Validation of the CERES Clear-Sky Surface Longwave Downward Radiation Products Under Air Temperature Inversion

Highlights

What are the main findings?

• The accuracy of CERES SLDR products is degraded heavily under ATI conditions.
• The concurrent atmospheric moisture inversion (AMI) compounds this degradation.
• It is urgent to refine CERES SLDR algorithms in the future.

What are the implications of the main findings?

• Advance our understanding of the clear-sky SLDR estimate mechanism.
• Provide valuable guidance and benefit the CERES SLDR product users.

Abstract

This study assessed the performance of the Clouds and the Earth’s Radiant Energy System (CERES) surface longwave downward radiation (SLDR) products under the atmospheric temperature inversion (ATI) conditions for the first time. Three years of ground-measured SLDRs from 409 globally distributed stations across four flux networks were employed, and the collocated MODIS atmospheric profile product was used to identify the ATI profiles at each flux station. All three SLDR estimate algorithms (Models A, B, and C) show a pronounced accuracy decline under ATI conditions, regardless of region (polar or non-polar) or time of day (daytime or nighttime). Under ATI conditions, the Bias/RMSE increases by approximately 10.0/5.0 W/m² for Models A and B, 5.0/1.0 W/m² for Model C. Sensitivity analysis reveals that the concurrent atmospheric moisture inversion (AMI) compounds this degradation; both the Bias and RMSE increase with the AMI intensity. These results underscore the need to refine CERES SLDR algorithms in the future.

1. Introduction

Surface longwave (LW) downward radiation (SLDR) is a crucial component of the surface radiation balance [1,2,3]. SLDR reflects the radiation exchange between the atmosphere and the surface, typically regulated by factors such as atmospheric temperature, water vapor, aerosols, and cloud cover [4,5]. An accurate estimate of SLDR has become one of the core tasks in climate and environmental research. Remote sensing has been acknowledged as a unique and effective means to map SLDR globally at a finer scale. A couple of typical global SLDR products have been released over the past three decades; for example, coarse-resolution (~1°) products like the Global Energy and Water Cycle Experiment–Surface Radiation Budget (GEWEX-SRB) [6], the International Satellite Cloud Climatology Project–Flux Data (ISCCP-FD) [7], and the Clouds and the Earth’s Radiant Energy System–Gridded Radiative Fluxes and Clouds (CERES-FSW) [8,9,10]; and the fine-resolution (1 km) Essential thermaL Infrared remoTe sEnsing (ELITE) SLDR product [11]. These SLDR products have benefited scientists in the field of climate change.

Air temperature inversion (ATI) is a common weather phenomenon that usually occurs at the near-surface boundary layer, especially in mountainous regions and the Antarctic Plateau [12]. The occurrence of the ATI alters the atmospheric state at the near-surface boundary layer and affects the accuracy of the clear-sky SLDR estimate, but is ignored in the production of the typical satellite SLDR products mentioned above. Dupont et al. [13] proposed a new parametric model for predicting the clear-sky SLDR by considering the variations in vertical distribution of water vapor (the ratio of screen-level to column-integrated water vapor density) and variations in the temperature lapse rate (the ratio of screen-level temperature to the minimum temperature during the day) based on an analysis of the parameterization schemes developed by Brutsaert [14] and Prata [15]. Yu et al. [16] developed a parameterization scheme that estimates the clear-sky SLDR using MODIS brightness temperature and integrated water vapor. The scheme can improve the SLDR estimate accuracy under high water vapor content and is claimed to be insensitive to temperature inversion. Cheng et al. [17] revealed the mechanism by which the ATI affects the estimate accuracy of SLDR and developed a correction method that substantially improves the accuracy of six typical parameterization schemes when validated at SURFRAD sites.

Despite these advances, critical issues in the SLDR estimate under ATI remain unresolved and require further exploration. One major unresolved issue is the lack of comprehensive validation of global satellite-based SLDR products under ATI conditions, including CERES SLDR products. Thus, this study evaluates the performance of the current SLDR estimate algorithm under ATI conditions, using the widely adopted CERES SLDR product algorithms as an example. The paper is structured as follows: Section 2 describes the ground measurements, satellite datasets and evaluation method; Section 3 presents the validation results; Section 4 is the Discussion, and Section 5 summarizes the main conclusions.

2. Materials and Methods

2.1. CERES-SSF SLDR Product

CERES is a National Aeronautics and Space Administration (NASA) satellite project that studies the role of clouds, aerosols, and radiation in the Earth’s climate system [10]. Each file of the Single Scanner Footprint top-of-atmosphere (TOA)/Surface Fluxes and Clouds (SSF) product contains one hour of instantaneous CERES data for a single scanner footprint (~20 km) from Terra and Aqua spacecraft observations. It combines CERES data with scene information from MODIS data on cloud and aerosol properties to derive instantaneous, all-sky SLDR. The CERES-SSF Edition 4A product was employed in this study.

The product provides three SLDR datasets derived from LW Models A, B, and C, suitable for different climatic conditions and radiative scenarios. As reported by Kratz et al. [9], the CERES LW Models for SLDR show global biases below 10 W/m² and RMSEs below 26 W/m² for clear-sky conditions, with larger biases in deserts (10–20 W/m²) and polar regions (~30 W/m²). Under cloudy-sky conditions, biases remain small (<6 W/m²), but RMSEs are substantially higher (20–30 W/m²), particularly in polar regions (up to 26.7–28.6 W/m²).

2.1.1. Model A

This algorithm was developed by Inamdar and Ramanathan [18], which calculates SLDR as the sum of the window and non-window components.

$$\mathrm{S}\mathrm{L}\mathrm{D}\mathrm{R}={\mathrm{F}}_{\mathrm{d},\mathrm{w}\mathrm{i}\mathrm{n}}+{\mathrm{F}}_{\mathrm{d},\mathrm{n}\mathrm{w}}$$

(1)

where ${\mathrm{F}}_{\mathrm{d},\mathrm{w}\mathrm{i}\mathrm{n}}$ and ${\mathrm{F}}_{\mathrm{d},\mathrm{n}\mathrm{w}}$ represent the window and non-window $\mathrm{S}\mathrm{L}\mathrm{D}\mathrm{R}$, respectively. The physics of radiative transfer in the window and non-window spectral regions form the basis of this algorithm; the TOA and surface longwave radiations are simulated using a broadband radiative transfer model, which uses soundings from ship sondes as input.

The atmospheric greenhouse effect (${\mathrm{G}}_{\mathrm{a}}$) is the key radiometric quantity in the parameterization since there is a close affinity between ${\mathrm{G}}_{\mathrm{a}}$ and $\mathrm{S}\mathrm{L}\mathrm{D}\mathrm{R}$.

$${\mathrm{G}}_{\mathrm{a}}={\displaystyle \underset{{\mathrm{z}}_{\mathrm{t}\mathrm{o}\mathrm{p}}}{\overset{0}{\int }}\mathrm{A}({\mathrm{z}}_{\mathrm{t}\mathrm{o}\mathrm{p}},{\mathrm{z}}^{\prime })\frac{\mathrm{d}\mathrm{B}({\mathrm{z}}^{\prime })}{\mathrm{d}{\mathrm{z}}^{\prime }}}\mathrm{d}{\mathrm{z}}^{\prime }$$

(2)

where $\mathrm{B}({\mathrm{z}}^{\prime })$ is the Planck function, $\mathrm{A}(\mathrm{z},{\mathrm{z}}^{\prime })$ is the atmospheric absorptance between the levels specified in the arguments, and ${\mathrm{z}}_{\mathrm{t}\mathrm{o}\mathrm{p}}$ is the altitude at the top of the atmosphere. Ramanathan [19] then described the greenhouse effect in terms of the difference

$${\mathrm{G}}_{\mathrm{a}}={\mathrm{F}}_{\mathrm{L}\mathrm{W}}^{\uparrow }(\mathrm{S}\mathrm{F}\mathrm{C})-{\mathrm{F}}_{\mathrm{L}\mathrm{W}}^{\uparrow }(\mathrm{T}\mathrm{O}\mathrm{A})$$

(3)

where ${\mathrm{F}}_{\mathrm{L}\mathrm{W}}^{\uparrow }(\mathrm{S}\mathrm{F}\mathrm{C})$ is the surface longwave upward radiation (SLUR), and ${\mathrm{F}}_{\mathrm{L}\mathrm{W}}^{\uparrow }(\mathrm{T}\mathrm{O}\mathrm{A})$ is the window upward longwave TOA radiation.

