1. Introduction

The planetary boundary layer (PBL) is the turbulent layer of the atmosphere nearest to Earth’s surface in which exchanges of energy and momentum between the surface and the atmosphere take place. An important parameter that measures the depth of this layer and is a key indicator in the mixing that takes place within the PBL is the PBL height (PBLH). The PBLH is used in many predictive and diagnostic methods and/or models to assess pollutant concentrations, and it is an important parameter in some atmospheric models (e.g., as a key term in some PBL mixing parameterizations).

The complicated interactions between the land/ocean surface and the PBL lead to large uncertainties in modeling the PBL, especially with regard to parameterization schemes, making it challenging to accurately simulate PBL thermodynamic structure and PBLH. These difficulties are perpetuated by the lack of globally comprehensive PBL observations, as well as the lack of both strategy and infrastructure to utilize all PBL-related observations cohesively. The importance of PBL dynamics was underscored by the 2017 Decadal Survey (National Academies of Sciences, Engineering, and Medicine 2018), which designated the PBL as an incubation class observable, and selected as priorities better observations of PBL temperature and water vapor profiles, and of PBLH. Following this designation, the PBL Incubation Study Team Report (Texeira et al. 2021) made clear that “a future global PBL observing system requires modeling and data assimilation as essential components.”

Data assimilation plays a critical role in combining a large volume of disparate observations with dynamical models to produce an optimal estimate of the atmospheric state, for use in research applications or to initialize model forecasts. The near-real-time NASA Goddard Earth Observing System (GEOS) data assimilation system uses approximately 4.5 million observations in each 6-h assimilation window, and overall, the observations have a significant impact on reducing model forecast errors. However, an assessment conducted on the effectiveness of the use of existing observing systems in the lower troposphere found that the data impacts are not satisfactory (Zhu et al. 2022). For example, satellite radiances have limited impact in the PBL, and model physics tends to return to its original free forecast mechanisms at 1-day forecast, retaining little observation information within the PBL. While several known factors contribute to the unsatisfactory data impact on the PBL, such as underutilization of GNSS-RO data and cloud-affected infrared radiances, our objective in the present study is to develop new PBL data assimilation capabilities in GEOS, focusing on the use of PBLH from multiple observing systems.

We aim to address the following key issues related to PBLH and its assimilation:

1. The PBLH itself is a key parameter in many applications, e.g., forecasting near-surface meteorology and air quality. Therefore, it is desirable to provide a global PBLH analysis and monitoring capability in GEOS. However, such an analysis requires utilizing PBLH derived from multiple sources with potentially disparate observables and retrieval algorithms. Key issues that need addressing in this context relate to the physical meanings and potential inconsistencies of the disparate PBLHs and the need for an optimal strategy for using these data together.

2. The second issue in need of addressing is related to the unique feature of a capping inversion. Traditional data assimilation methodologies tend to focus on amplitude error, whereas error in the PBL is often positional, such as a misplaced inversion height. As a consequence, current data assimilation algorithms tend to distort inversion structures (Fowler et al. 2012). In the present assimilation framework study, the use of PBLH data to capture the capping inversion and benefit all the other observations is explored.

3. A third issue is the retention of observation information by the forecast model. In the situation where PBLH is a prognostic variable, PBLH analysis increments as a result of PBLH data assimilation can be fed back to the forecast model in the same way as the increments of other control variables. However, PBLH is a diagnosed variable in GEOS. In addition, short process time scales within the PBL allow model parameterizations to rapidly compensate for analysis increments, limiting the impact of observations. One potential solution is model parameter estimation, frequently done along with analysis increment estimation in data assimilation (Zhu and Navon 1999). Methods to utilize the global PBLH analysis within the model parameterizations to better retain observation information will be discussed in detail in a follow-up paper.

To provide a comprehensive global estimate of PBLH and serve as the reference for PBLH data generated from both existing and new sensors, our overarching goal is to explore a global data synergy strategy for the assimilation of PBLH data from multiple observing systems and produce a global PBLH analysis in GEOS. Many groups have investigated and derived PBLH data from various observing systems (e.g., Lewis et al. 2013; McGrath-Spangler and Denning 2013; Molod et al. 2015; Ganeshan and Wu 2015; Palm et al. 2021a; Salmun et al. 2023). In this NASA PBL Incubation project, PBLH data from radiosondes, Global Navigation Satellite System-Radio Occultation (GNSS-RO), space-based lidars including the Cloud–Aerosol Lidar with Orthogonal Polarization (CALIOP) instrument aboard the Cloud–Aerosol Lidar and Infrared Pathfinder Satellite Observations (CALIPSO) and the Cloud–Aerosol Transport System (CATS), ground-based Micro-Pulse Lidar Network (MPLNET), and global radar wind profiler networks are used. However, only PBLH data from radiosondes and GNSS-RO are included in this paper for methodology development. Other PBLH data and statistical evaluations will be discussed in a follow-up article, and the developed methodology is expected to be applicable to these PBLH data with the additional consideration that they are also sensitive to hydrometeors and aerosols.

Model PBLH from GEOS or the Modern-Era Retrospective Analysis for Research and Applications, version 2 (MERRA-2; Gelaro et al. 2017), has been used in previous studies to compare to PBLH data estimated from various observing systems (Molod et al. 2015; Ganeshan and Yang 2019). Although these studies found comparable patterns between PBLH data and model PBLH, differences were also noticed. One of the major reasons for the differences lies in the mismatch of PBLH definitions (Seidel et al. 2010; McGrath-Spangler and Molod 2014). PBLH products are derived using observing systems and retrieval algorithms that may be sensitive to different key variables, and as a result, the derived PBLH data may represent different physical heights. For example, radiosonde-based PBLH is often based on thermodynamic structure and gradients, GNSS-RO PBLH data can be based on refractivity gradient, while lidar PBLH data often use aerosols as a tracer to identify the top of the surface-attached mixed layer (Lewis et al. 2013; McGrath-Spangler and Denning 2013; Palm et al. 2021b). In this study, consistent model PBLH definitions will be used to compare PBLH data from different sources and to compute departures of first guess from observations (OmF).

The traditional approach of the variational method to assimilate PBLH data is to construct an observation operator, which links control variables (or model prognostic variables) to PBLH data from each observing system and its tangent linear and adjoint operators. Amerault et al. (2024) use this approach in their variational assimilation of the PBLH data study and show clear sensitivities of PBLH with respect to potential temperature, water vapor mixing ratio, and wind. However, the observation operator can be very complicated and involves discontinuities, e.g., the lidar-based PBLH observation operator includes the total attenuated backscatter (ATB) lidar forward observation operator, the algorithm to derive PBLH from ATB, interpolation, and any calculations to handle the mismatch between observed and model scales. An alternative approach is to augment PBLH to the control variable vector. This approach technically moves part of the complicated forward observation operator (e.g., ATB calculation and PBLH retrieval algorithm) to the model, and as a result, it allows PBLH data to be assimilated with a straightforward observation operator, which consists of only interpolation and calculations handling the mismatch between observed and model scales, and generates an hourly global PBLH analysis. The correlations between PBLH and other control variables, which are embedded in the flow-dependent ensemble background error covariance, provide a mechanism for the PBLH data to impact other variables. Since the PBLH is a 2D variable, the requirement of extra computational resources will be minimal. This approach has been used in previous studies, e.g., the assimilation algorithm for radiosonde and aircraft PBLH data developed in the 2D variational Real-Time Mesoscale Analysis (RTMA) system for a dispersion modeling study (Tassone et al. 2012), and the PBLH assimilation study using lidar PBLH data from a field campaign in Greensburg, Kansas, which however employed a comparison of PBLH data with inconsistent model PBLH (Tangborn et al. 2021). Analogous to these applications, this approach of control variable augmentation was also used in the assimilation of radar reflectivity without tangent linear and adjoint of the nonlinear observation operator (Wang and Wang 2017). Because of its relative simplicity, this approach is adopted in the present study, and the original PBLH data assimilation algorithm used in the RTMA is adapted into the GEOS hybrid 4D ensemble-variational (EnVar) framework. In addition, with the PBLH data from different observing systems, special attention is paid to the consistency of PBLH data and model PBLH simulation. For example, a bulk Richardson number–based PBLH definition is used to compare the model and radiosonde data, and a refractivity gradient–based PBLH definition is used to compare the model and GNSS-RO data. Corresponding model PBLH definitions are augmented to the control variable vector.

