Atmospheric correction for retrieving ground brightness temperature at commonly-used passive microwave frequencies

Xiao-Jing Han; Si-Bo Duan; Zhao-Liang Li

doi:10.1364/OE.25.000A36

1. Introduction

Instantaneous satellite observations have been used in a variety of applications to estimate ocean and land surface temperatures, land surface emissivity, and regional evapotranspiration [1–6]. However, previous studies have shown that atmosphere has significant effects on surface parameter retrieval, especially from optical and infrared remotely-sensed data, that must be corrected [7–12]. Therefore, atmospheric correction plays a key role in retrieving surface parameters.

A series of operational satellite-based passive microwave sensors have been available since 1978 [13], and passive microwave remote sensing is considered to be a more effective tool for all-weather monitoring of the earth’s surface than optical and infrared remote sensing [14–16]. Passive microwave data have also been widely used for the retrieval of ocean and land surface parameters (e.g., ocean temperature, salinity, soil moisture, land surface temperature, vegetation optical depth and snow water equivalence) [17–22]. The passive microwave brightness temperature measured at the top of atmosphere (TOA) is related to soil and vegetation on the land surface and atmospheric conditions (e.g., water vapor (WV), oxygen (O₂), and cloud liquid water (CLW)) [23]. Hence, land surface parameters can be inferred more accurately if atmospheric effects are considered. Microwave indices (e.g., microwave polarization difference index and microwave vegetation index) are often used to retrieve land surface parameters due to their simplicity and practicability, but most studies ignored the effect of the atmosphere [24, 25]. In addition, several studies of physically-based algorithms for land surface parameter retrieval also ignored the effect of the atmosphere [22, 26]. However, ignoring atmospheric effects leads to larger errors in the retrieval of land surface parameters and thus limits the application of passive microwave data, especially at high frequencies. Qiu et al. (2007) [27] showed that the atmosphere directly influences land surface parameter retrieval; even on cold winter days under cloud-free atmospheric conditions, the atmospheric contribution is more than 0.86 K at the center of the Tibetan Plateau (for the frequencies of AMSR-E). Additionally, Liu et al. (2013) [9] indicated that atmospheric correction can improve the accuracy of land surface temperature retrieval with root mean square errors (RMSEs) from 6.04 K to 0.99 K. Hence, an analysis of the atmospheric impact on the ground brightness temperature is necessary because the accurate retrieval of land surface parameters is based on accurate ground brightness temperatures. Moreover, atmospheric corrections are needed to obtain accurate ground brightness temperatures. Most studies of atmospheric impacts and corrections have been performed under specific atmospheric conditions [28], in specific study areas [27, 29, 30] and at specific frequencies [9]. Little research has focused on the atmospheric impacts and corrections for all-weather conditions, all underlying surfaces and all commonly-used frequencies due to the complex microwave emissions and interactions between the atmosphere and the underlying land surface [31]. Therefore, an overall analysis of the atmospheric impact on ground brightness temperature and atmospheric correction is necessary for accurate land surface parameter retrieval, especially at commonly-used frequencies (i.e., 1.4 GHz, 6.93 GHz, 10.65 GHz, 18.7 GHz, 23.8 GHz, 36.5 GHz and 89.0 GHz) [13].

The objectives of this study are (1) to analyze the atmospheric impact on ground brightness temperature over numerous types of land surface under clear sky and cloudy (no rain) conditions and (2) to propose an atmospheric correction method for each commonly-used passive microwave frequency.

This article is organized as follows. Section 2 describes the data for the simulation and accuracy assessment. Section 3 provides an overall analysis of the atmospheric impact on ground brightness temperature, and Section 4 presents the atmospheric correction method and results. The discussion and conclusions are given in Sections 5 and 6, respectively.

2. Simulated data

To analyze the atmospheric impact on the passive microwave ground brightness temperature and develop appropriate atmospheric correction methods at each commonly-used frequency, a data set that includes a wide range of TOA brightness temperatures, ground brightness temperatures, atmospheric transmittances, atmospheric brightness temperatures (upwelling and downwelling), and the optical depths of WV, O₂ and CLW, which are obtained using a series of emissivities, land surface temperatures and atmospheric profiles, is needed. Two data sets were established: one for analyzing the atmospheric impact and developing the atmospheric correction method (Data set 1) and the other one for the accuracy assessment of the atmospheric correction method (Data set 2).