The model A explained the SLDR in the window and non-window region in terms of their respective components of ${\mathrm{G}}_{\mathrm{a}}$ and other residual variables:

$${\mathrm{F}}_{\mathrm{d},\mathrm{w}\mathrm{i}\mathrm{n}}-{\mathrm{a}}_0{\mathrm{g}}_{\mathrm{a},\mathrm{w}\mathrm{i}\mathrm{n}}={\mathrm{c}}_0+{\mathrm{F}}_{\mathrm{t}\mathrm{o}\mathrm{a},\mathrm{w}\mathrm{i}\mathrm{n}}{\displaystyle \sum _{\mathrm{i}}{\mathrm{c}}_{\mathrm{i}}{\mathrm{P}}_{\mathrm{i}}}$$

(4)

$${\mathrm{F}}_{\mathrm{d},\mathrm{n}\mathrm{w}}-{\mathrm{b}}_0{\mathrm{g}}_{\mathrm{a},\mathrm{n}\mathrm{w}}={\mathrm{d}}_0+{\mathrm{F}}_{\mathrm{t}\mathrm{o}\mathrm{a},\mathrm{n}\mathrm{w}}{\displaystyle \sum _{\mathrm{i}}{\mathrm{d}}_{\mathrm{i}}{\mathrm{P}}_{\mathrm{i}}}$$

(5)

where ${\mathrm{F}}_{\mathrm{d},\mathrm{w}\mathrm{i}\mathrm{n}}$ and ${\mathrm{F}}_{\mathrm{d},\mathrm{n}\mathrm{w}}$ are the window and non-window SLDR, respectively. ${\mathrm{g}}_{\mathrm{a},\mathrm{w}\mathrm{i}\mathrm{n}}$ and ${\mathrm{g}}_{\mathrm{a},\mathrm{n}\mathrm{w}}$ represent the window and non-window components of the greenhouse effect. ${\mathrm{F}}_{\mathrm{t}\mathrm{o}\mathrm{a},\mathrm{w}\mathrm{i}\mathrm{n}}$ and ${\mathrm{F}}_{\mathrm{t}\mathrm{o}\mathrm{a},\mathrm{n}\mathrm{w}}$ are the window and non-window upward longwave TOA radiation, respectively. ${\mathrm{P}}_{\mathrm{i}}$ is a combination of parameters such as the window (or non-window) SLUR, the total column water vapor ($\mathrm{w}$), the air temperature in the vicinity of the surface (${\mathrm{\theta }}_{\mathrm{a}}$), and the surface air temperature (${\mathrm{\theta }}_0$). ${\mathrm{a}}_0$, ${\mathrm{b}}_0$, ${\mathrm{c}}_{\mathrm{i}}$, and ${\mathrm{d}}_{\mathrm{i}}$ are regression coefficients which vary with the geographic region.

2.1.2. Model B

Model B builds upon the Langley Parameterized Longwave Algorithm (LPLA), developed from an accurate narrowband radiative transfer model [5]. The clear-sky SLDR is represented in terms of the water vapor and temperature distribution close to the surface according to extensive sensitivity tests:

$${\mathrm{SLDR}}_{\mathrm{d},\mathrm{c}}=({\mathrm{A}}_0+{\mathrm{A}}_1\mathrm{V}+{\mathrm{A}}_2{\mathrm{V}}^2+{\mathrm{A}}_3{\mathrm{V}}^3){{\mathrm{T}}_{\mathrm{e}}}^{3.7}$$

(6)

where $\mathrm{V}=\ln (\mathrm{W})$, $\mathrm{W}$ (unit: g/cm²) is the water vapor burden under the atmosphere and derived from the humidity profile, ${\mathrm{T}}_{\mathrm{e}}$ is an effective emitting temperature of the atmosphere, and ${\mathrm{A}}_{\mathrm{i}}$ are regression coefficients. ${\mathrm{T}}_{\mathrm{e}}$ is computed as:

$${\mathrm{T}}_{\mathrm{e}}={\mathrm{k}}_{\mathrm{s}}{\mathrm{T}}_{\mathrm{s}}+{\mathrm{k}}_1{\mathrm{T}}_1+{\mathrm{k}}_2{\mathrm{T}}_2$$

(7)

where ${\mathrm{T}}_{\mathrm{s}}$ is the surface skin temperature, ${\mathrm{T}}_1$ and ${\mathrm{T}}_2$ are the mean temperatures of the first and second atmospheric levels closest to the surface, which correspond to the surface–800 mb, and 800–680 mb, and ${\mathrm{k}}_1$, ${\mathrm{k}}_2$, and ${\mathrm{k}}_3$ are weighting factors determined from sensitivity analysis, with values of 0.60, 0.35, and 0.05, respectively.

2.1.3. Model C

The CERES science team adopted the method of Zhou et al. [20] as LW Model C to compute the clear-sky and cloudy-sky flux separately. The clear-sky SLDR is parameterized in terms of the $\mathrm{S}\mathrm{L}\mathrm{U}\mathrm{R}$ and $\mathrm{P}\mathrm{W}\mathrm{V}$ as

$$\mathrm{S}\mathrm{L}\mathrm{D}{\mathrm{R}}_{\mathrm{d},\mathrm{c}\mathrm{l}\mathrm{r}}={\mathrm{a}}_0+{\mathrm{a}}_1\mathrm{S}\mathrm{L}\mathrm{U}\mathrm{R}+{\mathrm{a}}_2\ln (1+\mathrm{P}\mathrm{W}\mathrm{V})+{\mathrm{a}}_3{[\ln (1+\mathrm{P}\mathrm{W}\mathrm{V})]}^2$$

(8)

where $\mathrm{S}\mathrm{L}\mathrm{U}\mathrm{R}$ represents the surface upwelling longwave flux computed from 2 m air temperature using Stefan–Boltzmann’s law assuming an emissivity equal to unity (W/m²), and $\mathrm{P}\mathrm{W}\mathrm{V}$ is the column precipitable water vapor (g/cm²). The current version of this model uses $\ln (1+\mathrm{P}\mathrm{W}\mathrm{V})$ terms replacing the origin $\ln (\mathrm{P}\mathrm{W}\mathrm{V})$ to approximate a near-linear relationship between PWV and SLDR when water vapor amounts are very low.

2.2. Ground Measurements

Globally distributed flux networks provide continuous measurements of surface longwave radiation, enabling the validation of a wide range of surface longwave radiation products. In this study, we collected three years of ground measurements from 409 flux stations at four flux networks situated in diverse climatic regions and biomes to verify the accuracy of the CERES-SSF clear-sky SLDR under ATI conditions. They range in altitude from −5 to 3513 m and in latitude from 70.65°S to 82.49°N. These stations include 186 stations from AmeriFlux [21], 126 stations from FLUXNET [22], 27 stations from Programme for Monitoring of the Greenland Ice Sheet (PROMICE) [23], and 70 stations from the Baseline Surface Radiation Network (BSRN) [24]. Longwave radiation at these stations was measured by the Kipp & Zonen net radiometers (CNR1, CNR4, and CNR1-lite), for which the manufacturer reports a sensor uncertainty of $~\pm$10% [23,25]. The global distribution of these stations is shown in Figure 1, and

Table A1. Detailed information of each site.