The second issue we address here is how to capture capping inversions in the PBL data assimilation. A recent study conducted by Hilton et al. (2020) employed multiple existing retrieval and one-dimensional variational (1D-Var) algorithms and found that, in general, only where 1D-Var retrievals used numerical weather prediction (NWP) as input and already had an inversion could the retrieval have an inversion too. Algorithms to account for positional error explicitly have been investigated (e.g., Ravela et al. 2007; Fowler et al. 2012); however, these algorithms tend to be complicated and sensitive to the factors including choice of displacement function, observation vertical resolution, and whether a capping inversion is present. In this study, we have explored new approaches to better capture the locations of capping inversions, through improving control variables’ ensemble background error covariances and vertical localization length scales based on the PBLH data information. This will benefit the assimilation of all other PBL-related observations such as direct observations of temperature, specific humidity, and radiances. Therefore, PBLH data can affect the PBL thermodynamic structure through improving both ensemble background error covariances and the aforementioned correlations between PBLH and other control variables prescribed by ensemble background error covariances. The separate and combined PBLH data impacts are investigated in this study.

The present paper constitutes a first step toward developing and testing a comprehensive PBLH data assimilation methodology. It includes discussion of issues related to PBLH data, corresponding model PBLH definitions, and methods of using PBLH data in the data assimilation followed by preliminary results. The optimization of our methods and impact of PBLH data from multiple observing systems and methods to utilize the PBLH analysis in the model parameterization schemes will be presented in subsequent papers. Here, following this introduction, section 2 gives a brief overview of the global GEOS data assimilation system, and section 3 introduces the methodologies for PBL data assimilation, including PBLH data from radiosonde and GNSS-RO observations and their corresponding model PBLH definitions and the methods to help capture capping inversions and correlations between PBLH and other control variables in the background error covariance. Section 4 describes preliminary results of assimilation studies to demonstrate the PBLH assimilation capabilities. Conclusions and future plans are discussed in section 5.

2. Overview of the global GEOS data assimilation system

The global GEOS data assimilation (DA) system is used routinely to support NASA space missions and field campaigns. While the system includes a coupled atmosphere–ocean configuration used for seasonal prediction, our study focuses on the atmosphere-only configuration composed of the GEOS atmospheric general circulation model (AGCM) and the atmospheric data assimilation system (ADAS). The GEOS AGCM is built around the finite-volume cubed-sphere dynamical core (Putman and Lin 2007) and has 72 vertical levels with 12 levels below about 1.7 km. The PBL parameterizations include the “Lock” K-profile scheme driven by surface and cloud-top buoyancy fluxes (Lock et al. 2000) and the “Louis” local scheme for stable conditions based on the Richardson number (Louis 1979). Together, these elements determine a profile of diffusivity coefficients that are used to calculate PBL tendencies. The analytical diffusivity profile shape in the Lock scheme is determined by parcels released upward from the surface and downward from stratocumulus cloud top. Above the mixed layer defined by the Lock surface plume, shallow cumulus convection is represented by the Park and Bretherton (2009) mass flux scheme. Additional parameterizations are summarized in Arnold et al. (2020). Aerosols are an integral part of the GEOS system, including estimates and forecasts from the Goddard Chemistry Aerosol Radiation and Transport (GOCART) model and assimilation of aerosol optical depth measurements.

The ADAS uses the Gridpoint Statistical Interpolation analysis system (GSI) (Kleist et al. 2009). In the hybrid 4D EnVar configuration (Todling and El Akkraoui 2018; Wang et al. 2007), two DA systems run concurrently: the ensemble ADAS (Whitaker et al. 2008) that provides the flow-dependent background error covariance information to the high-resolution deterministic ADAS, and the deterministic ADAS that provides its analysis to the recentering procedure of the ensemble ADAS and bias correction information.

The near-real-time GEOS system combines all operational observations coherently with the model physics to generate optimal analysis increments of the control variables for the subsequent model forecast. The resultant analysis increments are fed back to the forecast model through the 4D incremental analysis update (IAU) approach (Takacs et al. 2018), where the model is integrated for 6 h with the 4D IAU tendencies. This approach allows the model to ingest information from observations and adjust its states and related physical balances gradually.

In this study, a lower horizontal resolution version of the then-current operational GEOS configuration in 2022 is used, with the deterministic cycle at 25 km for model forecast and 50 km for analysis and the ensemble cycle at 100 km, with 72 vertical levels in all its components.

3. Strategy and methodologies for assimilation of PBLH data from multiple observing systems

The PBLH data products used in this study have different strengths and weaknesses. Satellite PBLH data can provide better global coverage and complement in situ PBLH data, while in general ground-based PBLH data provide a more complete temporal sampling of the PBL diurnal cycle. In this section, we first focus on the derivation of PBLH data and appropriate model PBLH definitions, which are used to assess PBLH data and calculate the departures of model PBLH from PBLH data, i.e., PBLH OmFs. Then, the strategies of effectively utilizing the PBLH data from multiple observing systems for the construction of the global PBLH data monitoring and analysis framework are presented. The methods to capture PBL capping inversions are explored, and the correlation relationships among PBLHs and other control variables are also examined to help improve the PBL thermodynamic structure in the analysis.

a. PBLH data and model PBLH definitions

A successful PBLH data assimilation requires that comparison between observations-derived PBLH data and model simulations be consistent. Due to the sensitivity of PBLH data to the observing method and choice of algorithm, it is important to use a model definition appropriate for each observation type to compute consistent OmFs of PBLH. In this paper, while PBLH is used to refer to PBLH definitions and data in general, ${\text{PBLH}}_{R_i}$ is used for bulk Richardson number–based PBLH with ${\text{PBLH}}_{R_i}^f$ indicating the model forecast background and ${\text{PBLH}}_{R_i}^o$ for observed data, and PBLH_RF is used for refractivity gradient–based PBLH with ${\text{PBLH}}_{\text{RF}}^f$ and ${\text{PBLH}}_{\text{RF}}^o$ indicating model and observed data, respectively.

1) Bulk Richardson number–based radiosonde and model ${\text{PBLH}}_{R_i}$ definition

Radiosondes offer in situ high accuracy measurements of temperature and humidity profiles, making them a de facto standard used to validate other PBLH data. Their greatest limitation stems from poor spatiotemporal sampling. Radiosondes are typically released at 0000 and 1200 UTC, and their spatial coverage contains large gaps over global oceans and the Southern Hemisphere. In our study, radiosonde observations are used to derive estimates of PBLH based on a bulk Richardson number defined in Seidel et al. (2012) as

$R_i=\frac{g}{{\theta }_{\upsilon }^{\text{sfc}}}\frac{({\theta }_{\upsilon }^z-{\theta }_{\upsilon }^{\text{sfc}})(z-z^{\text{sfc}})}{[{(u^z-u^{\text{sfc}})}^2+{({\upsilon }^z-{\upsilon }^{\text{sfc}})}^2]},$(1)

where g is the gravitational acceleration; ${\theta }_{\upsilon }^z$ and ${\theta }_{\upsilon }^{\text{sfc}}$ are the virtual potential temperature at the height z above the ground level and at the surface, respectively; and u and υ are the zonal and meridional winds, respectively. The winds at the surface are assumed to be zero. When humidity and winds are observed at different heights than those of temperature, the humidity and winds are linearly interpolated to the temperature heights. The ${\text{PBLH}}_{R_i}$ is evaluated based on differences between the surface and successively higher levels, assuming that the surface layer is unstable, and ${\text{PBLH}}_{R_i}$ is identified as the height at which R_i first reaches a critical value of 0.25. If the critical value is not found exactly at an observation or model level, then ${\text{PBLH}}_{R_i}$ is found by linearly interpolating between the highest unstable level and the first stable level.