2.1 Data simulation

MonoRTM is an atmospheric radiative transfer model that is designed to process one or a number of monochromatic wavenumber values. It is particularly useful in the microwave spectral region. The model utilizes the same physics and continuum model as the line-by-line radiative transfer model [32] and is suitable for calculating radiances associated with atmospheric absorption by molecules in all spectral regions and cloud liquid water in the microwave spectrum. The Monochromatic Optical Depth Model (MODM) module is the core component of MonoRTM and is used to calculate the molecular optical depths. MonoRTM (v5.2) was downloaded from AER (http://rtweb.aer.com/) and used for the data simulation in this study. The data simulation procedure is shown in Fig. 1. The incidence angle at the surface was set to 53° based on the configuration of commonly-used passive microwave sensors. Seven commonly-used frequencies (i.e., 1.4 GHz, 6.93 GHz, 10.65 GHz, 18.7 GHz, 23.8 GHz, 36.5 GHz and 89.0 GHz) were selected for the simulation. The inputs of MonoRTM are the atmospheric profile, emissivity and land surface temperature (LST), which is equal to the atmospheric bottom temperature T₀. The outputs of MonoRTM are the TOA brightness temperature T_b, ground brightness temperature T_g, atmospheric transmittance τ, atmospheric brightness temperature (upwelling T_ba↑ and downwelling T_ba↓), and optical depths of WV (A_V), O₂ (A_O) and CLW (A_L). To consider more land surface types, a reasonable variation of emissivity is needed; thus, the emissivity was varied from 0.6 to 1.0 at steps of 0.1.

Fig. 1 Flowchart for generating the simulated data.

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2.2 Atmospheric profiles

To ensure that the data sets were representative, the Thermodynamic Initial Guess Retrieval (TIGR) data set from the Laboratory of Dynamic Meteorology (LMD) was used. The TIGR data set is a climatological library of 2311 representative atmospheric situations that were selected by statistical methods from 80,000 radiosonde reports [33]. Each profile is described from the surface to the TOA by the temperature, WV and ozone concentrations on a given pressure grid. Eighty clear sky atmospheric profiles with nearly uniformly distributed WV values ranging from 0.09 to 6.02 cm were selected from the TIGR data set; the bottom temperatures of these atmospheric profiles varies from 232.3 to 308.3 K [Fig. 2]. To consider both clear sky and cloudy atmospheric conditions, reasonable CLW was added to the atmospheric layers at heights of less than 5 km; the CLW was added to a single atmospheric layer in each situation. The CLW ranged from 0 to 0.5 mm at steps of 0.1 in the atmospheric layers at heights less than 2 km, 0 to 0.4 mm at steps of 0.1 mm from 2 km to 4 km, and 0 to 0.3 mm at steps of 0.1 mm from 4 km to 5 km. Clear sky conditions occur when the CLW is equal to 0, and CLW values greater than 0 represent cloudy conditions. Data set 1, which was used to analyze the atmospheric impact and develop the atmospheric correction method, was generated based on Fig. 1 using 60 atmospheric profiles, and Data set 2, which was used for the accuracy assessment of the atmospheric correction method, was generated using the same procedure with the other 20 atmospheric profiles [Fig. 2].

Fig. 2 Scatter plot of WV vs T₀ of 80 selected atmospheric profiles (60 atmospheric profiles are used for the simulation, and 20 are used for the accuracy assessment).

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3. Atmospheric impact on the passive microwave ground brightness temperature

The microwave radiative transfer equation in the Rayleigh-Jeans approximation over a flat lossy surface can be expressed in terms of the TOA brightness temperature as follows [19, 34]:

T_{b} = T_{b a ↑} + τ T_{g}

the ground brightness temperature T_g is defined as:

T_{g} = ε T_{s} + (1 - ε) T_{b a ↓} + (1 - ε) τ T_{b c}

where T_b is the brightness temperature at the TOA, ε is the surface emissivity, T_s is the surface temperature, τ is the atmospheric transmittance, T_ba↑ and T_ba↓ are the upwelling and downwelling atmospheric brightness temperatures, respectively, and T_bc is the brightness temperature that is contributed by the cosmic background, which is usually treated as a constant of 2.75 K.