Network	ShortName	Latitude	Longitude	Period
AmeriFlux	BR-Npw	−16.50	−56.41	2014–2016
AmeriFlux	BR-Sa3	−3.02	−54.97	2001–2003
AmeriFlux	CA-ARB	52.69	−83.95	2012–2014
AmeriFlux	CA-ARF	52.70	−83.96	2012–2014
AmeriFlux	CA-Ca3	49.53	−124.90	2009–2011
AmeriFlux	CA-Cbo	44.32	−79.93	2009–2011
AmeriFlux	CA-DBB	49.13	−122.98	2016–2018
AmeriFlux	CA-Gro	48.22	−82.16	2009–2011
AmeriFlux	CA-LP1	55.11	−122.84	2009–2011
AmeriFlux	CA-MA2	50.17	−97.88	2009–2011
AmeriFlux	CA-MA3	50.18	−97.87	2009–2011
AmeriFlux	CA-Na1	46.47	−67.10	2003–2005
AmeriFlux	CA-Oas	53.63	−106.20	2007–2009
AmeriFlux	CA-Obs	53.99	−105.12	2007–2009
AmeriFlux	CA-Ojp	53.92	−104.69	2007–2009
AmeriFlux	CA-Qcu	49.27	−74.04	2007–2009
AmeriFlux	CA-Qfo	49.69	−74.34	2007–2009
AmeriFlux	CA-SCB	61.31	−121.30	2014–2017
AmeriFlux	CA-SCC	61.31	−121.30	2013–2016
AmeriFlux	CA-SF1	54.49	−105.82	2004–2006
AmeriFlux	CA-SF2	54.25	−105.88	2002–2004
AmeriFlux	CA-SF3	54.09	−106.01	2002–2004
AmeriFlux	CA-SJ1	53.91	−104.66	2007–2009
AmeriFlux	CA-SJ2	53.95	−104.65	2007–2009
AmeriFlux	CA-SJ3	53.88	−104.65	2007–2009
AmeriFlux	CA-TP4	42.71	−80.36	2007–2009
AmeriFlux	CA-TPD	42.64	−80.56	2013–2015
AmeriFlux	US-A03	70.50	−149.88	2015–2017
AmeriFlux	US-A10	71.32	−156.61	2013–2015
AmeriFlux	US-A32	36.82	−97.82	2016–2018
AmeriFlux	US-A74	36.81	−97.55	2016–2017
AmeriFlux	US-ALQ	46.03	−89.61	2016–2018
AmeriFlux	US-An1	68.99	−150.28	2009–2011
AmeriFlux	US-An2	68.95	−150.21	2009–2011
AmeriFlux	US-An3	68.93	−150.27	2009–2011
AmeriFlux	US-AR1	36.43	−99.42	2009–2012
AmeriFlux	US-AR2	36.64	−99.60	2009–2012
AmeriFlux	US-Aud	31.59	−110.51	2007–2009
AmeriFlux	US-Bi1	38.10	−121.50	2017–2019
AmeriFlux	US-Bi2	38.11	−121.54	2017–2019
AmeriFlux	US-Bkg	44.35	−96.84	2007–2009
AmeriFlux	US-Blk	44.16	−103.65	2004–2006
AmeriFlux	US-BMM	45.78	−110.78	2016–2018
AmeriFlux	US-Bo1	40.01	−88.29	2007–2009
AmeriFlux	US-Bo2	40.01	−88.29	2005–2007
AmeriFlux	US-Br1	41.97	−93.69	2007–2009
AmeriFlux	US-Br3	41.97	−93.69	2007–2009
AmeriFlux	US-CaV	39.06	−79.42	2007–2009
AmeriFlux	US-ChR	35.93	−84.33	2007–2009
AmeriFlux	US-CMW	31.66	−110.18	2007–2009
AmeriFlux	US-CPk	41.07	−106.12	2010–2012
AmeriFlux	US-CRT	41.63	−83.35	2011–2013
AmeriFlux	US-Ctn	43.95	−101.85	2007–2009
AmeriFlux	US-Cwt	35.06	−83.43	2012–2014
AmeriFlux	US-DFC	43.34	−89.71	2018–2020
AmeriFlux	US-Dia	37.68	−121.53	2011–2013
AmeriFlux	US-Dk1	35.97	−79.09	2005–2007
AmeriFlux	US-Dk2	35.97	−79.10	2005–2007
AmeriFlux	US-Dk3	35.98	−79.09	2005–2007
AmeriFlux	US-EDN	37.62	−122.11	2018–2020
AmeriFlux	US-EML	63.88	−149.25	2012–2014
AmeriFlux	US-Fmf	35.14	−111.73	2007–2009
AmeriFlux	US-FPe	48.31	−105.10	2007–2009
AmeriFlux	US-FR2	29.95	−98.00	2007–2009
AmeriFlux	US-FR3	29.94	−97.99	2007–2009
AmeriFlux	US-Fuf	35.09	−111.76	2007–2009
AmeriFlux	US-Fwf	35.45	−111.77	2007–2009
AmeriFlux	US-GBT	41.37	−106.24	2007–2009
AmeriFlux	US-Goo	34.25	−89.87	2007–2009
AmeriFlux	US-HB1	33.35	−79.20	2019–2019
AmeriFlux	US-HB2	33.32	−79.24	2019–2019
AmeriFlux	US-HB3	33.35	−79.23	2019–2019
AmeriFlux	US-HBK	43.94	−71.72	2017–2019
AmeriFlux	US-Hn3	46.69	−119.46	2017–2018
AmeriFlux	US-Ho2	45.21	−68.75	2007–2009
AmeriFlux	US-Ho3	45.21	−68.73	2007–2009
AmeriFlux	US-HRA	34.59	−91.75	2015–2017
AmeriFlux	US-HRC	34.59	−91.75	2016–2017
AmeriFlux	US-Ivo	68.49	−155.75	2004–2006
AmeriFlux	US-KM4	42.44	−85.33	2011–2013
AmeriFlux	US-KS3	28.71	−80.74	2017–2020
AmeriFlux	US-KUT	44.99	−93.19	2006–2008
AmeriFlux	US-MC1	48.19	−114.15	2015–2015
AmeriFlux	US-Me2	44.45	−121.56	2007–2009
AmeriFlux	US-Me3	44.32	−121.61	2006–2008
AmeriFlux	US-Me6	44.32	−121.61	2011–2013
AmeriFlux	US-MH1	45.92	−108.24	2015–2015
AmeriFlux	US-Mj1	46.99	−109.61	2013–2014
AmeriFlux	US-Mj2	47.00	−109.63	2014–2014
AmeriFlux	US-Mpj	34.44	−106.24	2009–2011
AmeriFlux	US-MRf	44.65	−123.55	2007–2009
AmeriFlux	US-MSR	47.48	−111.72	2016–2016
AmeriFlux	US-MtB	32.42	−110.73	2010–2012
AmeriFlux	US-NGB	71.28	−156.61	2013–2015
AmeriFlux	US-NGC	64.86	−163.70	2017–2019
AmeriFlux	US-Oho	41.55	−83.84	2007–2009
AmeriFlux	US-ONA	27.38	−81.95	2017–2019
AmeriFlux	US-OWC	41.38	−82.51	2015–2016
AmeriFlux	US-Prr	65.12	−147.49	2012–2014
AmeriFlux	US-Rls	43.14	−116.74	2015–2017
AmeriFlux	US-Rms	43.06	−116.75	2015–2017
AmeriFlux	US-Ro1	44.71	−93.09	2007–2009
AmeriFlux	US-Ro2	44.73	−93.09	2007–2009
AmeriFlux	US-Ro4	44.68	−93.07	2016–2018
AmeriFlux	US-Ro5	44.69	−93.06	2017–2019
AmeriFlux	US-Ro6	44.69	−93.06	2017–2019
AmeriFlux	US-Rwe	43.07	−116.76	2004–2006
AmeriFlux	US-Rwf	43.12	−116.72	2015–2017
AmeriFlux	US-Rws	43.17	−116.71	2015–2017
AmeriFlux	US-Seg	34.36	−106.70	2007–2009
AmeriFlux	US-Ses	34.33	−106.74	2007–2009
AmeriFlux	US-SFP	43.24	−96.90	2007–2009
AmeriFlux	US-Skr	25.36	−81.08	2007–2009
AmeriFlux	US-Slt	39.91	−74.60	2007–2009
AmeriFlux	US-Sne	38.04	−121.75	2016–2018
AmeriFlux	US-Snf	38.04	−121.73	2018–2020
AmeriFlux	US-SRC	31.91	−110.84	2010–2012
AmeriFlux	US-SRG	31.79	−110.83	2010–2012
AmeriFlux	US-SRM	31.82	−110.87	2007–2009
AmeriFlux	US-Srr	38.20	−122.03	2016–2017
AmeriFlux	US-Tw1	38.11	−121.65	2012–2014
AmeriFlux	US-Tw2	38.10	−121.64	2012–2013
AmeriFlux	US-Tw3	38.12	−121.65	2014–2016
AmeriFlux	US-Tw4	38.10	−121.64	2014–2016
AmeriFlux	US-Tw5	38.11	−121.64	2018–2020
AmeriFlux	US-Uaf	64.87	−147.86	2010–2012
AmeriFlux	US-UiA	40.06	−88.20	2009–2011