The GEOS model uses two PBLH definitions as length scales in the Louis turbulence scheme: the bulk R_i–based PBLH described above which is used over land, and a second definition based on the profile of thermal diffusivity is used over ocean. The latter PBLH is defined as the height at which the diffusivity drops to a threshold of 10% of the column maximum K_H. We use the bulk R_i–based model ${\text{PBLH}}_{R_i}^f$ definition everywhere for comparison with radiosonde ${\text{PBLH}}_{R_i}^o$ data. An example of ${\text{PBLH}}_{R_i}^o$ from global radiosonde data and of R_i-based model ${\text{PBLH}}_{R_i}^f$ definition at 0000 and 1200 UTC is shown in Fig. 1. Figures 1a and 1b correspond to ${\text{PBLH}}_{R_i}^o$ estimated from observations at 0000 and 1200 UTC, respectively, and Figs. 1c and 1d correspond to model estimates of ${\text{PBLH}}_{R_i}^f$ at those times. The large diurnal variation over land is evident in the figure. For example, while North America has a deep convective PBL at 0000 UTC, the same region displays shallow PBL features at 1200 UTC.

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(a),(b) Bulk R_i–based radiosonde ${\text{PBLH}}_{R_i}^o$ data and (c),(d) model ${\text{PBLH}}_{R_i}^f$ at (a),(c) 0000 UTC and (b),(d) 1200 UTC 9 Sep 2015 with clear diurnal variation over land.

Citation: Monthly Weather Review 153, 3; 10.1175/MWR-D-24-0141.1

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2) Refractivity gradient–based GNSS-RO and model PBLH_RF definition

Space-based observing systems, such as GNSS-RO, offer greater global data coverage than radiosonde or sparse ground-based networks. GNSS-RO is based on an atmospheric limb sounding technique that yields information at relatively coarse horizontal but high vertical resolution, which makes it potentially suitable for PBLH retrievals over oceans and flat land surfaces. The GNSS-RO ${\text{PBLH}}_{\text{RF}}^o$ is retrieved based on the gradients in the refractivity profile that result from temperature and moisture changes at the PBL top. As not all refractivity profiles reach the surface—depending on location and regime—the refractivity retrievals can be negatively biased below 2 km due to the presence of critical refraction layers. This bias, however, typically does not impact the ${\text{PBLH}}_{\text{RF}}^o$ retrieval (Ao et al. 2012; Guo et al. 2011).

In this study, we use a modified version of the algorithm of Ao et al. (2012) to calculate PBLH_RF. In the model, the refractivity of moist air (Cucurull et al. 2013) is computed using model profiles of temperature, humidity, and pressure. The algorithm of Ao et al. (2012) identifies ${\text{PBLH}}_{\text{RF}}^o$ as the height of the overall minimum refractivity gradient below 6 km. A minimum threshold of −40 N-unit km⁻¹ was also applied to the observed values to eliminate weak or ambiguous retrievals. A global snapshot of the original model ${\text{PBLH}}_{\text{RF}}^f$ is shown in Fig. 2a for 0000 UTC 9 September 2015.

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Model (a) original ${\text{PBLH}}_{\text{RF}}^f$, (b) ${\text{PBLH}}_{\text{RFLOW}}^f$, and (c) ${\text{PBLH}}_{\text{RFHGH}}^f$ at 0000 UTC 9 Sep 2015. White color in (c) indicates the locations where only one minimum refractivity gradient exists (i.e., ${\text{PBLH}}_{\text{RFLOW}}^f$ is equal to ${\text{PBLH}}_{\text{RFHGH}}^f$).

Citation: Monthly Weather Review 153, 3; 10.1175/MWR-D-24-0141.1

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Close inspection of the original ${\text{PBLH}}_{\text{RF}}^f$ revealed undesirable features. Substantial small-scale variability is apparent in the snapshot of Fig. 2a, with large jumps in ${\text{PBLH}}_{\text{RF}}^f$ over oceans and in regions of nocturnal boundary layers (Africa, Asia). The ensemble spread of ${\text{PBLH}}_{\text{RF}}^f$, which provides the flow-dependent background error covariance information for data assimilation in GEOS, was also unexpectedly large and noisy, again over areas of land during nighttime and much of the global ocean (Fig. 3a). These issues were attributed to the presence of multiple significant minima in the refractivity gradient for many profiles. Multiple minima were also commonly seen in observed profiles of refractivity gradient, as shown in Fig. 3b. In the affected columns, small variations in the thermodynamic profiles could cause the overall minimum (and PBLH_RF) to jump from one level to another. This can produce large variations across the ensemble and large observation-model differences.

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(a) Ensemble spread of original ${\text{PBLH}}_{\text{RF}}^f$ at 0000 UTC 9 Sep 2015 and (b) observed GNSS-RO profiles of refractivity divided by 10 and refractivity gradient (near 31.1°N, 179.6°W). Horizontal blue and red dashed lines indicate ${\text{PBLH}}_{\text{RFLOW}}^o$ and ${\text{PBLH}}_{\text{RFHGH}}^o$, respectively. Black dotted vertical line indicates −40 N-unit km⁻¹ threshold. (c) As in (a), but for ${\text{PBLH}}_{\text{RFLOW}}^f$.

Citation: Monthly Weather Review 153, 3; 10.1175/MWR-D-24-0141.1

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These findings led us to modify the refractivity gradient–based algorithm as follows. We first identify all local minima in the refractivity gradient exceeding a certain threshold. The threshold is a weighted average of the overall minimum (25%) and the average gradient below 6 km (75%). This effectively removes small wiggles in the profile from consideration, while retaining more significant minima. The two most negative minima are then identified and labeled PBLH_RFLOW and PBLH_RFHGH based on their relative heights. In cases with only one minimum exceeding the threshold, both PBLH_RFLOW and PBLH_RFHGH are set to the same value. In the example shown in Fig. 3b, the lower minimum at an altitude of approximately 1326 m is denoted as PBLH_RFLOW, while the upper minimum at 2356 m is denoted as PBLH_RFHGH. A flowchart is provided in Fig. 4 to illustrate the algorithm used for estimating PBLH_RFLOW and PBLH_RFHGH.

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Flowchart to illustrate the derivation algorithm for refractivity gradient–based PBLH_RFLOW and PBLH_RFHGH.

Citation: Monthly Weather Review 153, 3; 10.1175/MWR-D-24-0141.1

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A global snapshot of ${\text{PBLH}}_{\text{RFLOW}}^f$ for 0000 UTC 9 September 2015 is shown in Fig. 2b. The revised PBLH field is much smoother than the original, with more consistent shallow values in nocturnal regions. Similarly, Fig. 3c shows a significant decrease in the ensemble spread, particularly over the oceans, and the absence of much of the popcorn-like patterns observed in the original ${\text{PBLH}}_{\text{RF}}^f$ ensemble spread over land (see Fig. 3a).

The global snapshot of ${\text{PBLH}}_{\text{RFLOW}}^f$ (Fig. 2b) at 0000 UTC 9 September 2015 shows the closer agreement of these results with ${\text{PBLH}}_{R_i}^f$ of Fig. 1c. While both ${\text{PBLH}}_{R_i}^f$ and ${\text{PBLH}}_{\text{RFLOW}}^f$ definitions return a similar PBLH for continental convective boundary layers, we note that the definitions can still be sensitive to different aspects of the marine and nocturnal PBL profiles. A similar difference pattern between ${\text{PBLH}}_{\text{RFLOW}}^f$ and ${\text{PBLH}}_{R_i}^f$ is also observed at 1200 UTC. This emphasizes the importance of using a common definition when comparing model and observations.

In principle, both low and high values could be assimilated; however, in this study, we will focus on assimilating ${\text{PBLH}}_{\text{RFLOW}}^o$ and hereby refer to PBLH_RFLOW as PBLH_RF. This is primarily due to its greater similarity to ${\text{PBLH}}_{R_i}$; it appears to be similarly sensitive to the surface-driven mixed layer depth, and assimilation of ${\text{PBLH}}_{\text{RFLOW}}^o$ from GNSS-RO may offer a more robust constraint on that physical feature. We note that the modified definition is also applied to GNSS-RO data for consistency in addition to the minimum threshold of −40 N-unit km⁻¹ criterion.