Thus, the atmospheric impact on T_g can be written as:

Δ T = T_{g} - T_{b} = (1 - τ) T_{g} - T_{b a ↑}

The atmospheric impact on T_g is mainly reflected in τ and T_ba↑, which represent the atmospheric absorptions and emissions, respectively. Specifically, τ decreases as the atmospheric absorptions increase, and T_ba↑ increases as the atmospheric emissions increase. The means and ranges of τ and T_ba↑ at each frequency are shown in Fig. 3. The means of atmospheric absorptions and emissions increase as the frequency increases except for 23.8 GHz. Due to the strong absorption of WV at 23.8 GHz, the means of the atmospheric absorptions and emissions at 23.8 GHz are greater than those at 36.5 GHz. The ranges of atmospheric absorptions and emissions have similar variations as the means. Consequently, compared to the 7 frequencies shown in Fig. 3, the minimum variation of τ (from 0.9783 to 0.9865 with a mean of 0.9847) occurs at 1.4 GHz, while the maximum variation of τ (from 0.2665 to 0.8730 with a mean of 0.4962) occurs at 89.0 GHz. The minimum variation of T_ba↑ (from 3.71 K to 5.02 K with a mean of 4.02 K) occurs at 1.4 GHz, while the maximum variation of T_ba↑ (from 33.04 K to 213.13 K with a mean of 141.74 K) occurs at 89.0 GHz.

Fig. 3 Means and ranges of τ (a) and T_ba↑ (b) at each frequency (i.e., 1.4 GHz, 6.93 GHz, 10.65 GHz, 18.7 GHz, 23.8 GHz, 36.5 GHz and 89.0 GHz).

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To evaluate the atmospheric impact on T_g at each frequency, ΔT was estimated with Eq. (2) using Data set 1. Figure 4 shows the means and RMSEs of ΔT for each emissivity at each frequency. For a specific frequency, the atmospheric effect varies with the emissivity because different emissivities lead to different values of T_g. Figure 4(a) shows that the absolute values of the means of ΔT decrease as the emissivity increases from 0.6 to 0.9, whereas the opposite tendency is exhibited when the emissivity ranges from 0.9 to 1.0. Mean ΔT values of 0 occur when the emissivities are 0.9066, 0.9430, 0.9569, 0.9599, 0.9381, 0.9518 and 0.9146 at frequencies of 1.4 GHz, 6.93 GHz, 10.65 GHz, 18.7 GHz, 23.8 GHz, 36.5 GHz and 89.0 GHz, respectively. Figure 4(b) indicates that the RMSEs of ΔT decrease when the emissivity ranges from 0.6 to 0.9, whereas the variation of the RMSEs of $Δ T$ depends on frequency when the emissivity ranges from 0.9 to 1.0. The minimum RMSEs of ΔT, which are equal to the standard deviations of ΔT, are obtained when the emissivities are 0.9066, 0.9430, 0.9569, 0.9599, 0.9381, 0.9518, and 0.9146 at frequencies of 1.4 GHz, 6.93 GHz, 10.65 GHz, 18.7 GHz, 23.8 GHz, 36.5 GHz and 89.0 GHz, respectively. In other words, T_g can be approximately described using T_b with a relatively high accuracy at these emissivities and frequencies. Moreover, Fig. 4 clearly demonstrates that for a specific emissivity, the atmospheric effect varies with the frequency because different amounts of the passive microwave signal are absorbed at different frequencies. The higher the frequency, the larger the absolute values of the means of ΔT except for 23.8 GHz. The RMSEs of ΔT increase as the frequency increases except for 23.8 GHz. Due to the strong absorption of WV at 23.8 GHz, the absolute values of the means of ΔT at 23.8 GHz are greater than those at 36.5 GHz, and the RMSEs of ΔT at 23.8 GHz are greater than those at 36.5 GHz except for the emissivity of 0.9. Note that the minimum RMSEs of ΔT are obtained when the emissivity is approximately 0.9 for all frequencies; thus, the variation of the RMSEs of ΔT for the emissivity of 0.9 differs from those of the other emissivities. The RMSEs of ΔT increase as the frequency increases for the emissivity of 0.9.

Fig. 4 Means (a) and RMSEs (b) of ΔT for each emissivity (i.e., 0.6, 0.7, 0.8, 0.9 and 1.0) at each frequency (i.e., 1.4 GHz, 6.93 GHz, 10.65 GHz, 18.7 GHz, 23.8 GHz, 36.5 GHz and 89.0 GHz).