AmeriFlux	US-UM3	45.57	−84.67	2013–2014
AmeriFlux	US-UMB	45.56	−84.71	2007–2009
AmeriFlux	US-UMd	45.56	−84.70	2007–2009
AmeriFlux	US-Var	38.41	−120.95	2007–2009
AmeriFlux	US-Vcm	35.89	−106.53	2007–2009
AmeriFlux	US-Vcp	35.86	−106.60	2007–2009
AmeriFlux	US-Vcs	35.92	−106.61	2016–2018
AmeriFlux	US-WBW	35.96	−84.29	2003–2005
AmeriFlux	US-WCr	45.81	−90.08	2007–2009
AmeriFlux	US-Wdn	40.78	−106.26	2006–2008
AmeriFlux	US-Wgr	45.11	−122.66	2015–2016
AmeriFlux	US-Whs	31.74	−110.05	2010–2012
AmeriFlux	US-Wjs	34.43	−105.86	2008–2010
AmeriFlux	US-Wkg	31.74	−109.94	2007–2009
AmeriFlux	US-Wpp	44.14	−123.18	2014–2016
AmeriFlux	US-WPT	41.46	−83.00	2011–2013
AmeriFlux	US-Wrc	45.82	−121.95	2007–2009
AmeriFlux	US-xAB	45.76	−122.33	2018–2020
AmeriFlux	US-xAE	35.41	−99.06	2018–2020
AmeriFlux	US-xBA	71.28	−156.62	2018–2020
AmeriFlux	US-xBL	39.06	−78.07	2018–2020
AmeriFlux	US-xBN	65.15	−147.50	2018–2020
AmeriFlux	US-xBR	44.06	−71.29	2018–2020
AmeriFlux	US-xCL	33.40	−97.57	2018–2020
AmeriFlux	US-xCP	40.82	−104.75	2018–2020
AmeriFlux	US-xDC	47.16	−99.11	2018–2020
AmeriFlux	US-xDJ	63.88	−145.75	2018–2020
AmeriFlux	US-xDL	32.54	−87.80	2018–2020
AmeriFlux	US-xDS	28.13	−81.44	2018–2020
AmeriFlux	US-xGR	35.69	−83.50	2018–2020
AmeriFlux	US-xHA	42.54	−72.17	2018–2020
AmeriFlux	US-xHE	63.88	−149.21	2018–2020
AmeriFlux	US-xJE	31.19	−84.47	2018–2020
AmeriFlux	US-xJR	32.59	−106.84	2018–2020
AmeriFlux	US-xKA	39.11	−96.61	2018–2020
AmeriFlux	US-xKZ	39.10	−96.56	2018–2020
AmeriFlux	US-xMB	38.25	−109.39	2018–2020
AmeriFlux	US-xML	37.38	−80.52	2018–2020
AmeriFlux	US-xNG	46.77	−100.92	2018–2020
AmeriFlux	US-xNQ	40.18	−112.45	2018–2020
AmeriFlux	US-xNW	40.05	−105.58	2018–2020
AmeriFlux	US-xRM	40.28	−105.55	2018–2020
AmeriFlux	US-xSB	29.69	−81.99	2018–2020
AmeriFlux	US-xSC	38.89	−78.14	2018–2020
AmeriFlux	US-xSE	38.89	−76.56	2018–2020
AmeriFlux	US-xSJ	37.11	−119.73	2018–2020
AmeriFlux	US-xSL	40.46	−103.03	2018–2020
AmeriFlux	US-xSP	37.03	−119.26	2018–2020
AmeriFlux	US-xSR	31.91	−110.84	2018–2020
AmeriFlux	US-xST	45.51	−89.59	2018–2020
AmeriFlux	US-xTA	32.95	−87.39	2018–2020
AmeriFlux	US-xTE	37.01	−119.01	2018–2020
AmeriFlux	US-xTL	68.66	−149.37	2018–2020
AmeriFlux	US-xTR	45.49	−89.59	2018–2020
AmeriFlux	US-xUK	39.04	−95.19	2018–2020
AmeriFlux	US-xUN	46.23	−89.54	2018–2020
AmeriFlux	US-xWD	47.13	−99.24	2018–2020
AmeriFlux	US-xWR	45.82	−121.95	2018–2020
AmeriFlux	US-xYE	44.95	−110.54	2018–2020
BSRN	ABS	44.02	144.28	2021–2023
BSRN	ALE	82.49	−62.42	2008–2010
BSRN	ASP	−23.80	133.89	2009–2011
BSRN	BAR	71.32	−156.61	2009–2011
BSRN	BER	32.30	−64.77	2007–2009
BSRN	BIL	36.61	−97.52	2009–2011
BSRN	BON	40.07	−88.37	2009–2011
BSRN	BOS	40.13	−105.24	2009–2011
BSRN	BOU	40.05	−105.01	2009–2011
BSRN	BRB	−15.60	−47.71	2009–2011
BSRN	BUD	47.43	19.18	2020–2022
BSRN	CAB	51.97	4.93	2009–2011
BSRN	CAM	50.22	−5.32	2009–2011
BSRN	CAP	79.27	101.75	2016–2016
BSRN	CAR	44.08	5.06	2009–2011
BSRN	CLH	36.91	−75.71	2007–2009
BSRN	CNR	42.82	−1.60	2009–2011
BSRN	COC	−12.19	96.84	2009–2011
BSRN	DAA	−30.67	23.99	2016–2018
BSRN	DAR	−12.43	130.89	2009–2011
BSRN	DRA	36.63	−116.02	2009–2011
BSRN	DWN	−12.42	130.89	2009–2011
BSRN	E13	36.61	−97.49	2009–2011
BSRN	ENA	39.09	−28.03	2015–2016
BSRN	EUR	79.99	−85.94	2009–2011
BSRN	FLO	−27.60	−48.52	2014–2016
BSRN	FPE	48.32	−105.10	2009–2011
BSRN	FUA	33.58	130.38	2011–2013
BSRN	GAN	23.11	72.63	2015–2015
BSRN	GCR	34.25	−89.87	2009–2011
BSRN	GIM	46.72	−87.41	2019–2021
BSRN	GOB	−23.56	15.04	2016–2018
BSRN	GUR	28.42	77.16	2018–2018
BSRN	GVN	−70.65	−8.25	2009–2011
BSRN	HOW	22.55	88.31	2014–2018
BSRN	ILO	8.53	4.57	2002–2004
BSRN	INO	44.34	26.01	2021–2023
BSRN	ISH	24.34	124.16	2011–2013
BSRN	KWA	8.72	167.73	2009–2011
BSRN	LAU	−45.05	169.69	2009–2011
BSRN	LER	60.14	−1.18	2009–2012
BSRN	LIN	52.21	14.12	2009–2011
BSRN	LRC	37.10	−76.39	2015–2017
BSRN	LYU	22.04	121.56	2019–2021
BSRN	MAN	−2.06	147.43	2009–2011
BSRN	MNM	24.29	153.98	2011–2013
BSRN	NAU	−0.52	166.92	2009–2011
BSRN	NEW	−32.88	151.73	2017–2019
BSRN	NYA	78.92	11.93	2009–2011
BSRN	OHY	−12.05	−75.32	2017–2019
BSRN	PAL	48.71	2.21	2009–2011
BSRN	PAR	5.81	−55.21	2019–2020
BSRN	PAY	46.81	6.94	2009–2011
BSRN	PSU	40.72	−77.93	2009–2011
BSRN	PTR	−9.07	−40.32	2009–2011
BSRN	REG	50.21	−104.71	2009–2011
BSRN	RUN	−20.90	55.48	2020–2022
BSRN	SAP	43.06	141.33	2011–2013
BSRN	SBO	30.86	34.78	2009–2011
BSRN	SEL	15.78	−91.99	2020–2023
BSRN	SMS	−29.44	−53.82	2009–2011
BSRN	SON	47.05	12.96	2013–2015
BSRN	SOV	24.91	46.41	2000–2002
BSRN	SXF	43.73	−96.62	2009–2011
BSRN	SYO	−69.01	39.58	2009–2011
BSRN	TAM	22.79	5.53	2009–2011
BSRN	TAT	36.06	140.13	2009–2011
BSRN	TIR	13.09	79.97	2015–2019
BSRN	TOR	58.26	26.46	2009–2011
BSRN	XIA	39.75	116.96	2009–2011
FLUXNET	AT-Neu	47.12	11.32	2009–2011
FLUXNET	AU-Ade	−13.08	131.12	2009–2011
FLUXNET	AU-ASM	−22.28	133.25	2007–2009
FLUXNET	AU-Cpr	−34.00	140.59	2011–2013
FLUXNET	AU-Cum	−33.62	150.72	2012–2014
FLUXNET	AU-DaP	−14.06	131.32	2009–2011
FLUXNET	AU-DaS	−14.16	131.39	2009–2011
FLUXNET	AU-Dry	−15.26	132.37	2009–2011
FLUXNET	AU-Emr	−23.86	148.47	2011–2013
FLUXNET	AU-Fog	−12.55	131.31	2006–2008
FLUXNET	AU-Gin	−31.38	115.71	2011–2013
FLUXNET	AU-GWW	−30.19	120.65	2013–2014
FLUXNET	AU-How	−12.49	131.15	2009–2011
FLUXNET	AU-Lox	−34.47	140.66	2008–2009
FLUXNET	AU-RDF	−14.56	132.48	2011–2013
