–atmosphere coupling (CZil) show that the parameter adapts to offset the model errors; however the errors are not eliminated. We then deduce that the model exhibits either structural error or parametric error of a different kind (i.e., to explicitly modify the flux–profile relationships). Experiments omitting atypical observations such as soil and flux measurements lead to qualitatively similar deductions, showing the potential for applying assimilation common in situ observations as an inexpensive alternative to identify model errors.

Specific findings from experiments and analysis with a large suite of atypical observations (downwelling shortwave irradiance, radiometric skin temperature, and soil temperature and moisture) include the following:

  1. Assimilation of surface observations with an ensemble filter, in the absence of explicit treatment of advection, can result in skillful 1-h prediction of surface observations (Fig. 1). The DA constrains the model so that the nearby state can be analyzed.

  2. Prediction errors on the lowest atmospheric computational level (15 m AGL), in the soil, and at the surface can be systematically much larger than error in predicting surface observations (Figs. 2, 3, and 8). This suggests that the best surface predictions are obtained with poorly predicted surface fluxes, and confirms the results of Steeneveld et al. (2011).

  3. Analysis increments can make the biases at the first model level worse (Fig. 5) and increase the vertical temperature gradient. Allowing CZil by to vary (resulting in an increase) acts to reduce the vertical temperature gradient (Fig. 9). The estimated parameter offsets errors but cannot eliminate them.

  4. Analysis increments increase the momentum flux by increasing the vertical wind shear (Figs. 5 and 9), consistent with the need to reduce thermal fluxes.

  5. The results are qualitatively similar whether or not the soil is forced to agree with soil observations.

  6. Estimated values for CZil are greater than the WRF default and act to reduce thermal land–atmosphere coupling (Fig. 7). Remaining nonzero systematic errors indicate that the error remains in the model.

Ideally, simpler experimentation needing only widely available observations would lead to the same conclusion. Results show that this is indeed the case, but the errors are less easily measured and are not necessarily clear until parameter estimation is invoked. Specifically

  1. The vertical temperature gradient across the surface layer is at first glance well predicted (Figs. 10a and 11a).

  2. Estimated values for CZil are consistent with reducing the coupling.

  3. Applying analysis increments to the soil state increase the already too-strong thermal gradient across the surface layer. Estimates of CZil increase accordingly to reduce the coupling.

A common thread in these experiments is that biased covariances, and poor predictions, of the 15-m state and the surface state are possible while errors at shelter and anemometer height remain smaller than expected observation error. Two implications are clear. First, assimilation of surface observations to retrieve PBL profiles with great skill may be difficult. Earlier work shows potential (e.g., Hacker and Snyder 2005; Hacker and Rostkier-Edelstein 2007; Rostkier-Edelstein and Hacker 2010), but that work also shows that profile skill is limited by model errors. Second, poor covariance between the soil and surface observations indicate that using surface observations to spin up (force) land surface models can lead to biased land surface states.

The results suggest the possibility of a structural error in the equations governing the atmospheric surface layer. The Monin–Obukhov similarity theory and its extensions are rarely questioned during neutral and convectively unstable conditions, which dominated daytime during these experiments. Although we have not ruled out parameters controlling the stability functions ψ(z/L), it is possible that the errors are more fundamental and arise from ignoring the assumption of lower-boundary heterogeneity underlying MOST. Further work would be required to rule out the remaining parameters. It is not yet clear whether surface-observation predictions correlate strongly with any of those parameters. If not, then linear estimation methods are ineffective and a different approach would be required.

Although a closer examination may be warranted, we expect that an offline (uncoupled from an atmospheric model) Noah LSM will behave similarly to how it behaves in the DA experiments. The offline Noah can be forced by observed shelter and anemometer-height observations, and uses a similar surface-flux formulation as when coupled to the WRF. Accurate 1-h prediction of those forcing variables should produce a similar response in the LSM.

An additional question that arises is whether to apply analysis increments to the soil state. Because the soil evolves on slower time scales than the atmosphere, intuition would lead us to postulate that the best way to retain information from assimilating near-surface observations is to ensure that the soil state always follows the modeled relationship between the atmospheric state and the soil state. Incrementing the soil would achieve this, and the results here suggest this may be the case. This topic deserves further study.

This study examined only one location, and focused on determining whether DA with parameter estimation can inexpensively reveal errors in the surface layer. Specific error forms may be different depending on site and season, among other factors. But the fact that the surface, soil, and atmospheric data cannot be simultaneously fit shows a fundamental problem. Compared to observation programs, the approach here can be quickly and inexpensively deployed to many more locations and seasons.

Acknowledgments

This research was partially funded by the Office of Naval Research Award N0001410WX20059. Data were obtained from the Atmospheric Radiation Measurement (ARM) Program sponsored by the U.S. Department of Energy, Office of Science, Office of Biological and Environmental Research, Environmental Sciences Division. This work benefited tremendously from conversations with Jeff Anderson and Glen Romine, and help from the rest of the Data Assimilation Research Testbed group.

APPENDIX

The WRF Single-Column Model Capability

A basic capability for single-column modeling (SCM) in WRF is provided. This amounts to a group of registry variables, an input stream, routines to initialize and force the model, a prototype namelist, and prototype scripts to create the forcing files. A WRF-SCM was originally implemented for the Global Energy and Water Exchanges Project (GEWEX) second Atmospheric Boundary Layer Study (GABLS2) SCM comparison case (Svensson et al. 2011), and subsequently generalized by Hacker et al. (2009). Ensuing development led to its use in the GABLS3 case (Bosveld et al. 2012, manuscript submitted to Bound.-Layer Meteor.).

The WRF SCM is set up as a 3 × 3 horizontal stencil with periodic boundary conditions in both directions. The vertical grid is at the user’s discretion. The initial conditions are read from a text file by the ideal-case initialization module. Forcings during the run are read from a netCDF file using the built in auxiliary input capabilities of WRF. Temperature, moisture, and winds can be forced in the default implementation, and forcing capability is easily extensible to additional quantities (such as cloud water). Two sets of forcings, on an arbitrary vertical grid and with independent time scales, are provided. These can be used, for example, to represent advective forcings from the immediate surroundings with a short time scale, and simultaneously to nudge the model toward a desired state on a long time scale. Time-varying geostrophic wind can be specified.

If the user wants to run with specified surface fluxes, the values can be included in the input file. They are provided to the running model by a special surface-layer scheme (idealscmsfclay) invoked by a namelist option.

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