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Figure 5 also shows the ranges of ΔT for each emissivity at each frequency. For land surfaces with emissivities less than 0.9, T_g is cooler than T_b with a $Δ T$ less than 0 for all atmospheric conditions at all frequencies except for 89.0 GHz, which leads to atmospheric warming effect. For land surfaces with emissivities of 0.9 and 1.0, the atmospheric effect (warming and cooling) depends on the frequency and the atmospheric conditions.

Fig. 5 Ranges of ΔT for each emissivity (i.e., 0.6, 0.7, 0.8, 0.9 and 1.0) at each frequency (i.e., 1.4 GHz, 6.93 GHz, 10.65 GHz, 18.7 GHz, 23.8 GHz, 36.5 GHz and 89.0 GHz).

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Overall, the atmospheric effect on T_g is negligible at 1.4 GHz for land surfaces with emissivities greater than 0.7, at 6.93 GHz for land surfaces with emissivities greater than 0.8, and at 10.65 GHz for land surfaces with emissivities greater than 0.9 if an accuracy of less than 1 K is desired. If an accuracy of less than 2 K is required, the atmospheric effect is negligible at 1.4 GHz for all land surfaces, at 6.93 GHz for land surfaces with emissivities greater than 0.7, at 10.65 GHz for land surfaces with emissivities greater than 0.8, and at 18.7 GHz for land surfaces with emissivities greater than 0.9. Otherwise, errors are caused by atmospheric effects [Fig. 4(b)]; the atmospheric correction is necessary for these conditions.

4. Atmospheric correction method and results

Based on the microwave radiative transfer equation [Eq. (1)], an atmospheric correction method is developed by parameterizing T_ba↑ and τ.

4.1 Parameterization of τ

The extinction (absorption and scattering) of the atmospheric components (WV, O₂ and CLW) gives the total optical depth A_I (integrated in the observation direction):

A_{I} = A_{V} + A_{O} + A_{L}

where A_V, A_O and A_L represent the integrated optical depths (in the observation direction) of WV, O₂ and CLW, respectively. Hereafter, the subscripts V, O and L denote WV, O₂ and CLW, respectively.

Thus, the total transmittance can be expressed as:

τ = e^{- A_{I}} = e^{- A_{V}} \times e^{- A_{O}} \times e^{- A_{L}} = τ_{V} \times τ_{O} \times τ_{L}

As shown in Eq. (4), A_V, A_O and A_L should be parameterized to obtain τ.

To perform the parameterization of A_V, the means and standard deviations of A_V at each frequency were computed using Data set 1. The results [Fig. 6] indicate that the means of A_V vary with the frequency. The means of A_V at 1.4 GHz, 6.93 GHz and 10.65 GHz are small, whereas they are relatively large at other frequencies, especially at 89.0 GHz. Due to the strong absorption of WV at 23.8 GHz, the mean of A_V at 23.8 GHz is greater than that at 36.5 GHz. The variations of the standard deviations are similar to those of the means. Figure 6 also shows that the means and standard deviations of A_V at 1.4 GHz, 6.93 GHz and 10.65 GHz are significantly less than 0.025. Hence, the effect of WV may be negligible or treated as a constant at 1.4 GHz, 6.93 GHz and 10.65 GHz.

Fig. 6 Means and standard deviations of A_v at each frequency (i.e., 1.4 GHz, 6.93 GHz, 10.65 GHz, 18.7 GHz, 23.8 GHz, 36.5 GHz and 89.0 GHz).

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Theoretically, at a given frequency, A_V is a function of three atmospheric parameters: WV, temperature and pressure. Because A_V is essentially directly proportional to the WV content and the WV content can be obtained more easily than temperature and pressure, it is reasonable to express A_V using the WV content [35]. Figure 7 shows scatter plots of the total WV content in observation direction (L_wv) and A_V, which demonstrate that A_V and L_wv are directly proportional. Hence, A_V can be expressed as follows:

A_{V} = a_{V} \times L_{w v}

where L_wv is the total WV content (cm), which is obtained by the integration of the WV density in the observation direction, and α_V is the absorption coefficient of WV. The values of α_V at each frequency, which are determined by linear regression with Data set 1, are shown in Table 1. The values of α_V increase as the frequency increases except for 23.8 GHz. This implies that the sensitivity to WV varies with the frequency. Clearly, A_V is insensitive to the variation of L_wv at 1.4 GHz, 6.93 GHz and 10.65 GHz because α_V is small at these frequencies. This is consistent with the results in Fig. 6. Moreover, the estimated values of A_V and τ_V were calculated from Eqs. (5) and (4), respectively, using the coefficients α_V in Table 1. Table 1 also shows the RMSEs of A_V and τ_V. The results illustrate that good estimates of τ_V were obtained with RMSEs ranging from 0 to 0.0141.