FLUXNET	AU-Rig	−36.65	145.58	2011–2013
FLUXNET	AU-Rob	−17.12	145.63	2011–2013
FLUXNET	AU-Stp	−17.15	133.35	2011–2013
FLUXNET	AU-TTE	−22.29	133.64	2012–2014
FLUXNET	AU-Wac	−37.43	145.19	2006–2008
FLUXNET	AU-Whr	−36.67	145.03	2011–2013
FLUXNET	AU-Wom	−37.42	144.09	2011–2013
FLUXNET	AU-Ync	−34.99	146.29	2012–2014
FLUXNET	BE-Bra	51.31	4.52	2011–2013
FLUXNET	BE-Lon	50.55	4.75	2011–2013
FLUXNET	BR-Sa3	−3.02	−54.97	2011–2013
FLUXNET	CA-Gro	48.22	−82.16	2011–2013
FLUXNET	CA-Oas	53.63	−106.20	2008–2010
FLUXNET	CA-Obs	53.99	−105.12	2008–2010
FLUXNET	CA-Qfo	49.69	−74.34	2008–2010
FLUXNET	CA-SF1	54.49	−105.82	2003–2005
FLUXNET	CA-SF2	54.25	−105.88	2003–2005
FLUXNET	CA-SF3	54.09	−106.01	2003–2005
FLUXNET	CA-TP4	42.71	−80.36	2009–2011
FLUXNET	CA-TPD	42.64	−80.56	2012–2014
FLUXNET	CH-Cha	47.21	8.41	2012–2014
FLUXNET	CH-Dav	46.82	9.86	2012–2014
FLUXNET	CH-Fru	47.12	8.54	2012–2014
FLUXNET	CH-Oe1	47.29	7.73	2006–2008
FLUXNET	CN-Cng	44.59	123.51	2006–2008
FLUXNET	CN-Sw2	41.79	111.90	2010–2012
FLUXNET	CZ-BK1	49.50	18.54	2010–2012
FLUXNET	CZ-BK2	49.49	18.54	2010–2012
FLUXNET	CZ-wet	49.02	14.77	2010–2012
FLUXNET	DE-Akm	53.87	13.68	2010–2012
FLUXNET	DE-Geb	51.10	10.91	2010–2012
FLUXNET	DE-Gri	50.95	13.51	2010–2012
FLUXNET	DE-Hai	51.08	10.45	2010–2012
FLUXNET	DE-Kli	50.89	13.52	2010–2012
FLUXNET	DE-Lkb	49.10	13.30	2010–2012
FLUXNET	DE-Lnf	51.33	10.37	2010–2012
FLUXNET	DE-Obe	50.79	13.72	2010–2012
FLUXNET	DE-RuR	50.62	6.30	2011–2013
FLUXNET	DE-RuS	50.87	6.45	2011–2013
FLUXNET	DE-SfN	47.81	11.33	2012–2014
FLUXNET	DE-Spw	51.89	14.03	2010–2012
FLUXNET	DE-Tha	50.96	13.57	2010–2012
FLUXNET	DE-Zrk	53.88	12.89	2013–2014
FLUXNET	DK-Sor	55.49	11.64	2012–2014
FLUXNET	FI-Hyy	61.85	24.29	2012–2014
FLUXNET	FI-Lom	68.00	24.21	2007–2009
FLUXNET	FR-Gri	48.84	1.95	2012–2014
FLUXNET	FR-LBr	44.72	−0.77	2006–2008
FLUXNET	FR-Pue	43.74	3.60	2012–2014
FLUXNET	GF-Guy	5.28	−52.92	2012–2014
FLUXNET	GH-Ank	5.27	−2.69	2012–2014
FLUXNET	IT-BCi	40.52	14.96	2012–2014
FLUXNET	IT-CA1	42.38	12.03	2012–2014
FLUXNET	IT-CA2	42.38	12.03	2012–2014
FLUXNET	IT-CA3	42.38	12.02	2012–2014
FLUXNET	IT-Col	41.85	13.59	2012–2014
FLUXNET	IT-Isp	45.81	8.63	2013–2014
FLUXNET	IT-La2	45.95	11.29	2000–2002
FLUXNET	IT-Lav	45.96	11.28	2012–2014
FLUXNET	IT-MBo	46.01	11.05	2010–2012
FLUXNET	IT-Noe	40.61	8.15	2012–2014
FLUXNET	IT-Ren	46.59	11.43	2011–2013
FLUXNET	IT-Ro1	42.41	11.93	2006–2008
FLUXNET	IT-Ro2	42.39	11.92	2010–2012
FLUXNET	IT-SR2	43.73	10.29	2013–2014
FLUXNET	IT-SRo	43.73	10.28	2010–2012
FLUXNET	IT-Tor	45.84	7.58	2012–2014
FLUXNET	JP-MBF	44.39	142.32	2003–2005
FLUXNET	JP-SMF	35.26	137.08	2003–2005
FLUXNET	MY-PSO	2.97	102.31	2003–2005
FLUXNET	NL-Hor	52.24	5.07	2009–2011
FLUXNET	NL-Loo	52.17	5.74	2012–2014
FLUXNET	RU-Che	68.61	161.34	2003–2005
FLUXNET	RU-Cok	70.83	147.49	2012–2014
FLUXNET	RU-Fyo	56.46	32.92	2012–2014
FLUXNET	RU-Sam	72.37	126.50	2012–2014
FLUXNET	RU-SkP	62.26	129.17	2012–2014
FLUXNET	SE-St1	68.35	19.05	2012–2014
FLUXNET	SJ-Adv	78.19	15.92	2012–2014
FLUXNET	US-AR1	36.43	−99.42	2010–2012
FLUXNET	US-AR2	36.64	−99.60	2010–2012
FLUXNET	US-ARM	36.61	−97.49	2010–2012
FLUXNET	US-CRT	41.63	−83.35	2011–2013
FLUXNET	US-GBT	41.37	−106.24	2004–2006
FLUXNET	US-GLE	41.37	−106.24	2012–2014
FLUXNET	US-Goo	34.25	−89.87	2004–2006
FLUXNET	US-Ivo	68.49	−155.75	2004–2006
FLUXNET	US-Los	46.08	−89.98	2012–2014
FLUXNET	US-Me2	44.45	−121.56	2012–2014
FLUXNET	US-Me3	44.32	−121.61	2004–2006
FLUXNET	US-Me6	44.32	−121.61	2012–2014
FLUXNET	US-NR1	40.03	−105.55	2012–2014
FLUXNET	US-Oho	41.55	−83.84	2011–2013
FLUXNET	US-ORv	40.02	−83.02	2011–2011
FLUXNET	US-Prr	65.12	−147.49	2012–2014
FLUXNET	US-SRC	31.91	−110.84	2012–2014
FLUXNET	US-SRG	31.79	−110.83	2012–2014
FLUXNET	US-SRM	31.82	−110.87	2012–214
FLUXNET	US-Syv	46.24	−89.35	2012–2014
FLUXNET	US-Tw1	38.11	−121.65	2012–2014
FLUXNET	US-Tw2	38.10	−121.64	2012–2013
FLUXNET	US-Tw3	38.12	−121.65	2013–2014
FLUXNET	US-Tw4	38.10	−121.64	2013–2014
FLUXNET	US-UMd	45.56	−84.70	2012–2014
FLUXNET	US-Var	38.41	−120.95	2012–2014
FLUXNET	US-WCr	45.81	−90.08	2012–2014
FLUXNET	US-Whs	31.74	−110.05	2012–2014
FLUXNET	US-Wkg	31.74	−109.94	2012–2014
FLUXNET	US-WPT	41.46	−83.00	2011–2013
FLUXNET	ZA-Kru	−25.02	31.50	2011–2013
FLUXNET	ZM-Mon	−15.44	23.25	2004–2006
PROMICE	CEN	77.13	−61.03	2017–2020
PROMICE	EGP	75.62	−35.97	2017–2020
PROMICE	KAN_B	67.13	−50.18	2017–2020
PROMICE	KAN_L	67.10	−49.95	2017–2020
PROMICE	KAN_M	67.07	−48.84	2017–2020
PROMICE	KAN_U	67.00	−47.03	2017–2020
PROMICE	KPC_L	79.91	−24.08	2017–2020
PROMICE	KPC_U	79.83	−25.17	2017–2020
PROMICE	MIT	65.69	−37.83	2017–2020
PROMICE	NUK_K	64.16	−51.36	2017–2020
PROMICE	NUK_L	64.48	−49.54	2017–2020
PROMICE	NUK_N	64.95	−49.89	2012–2014
PROMICE	NUK_U	64.51	−49.27	2017–2020
PROMICE	QAS_A	61.24	−46.73	2012–2014
PROMICE	QAS_L	61.03	−46.85	2017–2020
PROMICE	QAS_M	61.10	−46.83	2017–2020
PROMICE	QAS_U	61.18	−46.82	2017–2020
PROMICE	SCO_L	72.22	−26.82	2017–2020
PROMICE	SCO_U	72.39	−27.23	2017–2020
PROMICE	TAS_A	65.78	−38.90	2017–2020
PROMICE	TAS_L	65.64	−38.90	2017–2020
PROMICE	TAS_U	65.70	−38.87	2012–2014
PROMICE	THU_L	76.40	−68.27	2017–2020
PROMICE	THU_U	76.42	−68.15	2017–2020
PROMICE	THU_U2	76.39	−68.11	2017–2020
PROMICE	UPE_L	72.89	−54.30	2017–2020
PROMICE	UPE_U	72.89	−53.58	2017–2020