Figure 2c shows ${\text{PBLH}}_{\text{RFHGH}}^f$ for the same analysis time as Figs. 2a and 2b. An additional noteworthy observation in Fig. 2c relates to the white areas that can be seen over many marine stratocumulus regions. These areas correspond to all the locations for which only one refractivity gradient minimum was found for that date, and these regions are usually associated with low ensemble spread (Fig. 3c). Additional evaluation may be needed in the future to determine more conclusively whether the model ensemble has too small variations in these regions.

b. PBLH data synergy strategy and quality control (QC)

One of our primary objectives is to build a global PBLH analysis and monitoring capability in the GEOS system. We seek to explore a strategy for assimilating PBLH data from various sources by using multiple PBLH control variables, i.e., the model ${\text{PBLH}}_{R_i}$ and PBLH_RF, and later, total attenuated backscatter gradient–based PBLH are augmented to the control variable vector and their analyses are generated along with those of other control variables at the end of the cost function minimization procedure. This strategy facilitates comparison of PBLH data with their corresponding model PBLH background in order to compute consistent OmFs for PBLH data assimilation. These PBLH control variables are highly correlated, their error correlations being provided by the 4D ensemble forecast, and the strong correlations enable PBLH data information from various sources to interact and combine coherently.

To successfully assimilate PBLH data, QC is a necessary and important step. Generally, QC procedures are unique for each PBLH dataset, e.g., the QC procedures for GNSS-RO PBLH data are different from those for PBLH data from space-based lidars. In this paper, only the QC procedures for PBLH data from radiosonde and GNSS-RO are presented. For both radiosonde and GNSS-RO PBLH data, while larger OmF errors are usually seen with deeper PBL, no obvious dependence of OmF errors on local time is found, and any PBLH data exceeding 6 km are excluded. For radiosonde ${\text{PBLH}}_{R_i}^o$ data, data from stations at elevations higher than 3 km are excluded. For GNSS-RO ${\text{PBLH}}_{\text{RF}}^o$ data, several QC procedures are designed. Data at locations with sharp gradient terrain or rapidly varying orography are excluded based on the subgrid standard deviation (STDV) of surface height (Fig. 5), with a critical threshold of 200 m. Data located in an atmospheric grid box over a mixed land/ocean surface are also rejected. In addition, any ${\text{PBLH}}_{\text{RF}}^o$ data corresponding to the refractivity profiles whose lowest observation level is higher than 500 m are excluded, and this QC procedure removes many large GNSS-RO PBLH OmFs.

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Subgrid surface height STDV (m) used to exclude GNSS-RO PBLH data at locations where the subgrid surface height STDV > 200 m.

Citation: Monthly Weather Review 153, 3; 10.1175/MWR-D-24-0141.1

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A final gross error check is applied to both radiosonde and GNSS-RO PBLH OmFs, where PBLH data will be rejected if the absolute value of the ratio of OmF to observation error exceeds a specified threshold, 5 in this study. This gross error check is modest and excludes only extreme outliers. As depicted in Fig. 6b for the 426 GNSS-RO ${\text{PBLH}}_{\text{RF}}^o$ data, 20 are removed due to mixed surface type and subgrid surface height STDV > 200 m, 175 are removed due to lowest observed level > 500 m, 9 are removed due to gross error check, and 222 data remain after all QC procedures. The effect of all the QC steps on radiosonde ${\text{PBLH}}_{R_i}^o$ and GNSS-RO ${\text{PBLH}}_{\text{RF}}^o$ data can also be seen in the PBLH OmF histograms of Fig. 7, where gray bars indicate values before applying QC procedures, and blue bars correspond to after. While only 5% of ${\text{PBLH}}_{R_i}^o$ data are excluded by the QC procedure, more than 50% of ${\text{PBLH}}_{\text{RF}}^o$ data, especially those on the right positive side of the OmF histogram, are rejected. This result is in agreement with Ao et al. (2012), who found that only about 50% of the COSMIC profiles were retained globally in their study, when only profiles that can penetrate down to within 500 m of the surface were used to retrieve the PBL heights. Overall, the data passing QC procedures have a good distribution centered around zero OmF.

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Difference in number of points of GNSS-RO ${\text{PBLH}}_{\text{RFLOW}}^o$ data that results from applying all QC procedures to the dataset: (a) before QC and (b) after QC for the analysis cycle of 0000 UTC 9 Sep 2015. The dot colors indicate OmF values, and the × marks denote removed observations after QC.

Citation: Monthly Weather Review 153, 3; 10.1175/MWR-D-24-0141.1

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OmF histograms of (a) ${\text{PBLH}}_{R_i}$ and (b) PBLH_RFLOW data for the period from 23 Aug to 17 Sep 2015. Gray bars correspond to the OmF values before QC criteria are used, and blue bars correspond to the OmF values obtained after QC criteria are applied. The bars filled with star pattern indicate total number of observations, where absolute values of OmF are greater than 3 km.

Citation: Monthly Weather Review 153, 3; 10.1175/MWR-D-24-0141.1

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Observation errors of ${\text{PBLH}}_{R_i}^o$ and ${\text{PBLH}}_{\text{RF}}^o$ data are specified as a piecewise linear function of PBLH data. Based on the scattering plot distribution pattern of PBLH OmF standard deviation with respect to PBLH observation bins, the observation error for ${\text{PBLH}}_{R_i}^o$ is set to be 200 m when ${\text{PBLH}}_{R_i}^o$ is less than 2 km and 500 m when ${\text{PBLH}}_{R_i}^o$ is larger than 4 km and varies linearly in between; for ${\text{PBLH}}_{\text{RF}}^o$, it is set to be 250 m when ${\text{PBLH}}_{\text{RF}}^o$ is less than 1.5 km and 800 m when ${\text{PBLH}}_{\text{RF}}^o$ is larger than 4 km and varies linearly in between. In addition, observation error for ${\text{PBLH}}_{\text{RF}}^o$ is inflated if the distance between adjacent observations is less than 125 km, with the inflation multiplier specified as the square root of ${\text{PBLH}}_{\text{RF}}^o$ data count within 125 km of each ${\text{PBLH}}_{\text{RF}}^o$ datum. The observation error for ${\text{PBLH}}_{R_i}^o$ is inflated by an empirical constant multiplier 1.2 if a station elevation is above 1 km.

c. Capturing capping inversion and correlation between PBLHs and thermodynamic states

In the GEOS hybrid 4D EnVar formulation, observation information OmFs are propagated by background error covariance and projected onto analysis variables. The background error covariance is composed of static and ensemble components, and the flow-dependent ensemble component is generally provided by the model forecast ensemble. The static part in this study is designed to be univariate for ${\text{PBLH}}_{R_i}$ and PBLH_RF control variables, background error variance is specified to be 30% of their respective physical field, which means the larger the PBLHs the larger the error variances, and horizontal correlation length is taken as the same as that for specific humidity q. The ensemble background error covariance represents the probability distribution of the system uncertainty. It is implicitly extracted from the ensemble control variables through the formulation of the multivariate increments from the ensemble part, which is calculated by taking the product of the control variable weights and the ensemble perturbations at those grid points.

The impact of PBLH data on the PBL thermodynamic structure is realized through the ensemble background error covariance and through improving the performance of model physical parameterization schemes by utilizing the PBLH analysis. The discussion of the latter is not included in this paper but will be described in a subsequent article. The ensemble forecast provides the horizontal and vertical correlation structure of background errors and relationships among control variables, and the localization length scales are used to reduce sampling errors of the background error covariance with undersized 32 ensemble members. The accurate ensemble background error covariance and vertical localization length scale are critical for the assimilation of all observation profiles in the lower troposphere when analyzing a boundary layer capping inversion. When a capping inversion is present, the ensemble background error covariance is characterized by a large ensemble spread around the inversion height and a decoupling between the errors in the PBL and the free atmosphere, as a strong inversion prevents the air masses from mixing. However, when there is a large discrepancy in inversion height between background and observation, the ensemble spread, which peaks around the background inversion height, will cause an incorrect mapping structure for the observation profile information OmFs. As a result, as shown in Fowler et al. (2012), traditional data assimilation algorithms tend to erroneously consider the large OmFs in the vicinity of the inversion as amplitude error, which in turn leads to the analysis smearing out and degrading the inversion structure.

Therefore, it is key to accurately represent the ensemble spread structure around the observed capping inversion height to increase the impact of PBLH observation, and more importantly, the impact of profile observations, on the state variables in the model levels closest to the observed PBLH and to confine the impacts of observations around the observed inversion height properly. In this study, we use PBLH data to better capture the location of a capping inversion directly through inflation of the ensemble spread and adjustment of the vertical localization length scale around the model levels closest to the PBLH data. This method is referred to as AdjEnsVloc hereafter. In this way, PBLH data are no longer just a single piece of information, and their utilization will effectively help to draw the analysis profile more closely to observations for capping inversions. In each 6-h assimilation window, we inflate the ensemble virtual temperature T_υ and relative humidity (RH) perturbations at the four neighboring ensemble analysis grids around each PBLH datum horizontally and on five model levels as discussed below. The inflation is applied to the two ensemble time levels adjacent to the time of each PBLH datum. The inflation coefficients, defined as a function of model vertical level k, are set as c = 1 + α × f(k) with Gaussian probability density function $f(k)=[1/\left(\sigma \sqrt{2\pi }\right)]e^{-(1/2){[(k-\mu )/\sigma ]}^2}$, where μ is the model level closest to each PBLH datum, σ is the standard deviation of the distribution, and α is a parameter to control the magnitude factor of the inflation. The inflation coefficients are applied to five model levels including the model level μ and two levels above and below. The plots of f(k) and inflation coefficients c are displayed in Fig. 8. For simplicity, σ² is set to be 1. Sensitivity experiments of inflation coefficient c are performed in section 4a(1) with different parameter values of α. With α value of 2.5, the inflation coefficient c is 2 at the model level μ, about 1.6 at the adjacent model levels (one level above and one level below), 1.1 at the next adjacent model levels (two levels above and below), and one for the rest of the model levels. The respective c values at these five model levels around a PBLH datum are 5, 3.4, and 1.5 when α is 10 and also 10, 6.5, and 2.3 when α is 22.5. This method will effectively increase the ensemble spread at desired model levels to allow the analysis to be drawn closer to observations near inversions. It should also be noted that, although this inflation method is applied on model levels in this preliminary study, it can be generalized without any technical difficulty within the context of physical variables (e.g., function of thickness/height/pressure), and optimal inflation magnitudes will also be explored in the future.

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(a) Probability density function f(k) and (b) inflation coefficient c that are used for ensemble virtual temperature or relative humidity perturbations at the model level closest to each PBLH datum and two model levels above and below.

Citation: Monthly Weather Review 153, 3; 10.1175/MWR-D-24-0141.1

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While the horizontal localization length scale is kept unchanged, adjustment is also made to the vertical localization length scale for the ensemble around inversions to ensure limited interactions between PBL and above air. The existing length scale varies at a different model level to mitigate spurious vertical error correlations but is a constant for the globe at each model level. It is specified in logarithmic pressure units for each model level and then converted to grid units used in recursive filter by scaling the length scale of each model level with the logarithmic pressure spacing of each model layer. In this study, PBLH data information is introduced into the process to allow the length scales to vary globally depending on the locations of PBLH data. Using the model’s geopotential height information at each model level, and latitude, longitude, and PBLH values at each PBLH observation location, we can identify the model level closest to the PBL top and the four neighboring horizontal ensemble grids surrounding the PBLH observation. In our preliminary tests, for simplicity, the vertical localization length scale is reduced by multiplication of 0.1 for the model level closest to the PBL top and two levels above and below, respectively, at these neighboring grids around each PBLH observation. A multiplication by the Gaussian-type function will be explored in the future.

Moreover, the correlation relationships among PBLHs and other control variables determine how the PBLH data information is projected and how PBLHs and other control variables interact with each other. All the PBLHs are tightly coupled with the PBL thermodynamic variables (e.g., as shown in Fig. 9), and thereby, the PBLH data will provide additional information for PBL temperature and moisture fields. For example, perturbation correlations between ${\text{PBLH}}_{R_i}$ and virtual temperature at low model levels exhibit a clear global PBLH diurnal cycle feature over land (not shown). The refractivity gradient–based model PBLH_RF is also closely related to atmospheric states. Although its perturbation correlations with virtual temperature over land are more complicated, the diurnal cycle accompanying the convective boundary layer is still obvious (Fig. 9, black arrows). In addition, very clear relationships are also seen persistently in marine stratocumulus regions on Fig. 9 (blue arrows) for the ensemble perturbation correlations between PBLH_RF and virtual temperature/specific humidity. Such relationships can also be illustrated in profiles of ensemble covariances between θ_υ, q and both ${\text{PBLH}}_{R_i}$ and PBLH_RF, as shown in the examples of Fig. 10. The profiles are taken from different regimes including a marine stratocumulus boundary layer beneath a strong inversion (Ocean StCu; black), a continental daytime convective boundary layer from Australia (Land CBL; red), and a tropical trade cumulus profile (Ocean Cu; blue) at 0000 UTC 5 September 2015. Covariances are normalized by the standard deviation of PBLH to indicate typical variations of θ_υ and q. Deeper PBLH of both definitions are clearly associated with cooling and moistening just at/above PBLH and drying within the subcloud mixed layer, except for the ensemble covariance between q and ${\text{PBLH}}_{R_i}$ in Ocean Cu condition which goes to opposite direction. In this Ocean Cu condition, ${\text{PBLH}}_{R_i}$ is much lower than PBLH_RF.

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Ensemble perturbation correlation coefficients (left) between PBLH_RF and virtual temperature and (right) between PBLH_RF and specific humidity at model level 9 at 0000, 0600, 1200, and 1800 UTC 9 Sep 2015. Black arrows illustrate the diurnal cycle, and blue arrows denote the strong correlation coefficients in the marine stratocumulus regions.

Citation: Monthly Weather Review 153, 3; 10.1175/MWR-D-24-0141.1

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Normalized ensemble covariance of thermodynamic quantities with PBLH in three boundary-layer regimes. (a),(c) θ_υ or (b),(d) q covariance with (a),(b) ${\text{PBLH}}_{R_i}$ or (c),(d) PBLH_RF (dashed lines), taken from a land convective boundary layer (red; 26°S, 145°E), a marine trade cumulus (blue; 11°S, 120°W), and a marine stratocumulus regime (black; 20°S, 72°W). Ensemble mean profiles of (e) virtual potential temperature and (f) specific humidity, with ensemble spread indicated by shading.

Citation: Monthly Weather Review 153, 3; 10.1175/MWR-D-24-0141.1

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4. Assimilation testing results

To demonstrate the capability of the newly developed PBLH data assimilation framework and methodology, and of the impact of PBLH data on the representation of the PBL thermodynamic structure, in section 4a, we randomly select an analysis cycle of 0000 UTC 9 September 2015 to investigate the PBLH data impacts on atmospheric state variables. A sensitivity study of the AdjEnsVloc mechanism is first conducted with global radiosonde data profiles and ${\text{PBLH}}_{R_i}^o$ data assimilation and examined in detail at a single radiosonde station. An additional control experiment (Control_Allob) using all operational observations is also performed to examine the impact of additional data on the analysis at this station. Then, the separate PBLH data impacts through each of the two mechanisms, i.e., AdjEnsVloc and multivariate aspects of the ensemble background error covariance, and their combined impact are assessed with the assimilation of all operational observations and GNSS-RO ${\text{PBLH}}_{RF}^o$ data. In section 4b, an evaluation for the period of 23 August–17 September 2015 is presented, where all operational observations are assimilated in the hybrid 4D EnVar configuration, and both radiosonde ${\text{PBLH}}_{R_i}^o$ and GNSS-RO ${\text{PBLH}}_{\text{RF}}^o$ data are used as well.