provides detailed information on site short names, latitude, longitude, and the time periods used.

2.3. MODIS Atmospheric Profile Product

The Moderate Resolution Imaging Spectroradiometer (MODIS) Level-2 Atmospheric Profile product (MxD07_L2) [27] is produced by the MODIS Atmosphere Science Team. The MxD07 utilizes 15 infrared channels to precisely retrieve high spatial resolution atmospheric temperature and moisture profiles, supporting detailed atmospheric analysis [27,28]. The dataset includes land surface temperature, pressure, elevation, cloud mask, and several other parameters such as dew point temperature, atmospheric temperature, and geopotential height for 20 fixed pressure levels: 1000, 950, 920, 850, 780, 700, 620, 500, 400, 300, 250, 200, 150, 100, 70, 50, 30, 20, 10, and 5 hPa. These parameters are generated continuously, day and night, at a 5 × 5 grid of 1 km pixels, provided that at least nine fields of view are cloud-free. This study uses the MODIS atmospheric temperature profile to identify ATI.

Since the MxD07 profiles are provided at 20 standard pressure levels, we implemented a truncation method to ensure physical consistency with the actual topography. For each profile, we utilized the surface pressure provided by MxD07 to truncate the profile. Any standard pressure levels where the pressure exceeded the surface pressure were removed. This is critical for ensuring that only valid atmospheric levels above the ground are used. When the surface pressure exceeded 1000 hPa, we extrapolated the profiles to the surface pressure level using the data from the two lowest available levels [17].

2.4. Methods

2.4.1. ATI Profile Identification

ATI occurs when atmospheric temperature increases with altitude from the surface, and there are several methods to identify it [29]. Kahl [30] and Serreze et al. [31] used the first-order temperature derivatives of sounding data to detect low-level inversions. Starting at the surface, for any two contiguous measurement levels, ${\mathrm{z}}_2$ (unit: m) and ${\mathrm{z}}_1$ (where ${\mathrm{z}}_2>{\mathrm{z}}_1$), with corresponding temperatures ${\mathrm{T}}_2$ and ${\mathrm{T}}_1$, the first-order temperature derivative is defined as follows:

$$\mathrm{\theta }(\mathrm{z})={\displaystyle \frac{\mathrm{d}\mathrm{T}}{\mathrm{d}\mathrm{z}}}={\displaystyle \frac{{\mathrm{T}}_2-{\mathrm{T}}_1}{{\mathrm{z}}_2-{\mathrm{z}}_1}}$$

(9)

The profile is designated as an ATI profile when any derivative is positive. The intensity of the ATI profile (K/100 m) is characterized as

$$\mathrm{A}\mathrm{T}\mathrm{I}={\displaystyle \frac{\mathrm{\Delta }\mathrm{T}}{\mathrm{\Delta }\mathrm{H}}}={\displaystyle \frac{{\mathrm{T}}_2-{\mathrm{T}}_1}{{\mathrm{H}}_2-{\mathrm{H}}_1}}\times 100$$

(10)

where ${\mathrm{T}}_2$ and ${\mathrm{T}}_1$ are the air temperature (K) at the bottom and the top of the inversion layer, and the height (m) at the corresponding levels are ${\mathrm{H}}_2$ and ${\mathrm{H}}_1$, respectively. Fochesatto [32] advanced this work by developing a numerical method for determining multi-layered temperature inversions that could deduce neutral thermal states, isothermal conditions, and surface-based inversions in a temperature profile. Combining the methodology proposed by Kahl [30] and Fochesatto [32], Zhang et al. [33] introduced a more comprehensive detection framework for ATIs from the surface up to 3 km. Their methodology embedded non-inversion layers less than 100 m in thickness within the overall inversion structure, while requiring the inversion depth to exceed 20 m and strength stronger than 0.5 K. The high-resolution vertical temperature profiles from radiosondes were widely employed to detect inversions. Despite the merit of sounding data in capturing atmospheric vertical structure, the radiosonde is sparse and unevenly distributed. In addition, this approach is labor-intensive, costly, and insufficient for monitoring large areas of the atmosphere. The advent of remote sensing technology has revolutionized the identification of ATIs on regional and global scales. Satellite-based methods now offer a highly effective method to obtain continuous atmospheric profiles with superior spatiotemporal resolution.

In this study, a method was developed to detect ATIs from the MODIS atmospheric profile product. The temperature lapse rate between the three lowest effective atmospheric temperature levels from the surface is first calculated. The lapse rates between the first three levels are given by

$${\mathrm{\gamma }}_1={\displaystyle \frac{{\mathrm{T}}_2-{\mathrm{T}}_1}{{\mathrm{z}}_2-{\mathrm{z}}_1}} \mathrm{a}\mathrm{n}\mathrm{d} {\mathrm{\gamma }}_2={\displaystyle \frac{{\mathrm{T}}_3-{\mathrm{T}}_2}{{\mathrm{z}}_3-{\mathrm{z}}_2}}$$

(11)

where ${\mathrm{T}}_{\mathrm{i}}$ and ${\mathrm{z}}_{\mathrm{i}}$ denote the temperature and altitude of the $\mathrm{i}-\mathrm{t}\mathrm{h}$ level, respectively. If the two continuous lapse rates are greater than or equal to zero $({\mathrm{\gamma }}_1\ge 0 \mathrm{a}\mathrm{n}\mathrm{d} {\mathrm{\gamma }}_2\ge 0)$, an ATI profile is identified.

The inversion top is defined as the level showing a lapse rate less than zero. We obtained 363,396 normal temperature profiles and 90,914 ATI profiles collocated with the geographical locations of the 409 flux stations. Figure 2 shows an example of an ATI profile. The air temperature increases with altitude near the surface boundary.

2.4.2. Data Processing

Only the pixels labeled as “confident clear” in the MxD07 cloud mask were retained to identify the clear-sky near-surface ATI. The ground-measured SLDR was compared to the collocated CERES SLDR without taking any temporal and spatial averaging.

Several steps are required to spatiotemporally match the ground measurements with CERES-SSF SLDR data. For temporal matching, the time difference between the station observations and the overpass time of satellite data should not exceed 15 min. Initially, we downloaded the MxD07 data according to the location of flux stations. Secondly, using the timestamps from the MODIS–ground temporal matching as the benchmark, we selected the clear-sky CERES SSF SLDR according to the MxD07 clear-sky pixels. The MODIS atmospheric profile, the SSF clear-sky SLDR, and the ground-measured SLDR at the same time are obtained. Spatial matching was performed by calculating the great-circle distance between the station geolocation and the satellite pixel. For the SSF product, which consists of multiple footprints, we selected all footprints with centers located within a 20 km radius of a ground station, whereas for the MxD07, which is stored in a latitude and longitude grid, each station is exclusively matched up with MODIS pixels within 5 km of the spherical distance.

$$\mathrm{d}=2\mathrm{R}·\arcsin \left[\sqrt{{\sin }^2({\displaystyle \frac{{\mathrm{\varphi }}_2-{\mathrm{\varphi }}_1}{2}})+\cos {\mathrm{\varphi }}_1\cos {\mathrm{\varphi }}_2{\sin }^2({\displaystyle \frac{{\mathrm{\phi }}_2-{\mathrm{\phi }}_1}{2}})}\right]$$

(12)

where $\mathrm{d}$ is the great circle distance between two points, $\mathrm{R}$ is the Earth’s radius (approximately 6378.14 km), and ${\mathrm{\varphi }}_{\mathrm{i}}$ and ${\mathrm{\phi }}_{\mathrm{i}}$ are the longitude and latitude of the point (in radians).

To minimize the effects of cloud contamination and ground measurement uncertainties [4,34], and to obtain robust statistics for SLDR validation, we removed outliers from the SLDR matched values using the “3σ Hampel identifier” method [35]. Specifically, we calculated the difference between the clear-sky SSF SLDR and the ground-measured SLDR. If the sample deviates more than three standard deviations from the median, it is considered an outlier and removed from the dataset. The standard deviation is calculated as follows:

$$\mathrm{S}=1.4826\times \mathrm{m}\mathrm{e}\mathrm{d}\mathrm{i}\mathrm{a}\mathrm{n}\left\{\left|{\mathrm{x}}_{\mathrm{i}}-{\mathrm{x}}_{\mathrm{m}}\right|\right\}$$

(13)

where ${\mathrm{x}}_{\mathrm{i}}$ represents the difference between the paired clear-sky CERES-SSF SLDR value and the ground-measured value, and ${\mathrm{x}}_{\mathrm{m}}$ signifies the median of $\left\{{\mathrm{x}}_{\mathrm{i}}\right\}$. The coefficient 1.4826 converts the median absolute deviation into an unbiased standard deviation estimate for normally distributed data. The term $\mathrm{S}$ refers to the standard deviation. Additionally, SLDR samples with ${\mathrm{x}}_{\mathrm{i}}$ less than ${\mathrm{x}}_{\mathrm{m}}-3\mathrm{S}$ or greater than ${\mathrm{x}}_{\mathrm{m}}+3\mathrm{S}$ are considered outliers and excluded. Note that the quality control and outlier removal process was performed independently for each CERES model (A, B, and C) on a site-by-site basis, resulting in varying final sample sizes. Consequently, we obtained 4264, 34,771, and 35,003 matchups for Models A, B, and C under ATI conditions.