Although PBLH data contain unique height location information of the gradient extrema in air properties across the inversion, which is challenging for the assimilation of profile data to capture in data assimilation systems, we intend to use PBLH data conservatively. If the profiles that are used to derive PBLH data are separately assimilated in the GEOS system, we will confine the impact of the PBLH data to the PBLH variable only, i.e., only PBLH univariate static background error covariance is used for the assimilation of this PBLH data type. This is the case for ${\text{PBLH}}_{R_i}$ control variable using radiosonde ${\text{PBLH}}_{R_i}^o$ data. Regarding GNSS-RO bending angle data, although they are assimilated extensively in GEOS, very few GNSS-RO bending angle data are used in the lower troposphere, and these data are also assigned with large observation errors. Hence, tests conducted with ${\text{PBLH}}_{\text{RF}}^o$ data will be examined with and without PBLH_RF ensemble contribution included in the background error covariance.

a. A case study: 0000 UTC 9 September 2015

1) Assimilation of radiosonde ${\text{PBLH}}_{R_i}^o$ data

First, we examine the capability to assimilate radiosonde ${\text{PBLH}}_{R_i}^o$ data with only ${\text{PBLH}}_{R_i}$ univariate static term of the background error covariance to produce the ${\text{PBLH}}_{R_i}$ analysis and assess the impact of AdjEnsVloc (i.e., the adjustments of T_υ and RH ensemble spread and the vertical localization length scale) using radiosonde ${\text{PBLH}}_{R_i}^o$ data on temperature and specific humidity analyses when assimilating radiosonde data. For this purpose, four experiments are conducted (Table 1): Control is the hybrid 4D EnVar GEOS configuration using only radiosonde data profiles; Control_AllOb is the same as control, but all operational observations used in the GEOS including radiosonde profile data are assimilated; experiment Inf2T2Q_0.1VLoc is the same as control, but radiosonde ${\text{PBLH}}_{R_i}^o$ data are also assimilated and used for ensemble spread inflation and vertical localization adjustment with the configuration described in section 3c, where inflation coefficient c is set to be 2 at the model level μ closest to ${\text{PBLH}}_{R_i}^o$ data for both virtual temperature and relative humidity and multiplication of 0.1 to vertical localization length scale; and experiment Inf5T10Q_0.1VLoc is the same as Inf2T2Q_0.1VLoc, but the inflation coefficient c is 5 for virtual temperature and 10 for relative humidity at the model level μ.

Table 1.

Experiments for the assimilation of radiosonde ${\text{PBLH}}_{R_i}^o$ data.

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The first guess and analysis of variables of the four experiments are interpolated to radiosonde station 91680 and then are compared with radiosonde data profiles and ${\text{PBLH}}_{R_i}^o$. This station is selected because of the presence of a strong capping inversion. The observation time at 850 hPa at this station is 2257 UTC 8 September 2015. Figure 11 shows vertical profiles of radiosonde observations (black solid line), first guess (red solid line), and analyses for virtual potential temperature, specific humidity, and temperature subtracted by the lapse rate from the four experiments at the analysis cycle of 0000 UTC 9 September 2015. Radiosonde data profiles indicate a convective boundary layer with strong inversion at 1675 m (around 832 hPa; horizontal black dashed line), whereas first guess fields exhibit a shallower boundary layer (horizontal red dashed line) and mild inversion. Because the radiosonde data used in this study from prepBUFR data files have reduced vertical coverage, the full-resolution specific humidity observations from the University of Wyoming site¹ and the corresponding first guess are provided in Fig. 11c to show a more complete representation of the boundary layer in the first guess. It is seen that the first guess at full-resolution observation locations is closer to observations at 760 and 890 hPa, but the patterns of first guess at the observed inversion in Figs. 11b and 11c that are similar as critical observations around the inversion height are already included in the observation data file used in this study.

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Vertical profiles of radiosonde observation (black solid), first guess (red solid), and analyses (Control: orange solid; Control_AllOb: green dotted; Inf2T2Q_0.1VLoc: light blue solid; Inf5T10Q_0.1VLoc: violet solid) of (a) virtual potential temperature (K), (b) specific humidity (g kg⁻¹), (c) full-resolution specific humidity observations from the University of Wyoming site and corresponding first guess, and (d) temperature subtracted by lapse rate (K) at the radiosonde station 91680 (lat: 17.8°S, lon: 177.5°E) for the analysis cycle of 0000 UTC 9 Sep 2015. Horizontal dashed lines indicate ${\text{PBLH}}_{R_i}$ from observation (black dashed), first guess (red dashed), and analyses (blue dashed).

Citation: Monthly Weather Review 153, 3; 10.1175/MWR-D-24-0141.1

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In both of the control runs, the radiosonde data for temperature T and specific humidity q profiles are assimilated. While small improvements in T and q analyses are observed, the analyses of T and θ_υ become worse than the first guess at and above the observed PBL top, and the PBL height is hardly moved. This is because the original ensemble spread (Fig. 12, red line) from the ensemble perturbations is small with little variation below 900 hPa and exhibits a slight increase from 900 hPa till peaking around 860 hPa with a shallower PBL than the observed. Compared with the analyses of control using only radiosonde data, the use of all available observations in Control_Allob has little additional benefits at this station; in fact, it moves analyses of T and θ_υ even farther away from observations at and above the observed PBL top.

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Vertical profiles of ensemble spread of (a) virtual temperature and (b) relative humidity before (red line) and after (blue line) ensemble spread inflation in the Inf2T2Q_0.1VLoc experiment at the radiosonde station 91680 at 2300 UTC 8 Sep 2015. Black lines indicate first guess of virtual temperature (K) in (a) and specific humidity (g kg⁻¹) in (b) at the corresponding time.

Citation: Monthly Weather Review 153, 3; 10.1175/MWR-D-24-0141.1

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In the other two experiments where ${\text{PBLH}}_{R_i}^o$ data are used, while we generate ${\text{PBLH}}_{R_i}$ analysis, we also apply adjustments to virtual temperature T_υ and RH ensemble spread and the vertical localization length scale around the observed PBL top. The vertical profiles of the ensemble spread of T_υ (left) and RH (right) after ensemble spread inflation (blue line) are provided in Fig. 12, and it is shown that the ensemble spread is now increased and peaks at model layer 12, while the original ensemble spread peaks at model layer 10. The impact of ensemble spread inflation is investigated in experiments Inf2T2Q_0.1VLoc (light blue solid) and Inf5T10Q_0.1VLoc (violet solid) (Fig. 11). The same adjustment multiplier of 0.1 to the vertical localization length scale is used in these two experiments to limit the influence of ${\text{PBLH}}_{R_i}^o$ data across model vertical layers near the PBL top. The analysis results indicate that the PBLH assimilation framework is working as expected (Fig. 11), and the ${\text{PBLH}}_{R_i}$ analysis (horizontal blue dashed line) is drawn closer to radiosonde ${\text{PBLH}}_{R_i}^o$ data compared to the first guess. Moreover, analyses of T and q profiles and the inversion are improved near the observed inversion height after applying the adjustments, and the larger the ensemble spread inflation, the more significant improvement can be obtained.

Small additional improvements are observed when further adjustment of vertical localization length scale is applied. Smaller vertical localization length scale can better represent the strong inversion at this station, and the analysis increments are more confined around the PBL top (figure not shown).

The 2D map analysis increments of virtual temperature and specific humidity below and above the PBL top, i.e., at model levels 11 and 13, are examined in Fig. 13. Compared with the control run, as expected, the adjustments to T_υ and RH ensemble spread and the vertical localization length scale make the increments larger in magnitude in the vicinity of this radiosonde station, and maximum location of specific humidity increments is also shifted to the west. In addition, compared to the control run, the interactions between model levels 11 and 13 are decreased in Inf2T2Q_0.1VLoc experiment due to the reduction of the vertical localization length scale. The large negative OmFs below the PBL top produce larger negative analysis T_υ increment at model level 11 but have smaller impact on model level 13, where small positive T_υ increments are now present in Inf2T2Q_0.1VLoc as a result of the positive OmFs above PBL top. Similarly, the specific humidity variable exhibits larger opposite signs of analysis increments at the two model levels based on the respective OmFs information below/above PBL top. The impact of the reduced vertical localization length scale can also be observed in the cross section of analysis increments in Fig. 14, for example around 18°S, there is less interaction between model levels 11 and 13, and opposite signs of analysis increments are increased at these two model levels.

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The analysis increments at model levels 11 (left four panels) and 13 (right four panels) for (a),(c),(e),(g) virtual temperature and (b),(d),(f),(h) specific humidity in (top) Control and (bottom) Inf2T2Q_0.1VLoc experiments at 2300 UTC 8 Sep 2015. The star indicates the location of radiosonde 91680.

Citation: Monthly Weather Review 153, 3; 10.1175/MWR-D-24-0141.1

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As in Fig. 13, but showing vertical cross section of analysis increments for (a),(c) virtual temperature and (b),(d) specific humidity in the (top) Control and (bottom) Inf2T2Q_0.1VLoc experiments along a longitudinal line at 177°E, which is shown by the red dashed line in the inset at the top-right corner of (a).

Citation: Monthly Weather Review 153, 3; 10.1175/MWR-D-24-0141.1

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Overall, the PBLH assimilation algorithm is working, producing PBLH analysis and improving the representation of PBL inversion. Inf2T2Q_0.1VLoc AdjEnsVloc setup will be used, hereafter, for conservative adjustments. Further studies are required to obtain optimal ensemble spread inflation and adjustment of the vertical localization length scale. It is also noticed that little improvement is observed near the surface. One of the possible causes may be associated with the small ensemble spread in the lowest model levels, which may be assessed in future stochastic model physics studies.

2) Assimilation of GNSS-RO ${\text{PBLH}}_{\text{RF}}^o$ data

Unlike the use of radiosonde ${\text{PBLH}}_{R_i}^o$ data, we assimilate ${\text{PBLH}}_{\text{RF}}^o$ data with hybrid mode, i.e., using both PBLH_RF univariate static and ensemble parts of background error covariance. When only the PBLH_RF univariate static part is used, other observations have no impact on PBLH_RF analysis, but ${\text{PBLH}}_{\text{RF}}^o$ data can change the analyses of other variables via adjustments to their ensemble spread and vertical localization length scale, i.e., AdjEnsVLoc. However, when the PBLH_RF ensemble part of background error covariance (BEC) is activated, i.e., hybrid BEC mode, PBLH_RF, and other control variables (especially T_υ and RH) would interact with each other through their correlation relationships embedded in the forecast ensemble, in addition to the impact from AdjEnsVLoc if this functionality is turned on. At the same time, not only the static part but also the ensemble part of BEC (through linear combination of ensemble perturbations) contributes to the analysis increment of PBLH_RF.

A set of experiments are performed to evaluate the effects of the two mechanisms on how ${\text{PBLH}}_{\text{RF}}^o$ data affect T_υ and q variables (Table 2). All operational observations are used in the GEOS hybrid 4D EnVar configuration, and GNSS-RO ${\text{PBLH}}_{\text{RF}}^o$ data are also assimilated in these experiments. PBLH_RF ensemble perturbations over land are not considered in these experiments; as we focus here on areas over ocean, the inflation parameter α is set as 2.5 for both T_υ and RH ensemble spread, and a 0.1 multiplier is applied to the vertical localization length scale if AdjEnsVLoc is turned on at the used 162 GNSS-RO ${\text{PBLH}}_{\text{RF}}^o$ data locations over ocean. Analysis increments from these experiments are examined. Because only univariate static BEC is employed for ${\text{PBLH}}_{\text{RF}}^o$ data in Exp1, the assimilation of ${\text{PBLH}}_{\text{RF}}^o$ data does not affect any other variables; hence, Exp1 essentially produces the same analyses for all other control variables as the original version of GEOS without ${\text{PBLH}}_{\text{RF}}^o$. The differences between Exp2 and Exp1 indicate the impact of using observed ${\text{PBLH}}_{\text{RF}}^o$ data to correct ensemble BEC, the differences between Exp3 and Exp1 arise from the additional thermodynamic information provided by ${\text{PBLH}}_{\text{RF}}^o$ data through multivariable relationships in the original ensemble BEC, and the differences between Exp4 and Exp1 are attributed to both AdjEnsVLoc and the multivariate correlation relationships in the improved ensemble BEC.

Table 2.

GNSS-RO ${\text{PBLH}}_{\text{RF}}^o$ experiments (with all operational observations in hybrid 4D EnVar configuration).

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It is shown in Fig. 15 that the impacts of AdjEnsVLoc (upper row) are different from the impacts of the multivariable relationships in original BEC (middle row). The differences of analysis increments between Exp2 and Exp1 are as expected, PBLH_RF increments are the same for the two experiments, but the increments for T_υ and q are changed due to AdjEnsVLoc based on the observed ${\text{PBLH}}_{RF}^o$ information. The use of hybrid BEC in Exp3 for PBLH_RF introduces changes to increments of PBLH_RF, T_υ, and q due to the effect of multivariable relationships embedded in the original unadjusted ensemble BEC, and the patterns of the changes for T_υ and q are different from those due to the effect of AdjEnsVLoc. The differences between Exp4 and Exp1 (lower row) are not just a simple combination of the two impacts, but a combined impact of AdjEnsVLoc and multivariable relationships in the corrected ensemble background error covariance. However, the fact that Exp2 and Exp4 results appear very similar indicates that the impact of AdjEnsVLoc is more important, rather than the multivariate aspects of ensemble covariances. Comparing the differences of analysis increments of Exp3 and Exp4 in Fig. 15, we can see that the patterns for PBLH_RF are very similar, but differences are still noticed such as around 42°S, 165°W; on the contrary, the patterns for T_υ/q are quite different, and the impacts from the adjustments of ensemble BEC based on observed ${\text{PBLH}}_{\text{RF}}^o$ data are clearly notable.

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Differences of analysis increments of (left) PBLH_RF (m), (middle) virtual temperature (K), and (right) specific humidity (g kg⁻¹) at model level 9 (around 880–910 hPa over tropical oceans) between (a),(b),(c) Exp2 and Exp1; (d),(e),(f) Exp3 and Exp1; and (g),(h),(i) Exp4 and Exp1.

Citation: Monthly Weather Review 153, 3; 10.1175/MWR-D-24-0141.1

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The combined impact of AdjEnsVLoc and hybrid BEC of PBLH_RF on the analysis is also reflected in the profiles of the mean of the normalized observation minus analysis (OmA) and its root-mean-square difference (RMSD) for GNSS-RO bending angle data for Exp1 and Exp4 (Fig. 16). The maximum impact on the analysis of thermodynamic variables can be expected at where ${\text{PBLH}}_{\text{RF}}^o$ data are located. The normalized OmA means of Exp4 are smaller than those of Exp1 below 3500-m impact height, and the normalized OmA RMSD of Exp4 is reduced below 5000-m impact height shown as negative change of RMSD in Fig. 16, which shows that the analysis of Exp4 is drawn closer to GNSS-RO bending angle observations due to the use of ${\text{PBLH}}_{\text{RF}}^o$ data, although same small weights are given to these bending angle observations as in Exp1 in the lower troposphere in the GEOS. These results indicate that using ${\text{PBLH}}_{\text{RF}}^o$ data with AdjEnsVLoc and hybrid BEC is beneficial for GEOS analysis in the lower troposphere, and longer-term experiments and evaluation are presented in the following section.

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(a) The number of assimilated GNSS-RO bending angles in the analysis cycle of 0000 UTC 9 Sep 2015, (b) the mean of normalized OmA of bending angles, and (c) RMSD of normalized OmA (black solid: Exp1; blue dashed: Exp4) of bending angles as a function of impact height. Red line indicates change in RMSD of Exp4 from Exp1 (%). Statistics are obtained for each 300-m impact height interval by aggregating the data within those intervals, and there are fewer used data near the surface.

Citation: Monthly Weather Review 153, 3; 10.1175/MWR-D-24-0141.1

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b. Assessment for 23 August–17 September 2015

So far, we have investigated the two mechanisms through which PBLH data affect the PBL thermodynamic structure, AdjEnsVloc using ${\text{PBLH}}_{R_i}^o$ or ${\text{PBLH}}_{\text{RF}}^o$ data, and multivariate aspects of hybrid BEC of PBLH_RF, in a single cycle case. To provide more reliable results, a longer period from 23 August to 17 September 2015 is considered in this section for the assimilation of both radiosonde ${\text{PBLH}}_{R_i}^o$ data and GNSS-RO ${\text{PBLH}}_{\text{RF}}^o$ data together in addition to all operational observations used in the GEOS.

Three stand-alone analysis experiments shown in Table 3 are conducted to access the impact of our algorithms. While all experiments assimilate radiosonde ${\text{PBLH}}_{R_i}^o$, GNSS-RO ${\text{PBLH}}_{\text{RF}}^o$, and all operational observations, the experiments differ in whether AdjEnsVLoc is turned on and whether hybrid or static BEC mode is used for PBLH_RF. The hybrid 4D EnVar configuration is used in all experiments, but univariate static BEC is used for ${\text{PBLH}}_{R_i}$. Control experiment (CNTL) employs the univariate static BEC for PBLH_RF with AdjEnsVloc turned off; hence, all PBLH data have no impact on atmospheric state variables; experiment AdjStat is the same as CNTL except that AdjEnsVloc is turned on, and experiment AdjHybr is the same as AdjStat, but hybrid BEC is employed for PBLH_RF. Therefore, a comparison between AdjStat and CNTL is expected to reveal the impact of AdjEnsVLoc, and the comparison between AdjHybr and CNTL is expected to reveal the combined impact of AdjEnsVloc and multivariate aspects of hybrid BEC.