2.4.3. Evaluation Metrics

In this paper, the mean difference between the predicted and observed SLDRs (Bias), the Root Mean Square Error (RMSE), and the determination coefficient (R²) were employed as evaluation metrics for assessing the performance of SLDR estimates. The formulas are as follows:

$$\mathrm{B}\mathrm{i}\mathrm{a}\mathrm{s}={\displaystyle \frac{1}{\mathrm{N}}}{\displaystyle \sum _{\mathrm{i}=1}^{\mathrm{N}}({\mathrm{y}}_{\mathrm{s}}-{\mathrm{y}}_{\mathrm{o}})}$$

(14)

$$\mathrm{R}\mathrm{M}\mathrm{S}\mathrm{E}={\left({\displaystyle \frac{1}{\mathrm{N}}}{{\displaystyle \sum _{\mathrm{i}=1}^{\mathrm{N}}({\mathrm{y}}_{\mathrm{s}}-{\mathrm{y}}_{\mathrm{o}})}}^2\right)}^{1/2}$$

(15)

$${\mathrm{R}}^2=1-{{\displaystyle \sum _{\mathrm{i}=1}^{\mathrm{N}}({\mathrm{y}}_{\mathrm{s}}-{\mathrm{y}}_{\mathrm{o}})}}^2/{\displaystyle \sum _{\mathrm{i}=1}^{\mathrm{N}}{({\mathrm{y}}_{\mathrm{o}}-\overline{{\mathrm{y}}_{\mathrm{o}}})}^2}$$

(16)

where $\mathrm{N}$ is the number of observations, ${\mathrm{y}}_{\mathrm{s}}$ represents the CERES-SSF clear-sky SLDR values, and ${\mathrm{y}}_0$ is the ground-measured SLDR values.

3. Results

3.1. Overall Performance

Figure 3 shows the overall performance of the three models. Under ATI conditions, the SLDR distribution range is relatively narrow, clustering primarily between 100 and 250 W/m². This prevalence of lower values is expected, as ATI events occur predominantly in polar regions and during the nighttime. The CERES clear-sky SLDR products are significantly underestimated, resulting in larger negative Biases of −14.63 W/m², −20.81 W/m², and −9.67 W/m² for Models A, B, and C, respectively; the corresponding RMSE values are 22.44 W/m², 29.15 W/m², and 22.79 W/m². On the contrary, the SLDR samples under normal atmospheric temperature conditions span a broader range (100–450 W/m²) and are symmetrically distributed around the 1:1 line, with Biases of −5.38 W/m², −11.82 W/m², and −5.7 W/m², and RMSE values of 17.36 W/m², 23.18 W/m², and 21.71 W/m², respectively. The validation results under normal conditions are consistent with those reported in Kratz et al. [9].

In summary, ATI degrades the clear-sky SLDR estimate accuracy. The degradation in Bias/RMSE is approximately 10.0/5.0 W/m² for Models A and B, whereas the values for Model C are approximately 5.0/1.0 W/m². In contrast to Models A and B, the impact of ATI on Model C is less pronounced, as it possesses a certain resistance to ATI.

Figure 4 shows the spatial distribution of the validation results at each flux station under ATI conditions except for those with too few samples (the number of samples is less than 30). In North America, the Biases of Model A and Model B are mainly within the range of 0 to –20 W/m², and that of Model C predominantly falls within ±10 W/m². Accordingly, the RMSE values of Models A and C are mostly distributed between 5 and 20 W/m² and 10 and 25 W/m², respectively, while the RMSE value of Model B is primarily concentrated in 5–30 W/m². In Europe, the Biases of Models A are primarily between –20 and −10 W/m², with many of Model B’s Biases clustered below –30 W/m². In contrast, Model C’s Bias ranges from –30 to –10 W/m². The corresponding RMSE values are generally 15–20 W/m² for Model A; 15–40 W/m² for Models B and C, with values above 30 W/m² at most stations. Despite the limitation of a single flux site in Australia, all Models exhibit a pronounced negative Bias (exceeding –10 W/m²) in SLDR, accompanied by RMSEs of 15–30 W/m². For the African sites, Model biases are predominantly in the –20 to –10 W/m² range, with respective RMSE values of 15–20, 10–20, and 15–25 W/m². In northeastern Asia (120°–180°E), a similar pattern shows Biases exceeding –20 W/m². While the RMSE of Models A and B exceeds 30 W/m², that of Model C falls between 15 and 30 W/m². In Antarctica, Models A, B, and C exhibit Biases of −5 to 0, 0 to −20, and −10 to 10 W/m², respectively. The corresponding RMSE ranges are 5–10 W/m² for Model A and 10–20 W/m² for both Models B and C. These findings are consistent with the overall accuracies presented in Figure 3a,c,e.

3.2. Results for Polar and Non-Polar Region

The pronounced vulnerability and intricate environmental dynamics of polar regions (>60° latitude), distinct from mid-latitudes, necessitate a thorough assessment of the effects of ATI on CERES SLDR products. The validation results are provided in Figure 5, which shows the scatter density distribution of the validation results for polar and non-polar regions under ATI conditions. For Model A, a distinct characteristic is observed during polar ATI events, where SLDR values tend to cluster within a relatively narrow and lower range (approximately 100–250 W/m²). This feature is less pronounced in Models B and C. This prevalence of low SLDR values stems from the intense surface LW radiative cooling, which is compounded by the inherently low ambient atmospheric temperatures and scarce water vapor content characteristic of polar environments.

Table 1. The accuracy of Models A, B, and C for polar and non-polar regions under ATI and normal conditions, respectively.

SLDR	Metric	Polar		Non-Polar
		ATI	Normal	ATI	Normal
Model A	Bias	−21.47	−13.32	−9.48	−4.70
	RMSE	28.75	18.05	16.14	17.30
	R2	0.76	0.95	0.94	0.89
	No.	1832	3573	2432	41,963
Model B	Bias	−23.04	−22.05	−12.25	−9.26
	RMSE	31.04	30.23	20.35	21.06
	R2	0.66	0.85	0.92	0.90
	No.	27,574	32,134	7197	128,734
Model C	Bias	−10.54	−10.93	−6.34	−4.37
	RMSE	23.53	24.92	19.72	20.81
	R2	0.65	0.82	0.90	0.88
	No.	27,727	32,550	7276	127,568

shows the performance of CERES clear-sky SLDR products in polar and non-polar regions under ATI and normal conditions, respectively. ATI further degrades product accuracy in both polar and non-polar regions, as evidenced by increased negative Biases and RMSEs.

In polar regions, the Bias/RMSE values for Models A, B, and C are −13.32/18.05 W/m², −22.05/30.23 W/m², and −10.93/24.92 W/m² under normal atmospheric temperature conditions. Under ATI conditions, these values change to −21.47/28.75 W/m², −23.04/31.04 W/m², and −10.54/23.53 W/m². A comparison of these conditions reveals distinct model behaviors. Model A exhibits the most significant performance degradation, with its absolute Bias and RMSE increasing by 8.15 W/m² and 10.70 W/m², respectively. Model B shows a minor decline, with increases of 0.99 W/m² in Bias and 0.81 W/m² in RMSE. In contrast, Model C demonstrates exceptional stability and robustness; its absolute Bias remains nearly unchanged, while its RMSE improves by 1.39 W/m² under ATI conditions. These results suggest that Model C is the most robust choice for operational retrieval in polar regions prone to temperature inversions.

In non-polar regions, the impact of ATI is primarily manifested in the intensification of systematic negative biases. Models A, B, and C saw their negative Biases increase by 4.78 W/m², 2.99 W/m², and 1.97 W/m², respectively. Notably, Model A in non-polar regions shows a twofold increase in negative bias when ATI occurs, suggesting that the model’s deficiency is physically linked to the vertical temperature structure.

Moreover, a notable pattern emerges: CERES models substantially underestimate SLDR in polar regions compared to non-polar regions, regardless of ATI presence. This occurs because clear-sky SLDR is influenced by both near-surface air temperature and water vapor content. Due to the extremely low water vapor content in polar regions [36], SLDR tends to be underestimated there, and as shown in Table 1, the presence of ATIs further exacerbates this underestimation [9].

3.3. Results for Daytime and Nighttime

The accurate estimation of SLDR is subject to diurnal variations in atmospheric processes. While daytime conditions are largely governed by solar radiative input, nighttime atmospheric stability often promotes the development of ATIs, which can substantially alter the near-surface radiative environment. Evaluating daytime and nighttime accuracy separately allows for a more nuanced understanding of how ATIs influence the Models.

Detailed validation results for daytime and nighttime are presented in

Table 2. The accuracy of Models A, B, and C for daytime and nighttime under ATI and normal conditions, respectively.