Table 3.

Radiosonde ${\text{PBLH}}_{R_i}^o$ and GNSS-RO ${\text{PBLH}}_{\text{RF}}^o$ DA experiments.

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The RMSDs of thermodynamic variable analysis departures from radiosonde observation data (Fig. 17) clearly demonstrate the direct impact of adjusting ensemble BEC of thermodynamic variables and vertical localization length scales on the analyses of thermodynamic variables at radiosonde observation locations. While the RMSDs of radiosonde temperature and specific humidity OmA below 700 hPa are displayed in Figs. 17a and 17b for all experiments, the corresponding OmA RMSD changes of T and q for experiments AdjStat (red solid line) and AdjHybr (blue dashed line) with respect to CNTL below 300 hPa are displayed in Figs. 17c and 17d. It is shown that the results of AdjStat and AdjHybr are basically the same at the radiosonde data locations because the multivariate aspect of BEC is lacking due to the fact that only univariate BEC is employed for ${\text{PBLH}}_{R_i}$. Moreover, improvements are observed in both temperature and specific humidity OmA RMSDs below 400 hPa, with the maximum improvements around 750 hPa.

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The profiles of RMSD of analysis departures OmA of (a) temperature and (b) specific humidity from CNTL (black solid), AdjStat (red dashed), and AdjHybr (blue dotted) experiments, respectively, against radiosonde observations below 700 hPa from 23 Aug to 17 Sep 2015. The profiles of percentage change in OmA RMSD of (c) temperature and (d) specific humidity of AdjStat (red solid) and AdjHybr (blue dashed) from CNTL below 300 hPa. Reduction of RMSD (negative) indicates improvement over CNTL experiment. The right y axis indicates the number of observations aggregated in each pressure level interval for statistics. The corresponding shading indicates the confidence intervals with the 95% significant level.

Citation: Monthly Weather Review 153, 3; 10.1175/MWR-D-24-0141.1

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The normalized GNSS-RO bending angle OmA RMSDs and the percentage changes from CNTL are presented in Fig. 18. GNSS-RO data used in this period are aggregated into 300-m impact height intervals to calculate the bending angle OmA RMSD. Here, bending angle OmA is normalized by the corresponding bending angle observation. Although adjustments are made to ensemble BEC of thermodynamic variables in the AdjEnsVloc method, the analysis fits GNSS-RO bending angle data also show improvement mostly below 4000 m, and the largest improvement happens around 2400 m. There is a slight degradation near surface, but this corresponds to much fewer GNSS-RO data compared to available data counts at other heights. Moreover, it is noticed that a majority of the improvements has already been obtained in experiment AdjStat, and only small additional impact is introduced into GEOS by AdjHybr. This suggests that the impact through AdjEnsVloc is more effective than the multivariate correlation aspects of hybrid BEC. Nevertheless, when relying on the ensemble to constrain variables, the multivariate aspects are expected to play an important role to use PBLH observations to constrain the model states. Although we have noticed very small PBLH_RF ensemble spread in Fig. 3c over many of the marine stratocumulus regions where high ensemble perturbation correlations are observed as shown in Fig. 9, the real cause of the smaller ${\text{PBLH}}_{\text{RF}}^o$ data impact through the multivariate aspects of hybrid BEC is unclear. Further investigation will be conducted in the future.

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As in Fig. 17, but for the normalized bending angle OmA RMSD as a function of impact height.

Citation: Monthly Weather Review 153, 3; 10.1175/MWR-D-24-0141.1

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In addition, the confidence intervals with the 95% significant level are calculated from a bootstrap resampling method with a resampling size of 2000 and are shown as shading in Figs. 17 and 18.

5. Conclusions and future work

An accurate PBLH representation is essential for many applications such as forecasting near-surface meteorology, air quality, water and energy cycles, aerosol and cloud chemistry, and climate. This paper focuses on developing a methodology of utilizing PBLH data to improve the PBL thermodynamic representation in the GEOS data assimilation system and building the framework of a global PBLH analysis and monitoring capability for multiple observing systems to help support the assessment of program of record and future PBL observations as recommended in the 2017 Decadal Survey Report.

We first derived PBLH from radiosonde and GNSS-RO refractivity data and discussed the necessity of using consistent model PBLH definitions for comparison with each observing system, since PBLH is sensitive to the observing method and retrieval algorithm. To this end, a new refractivity gradient–based ${\text{PBLH}}_{\text{RF}}^f$ was implemented in the GEOS model, in addition to the existing bulk Richardson number–based ${\text{PBLH}}_{R_i}^f$ definition. Although the patterns of ${\text{PBLH}}_{R_i}^f$ and ${\text{PBLH}}_{\text{RF}}^f$ are similar in convective boundary layer regimes, differences are observed. These developments have helped us to understand the PBLH data from various observing systems from physical and instrument-wise perspectives and allowed us to use a model PBLH definition appropriate for each data type in the assessment of PBLH data and calculation of PBLH OmFs.

The strategy and framework for assimilating PBLH data from multiple observing systems were then developed. Unlike the traditional data assimilation method to construct observation operators connecting control/state variables and PBLH variables, both ${\text{PBLH}}_{R_i}$ and PBLH_RF were added as new control variables in GEOS; thus, their analyses are generated together with those of other control variables in the data assimilation procedure. Quality control procedures were also carefully designed for radiosonde ${\text{PBLH}}_{R_i}^o$ data and GNSS-RO ${\text{PBLH}}_{\text{RF}}^o$ data. The framework developed in this study introduces a new capability of global analysis and monitoring of any program of record or future PBLH data and provides a platform to test and evaluate PBLH data from multiple observing systems.

With the PBLH data assimilation approach adopted in this study, the impacts of PBLH data on the PBL thermodynamic structure can be realized through interaction with other variables via correlation relationships reflected in ensemble BEC. More importantly, PBLH data can help to capture capping inversions, which may present the biggest challenge in PBL profile assimilation. This challenge stems from the uncertainties of BEC, particularly the accuracy of ensemble BEC and the vertical localization length scale. These elements are key to a good representation of the PBL thermodynamic structure. Therefore, we developed the AdjEnsVloc method in this study to improve ensemble BEC by inflating ensemble spread of T_υ and RH and adjusting the vertical localization length scale based on observed PBLH data. Preliminary results are promising, demonstrating an improved representation of capping inversions and beneficial impact on the use of other observations in the lower troposphere. The results also indicate that, in the lower troposphere, the impacts of PBLH data on T_υ and q through AdjEnsVloc are different from their impacts through multivariate correlation relationships in the original BEC and are more important. The combined PBLH data impact through AdjEnsVloc and multivariate aspects of the improved BEC provides the largest benefits to PBL data assimilation.

The present paper is a first step toward developing and testing a comprehensive PBLH data assimilation methodology. One specific follow-on step is to continue to explore the optimal adjustments of ensemble spread and the vertical localization length scale, either from climatology or a physical perspective. A follow-up study will focus on including PBLH data derived from space- and ground-based lidar observations (McGrath-Spangler and Denning 2012; Palm et al. 2021b) as well as from global radar wind profiler measurements (Molod et al. 2015). In addition, methods to retain observation information through using PBLH analysis to adjust model PBL parameterization schemes will be addressed in the follow-up studies.

Data availability statement.

The GEOS model and data assimilation system and supporting software are maintained by the Global Modeling and Assimilation Office and publicly available through the NASA Open Source Agreement (NOSA) on GitHub (https://github.com/GEOS-ESM). Changes to GEOS software resulting from this project and used in publications are also available in the GitHub PBLH branch. The data generated in this study include PBL height data and output from the GEOS model and data assimilation system. All the unprocessed output and data are archived and can be accessed on the NCCS Discover Centralized Storage System.