SLDR	Metric	Daytime		Nighttime
		ATI	Normal	ATI	Normal
Model A	Bias	−12.93	−3.17	−15.20	−7.77
	RMSE	21.53	17.20	22.74	17.53
	R2	0.81	0.93	0.95	0.89
	No.	1077	23,686	3187	21,850
Model B	Bias	−15.82	−11.29	−21.98	−12.53
	RMSE	26.10	23.30	29.83	23.02
	R2	0.81	0.92	0.84	0.90
	No.	6614	92,731	28,157	68,137
Model C	Bias	−4.44	−4.68	−10.86	−7.06
	RMSE	22.09	21.65	22.94	21.78
	R2	0.79	0.91	0.83	0.88
	No.	6491	90,961	28,512	69,157

. In general, the occurrence of ATI leads to a degradation in SLDR estimate accuracy, but the magnitude of this impact varies significantly by Model and time of day. During the daytime, Models A and B exhibit a clear decline in performance under ATI conditions. For instance, the negative Bias for Model A widened from −3.17 to −12.93 W/m², while its RMSE increased from 17.20 W/m² to 21.53 W/m². Similarly, Model B sees its negative Bias exacerbate from −11.29 to −15.82 W/m² with RMSE rising from 23.30 to 26.10 W/m². In contrast, Model C demonstrates remarkable stability during the day, with its Bias remaining virtually unchanged (from −4.68 to −4.44 W/m²) and only a negligible increase in RMSE (from 21.65 to 22.09 W/m²), suggesting it is robust against daytime ATI events.

During nighttime, however, the negative impact of ATI becomes more systematic across all models. Model A’s Bias drops further from −7.77 to −15.20 W/m². Model B experiences the most severe degradation, with its Bias plunging from −12.53 to −21.98 W/m² and RMSE increasing significantly from 23.02 to 29.83 W/m². Even Model C, which was stable during the day, shows a noticeable decline at night, with Bias shifting from −7.06 to −10.86 W/m², although its RMSE increase remains moderate (1.16 W/m²).

Furthermore, ATI events are found to occur at significantly higher frequency during the nighttime compared to the daytime. This may be attributed to the imbalance among SLUR, shortwave downward radiance (SWDR) and SLDR [37]. In the absence of SWDR input at night, the surface typically experiences a net loss of energy due to continuous SLUR exceeding the SLDR, leading to surface cooling rapidly, fostering the development of a stable ATI where the surface is colder than the air directly above it.

4. Discussion

SLDR is dominated by near-surface air temperature and water vapor content under clear-sky conditions [5]. This is the physical basis for LW models. However, nighttime radiative cooling and extremely dry conditions in polar and high-altitude regions can cause the surface to cool faster than the air above, forming strong temperature inversions, particularly in polar and high-altitude regions. The surface skin temperature can be up to 20–30 K lower than that of the lowest atmospheric layers, as the extremely dry air allows the surface to radiate almost directly to space, resulting in a substantial underestimation of SLDR [9]. Generally, the CERES Models implicitly assume a typical lapse-rate structure, and the ATI condition was not incorporated into the developed LW Models. Hence, they systematically underestimate SLDR in regions where strong, persistent ATIs are common. To assess the influence of ATI on the SLDR estimate, we performed a sensitivity analysis utilizing version 2.4 of the Santa Barbara DISORT Atmospheric Radiative Transfer (SBDART) model [38]. The atmospheric profile inputs for these simulations are selected from the latest Thermodynamic Initial Guess Retrieval (TIGR2000) [39] database, which encompasses a wide range of global atmospheric conditions, featuring water vapor content from 0.1 to 8 g/cm² and surface temperature from 231 to 315 K. The profiles with relative humidity greater than 90% at any single level or greater than 85% at two consecutive levels were removed from the TIGR database [17]. The remaining clear-sky profiles were subsequently categorized as either normal or ATI according to the near-surface air temperature gradient, in accordance with the algorithm outlined in Section 2.4. Ultimately, we obtained 349 ATI profiles and 1032 normal profiles under the clear-sky conditions, respectively. Due to the limited variation in ATIIs in the TIGR atmospheric profiles, which is inadequate for an extensive sensitivity analysis, we did not use the selected ATI profiles but manually altered the normal atmospheric temperature profile to an anomalous one. For each normal profile, the initial bottom-level air temperature was preserved. Then, the temperatures at the first and second atmospheric levels immediately above the surface were calculated based on the air temperature inversion intensity (ATII) and replaced the original values. Subsequently, we retrieved the SLDR with both the normal and modified ATI atmospheric profiles, adopting the coefficients from the CERES LW Models as detailed in Section 2.1. The SDLR simulated by SBDART was treated as the true value and used to evaluate the SDLR calculated by the CERES LW algorithms.

Figure 6 displays the resultant Biases and RMSEs of SLDR retrieval for different ATIIs. The Bias of models shows a distinct sensitivity to the ATII. Model B exhibits a gradual reduction in Bias from 0.79 W/m² under normal conditions to −4.96 W/m² at the ATII of 5 K/100 m. A more apparent tendency is evident for Model C, of which Bias transits from a considerable positive value of 15.94 W/m² to a notable negative value of −13.17 W/m². The continual drop in Bias, resulting in negative Biases at high ATI strengths, suggests that both models tend to underestimate SLDR under robust inversion conditions.

Concerning the RMSE, Model B displays a consistent increase with ascending ATII, rising from 18.65 W/m² to 22.75 W/m². For Model C, the SLDR estimate remains nearly unchanged since the input variables (surface temperature and column precipitable water vapor) are consistent throughout. As the ATII escalates, the simulated “true” SLDR value rises, contributing to a reduction in the model’s initial overestimation, which ultimately shifts to a significant underestimation. Consequently, the Bias transitions from positive to negative. This tendency is explicitly reflected in the RMSE, which first diminishes from 17.56 W/m² to a minimum of 11.13 W/m² at the ATII of 2.5 K/100 m. Afterward, as ATII further advances, the RMSEs sharply rise to around 20 W/m².

The rise in air temperature causes increased emissions from the atmosphere, the effective atmospheric temperature derived from constant weighted air temperature in Model B fails to capture these changes, making it underestimate the SLDR. Meanwhile, Model C omits the thermal radiation emitted by the atmosphere, resulting in a systematic negative bias.

The moisture inversion frequently coincides with atmospheric temperature inversion. We investigated the resulting changes in SLDR during the simultaneous occurrence of both inversion types. The atmospheric moisture inversion intensity (AMII, %/100 m) is defined as:

$$\mathrm{A}\mathrm{M}\mathrm{I}\mathrm{I}={\displaystyle \frac{{\mathrm{W}}_2-{\mathrm{W}}_1}{{\mathrm{W}}_1}}{\displaystyle \frac{1}{({\mathrm{H}}_2-{\mathrm{H}}_1)}}\times 100$$

(17)

where ${\mathrm{W}}_1$ and ${\mathrm{W}}_2$ is the water vapor content (g/m³) at the bottom and the top of the inversion layer, the height (m) at the corresponding level is ${\mathrm{H}}_2$ and ${\mathrm{H}}_1$, respectively.

Figure 7 shows the resulting Biases and RMSEs of the retrieved SLDR for various AMIIs. It demonstrates that the RMSEs of both Models grow when the AMI intensifies, and this upward trend is more evident at higher ATI intensities. Both models’ Biases follow a trend of increasing negatively with the strength of the inversion. These results emphasize the complexities of obtaining SLDR in atmospheres with anomalous temperature and moisture profiles. The contemporaneous existence of these inversions significantly reduces the accuracy of SLDR estimates, underlining the necessity for retrieval methods that can properly account for such anomalous atmospheric circumstances.

5. Conclusions

This paper comprehensively investigated the accuracy of the clear-sky CERES SLDR products under ATI conditions, utilizing ground-measured SLDR collected from 409 flux stations across four radiation flux networks. Under ATI conditions, the accuracy of all three Models degraded in the polar and non-polar regions, as well as during the daytime and nighttime.

Under ATI conditions, the Bias/RMSE increases by approximately 10.0/5.0 W/m² for Models A and B, 5.0/1.0 W/m² for Model C, compared to those obtained under normal conditions. Both the Bias and RMSE of Models B and C are sensitive to the ATI intensity. The Bias decreases with the increase in ATI intensity, but RMSE behaves differently with the increase in ATI intensity. Both the Bias and RMSE deteriorate with the increase in the concurrent AMII.

This efforts in this study, combined with existing CERES SLDR product validation, provides a thorough evaluation of the CERES SLDR product, offering strong guidance for its users. Furthermore, these findings mark a crucial step toward refining our understanding of ATI’s impact on clear-sky SLDR estimates. We must make efforts to advance the current algorithm, most critically, by systematically integrating additional, well-quantified influencing factors.