Fig. 7 Scatter plots of L_wv vs A_v at each frequency (i.e., 1.4 GHz, 6.93 GHz, 10.65 GHz, 18.7 GHz, 23.8 GHz, 36.5 GHz and 89.0 GHz).

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Table 1. Coefficients and RMSEs for the relationships between L_wv and A_v at each frequency (i.e., 1.4 GHz, 6.93 GHz, 10.65 GHz, 18.7 GHz, 23.8 GHz, 36.5 GHz and 89.0 GHz).

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To perform the parameterization of A_O, Fig. 8 exhibits the means and standard deviations of A_O at each frequency, which indicate that the means of A_O increase with increasing frequency; namely, the effect of O₂ increases with increasing frequency. The standard deviations of A_O are much smaller than the means of A_O, which indicates that the effect of O₂ is relatively stable under different atmospheric conditions. Figure 9 shows the scatter plots of L_wv vs A_O at each frequency, which illustrates that A_O is slightly varied with the variation of L_wv when L_wv is less than 2.0 cm, and there is no significant variation of A_O when L_wv is larger than 2.0 cm. This is consistent with the results of Fig. 8. Figure 9 also shows that the variation of A_O at each frequency is much smaller than that of A_V, thus, to simplify the parameterization, A_O can be approximated as a constant with an acceptable accuracy:

A_{O} = b_{O}

where b_O is the mean of A_O at each frequency. Hence, the estimated values of A_O and τ_O were computed using Eqs. (6) and (4). The values of b_O and the RMSEs of A_O and τ_O are shown in Table 2. The results show that Eq. (6) performs well for estimating A_O and that τ_O was obtained accurately with RMSEs ranging from 0.0018 to 0.0128.

Fig. 8 Means and standard deviations of A_O at each frequency (i.e., 1.4 GHz, 6.93 GHz, 10.65 GHz, 18.7 GHz, 23.8 GHz, 36.5 GHz and 89.0 GHz).

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Fig. 9 Scatter plots of L_wv vs A_O at each frequency (i.e., 1.4 GHz, 6.93 GHz, 10.65 GHz, 18.7 GHz, 23.8 GHz, 36.5 GHz and 89.0 GHz).

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Table 2. Values of b_O and RMSEs for Eq. (6) at each frequency (i.e., 1.4 GHz, 6.93 GHz, 10.65 GHz, 18.7 GHz, 23.8 GHz, 36.5 GHz and 89.0 GHz).

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To perform the parameterization of A_L, the means and standard deviations of A_L at each frequency were computed with Data set 1. The results [Fig. 10] indicate that the means of A_L increase as the frequency increases; namely, the effect of CLW increases with increasing frequency. The variations of the standard deviations of A_L are also similar to those of the means. Moreover, the means and standard deviations at 1.4 GHz, 6.93 GHz and 10.65 GHz are significantly less than 0.01 and are much smaller than those at the other frequencies. Hence, the effect of CLW may be negligible or treated as a constant at 1.4 GHz, 6.93 GHz and 10.65 GHz.

Fig. 10 Means and standard deviations of A_L at each frequency (i.e., 1.4 GHz, 6.93 GHz, 10.65 GHz, 18.7 GHz, 23.8 GHz, 36.5 GHz and 89.0 GHz).

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Theoretically, because the cloud droplet diameter (0.001-0.1 mm) is small compared to the microwave wavelengths considered in this study, the optical depth A_L is given by the Rayleigh scattering approximation [35]:

A_{L} = \frac{0.6 π L_{c l w}}{λ} i m (\frac{1 - ε^{'}}{2 + ε^{'}})

where λ is the radiation wavelength (cm), L_clw is the total cloud liquid water (mm), and εˊ is the water complex dielectric constant.

Hence, at a given frequency, A_L can be described using L_clw and εˊ. Previous study showed that εˊ varies with the water temperature at a certain frequency [36]. Thus, A_L/L_clw is a function of the mean temperature of the cloud T_clw. To analyze the relationship between T_clw and A_L/L_clw, Fig. 11 shows scatter plots of T_clw and A_L/L_clw at each frequency. The results demonstrate that linear relationships can approximate the relationships between T_clw and A_L/L_clw. Thus,

Fig. 11 Scatter plots between T_clw vs A_L/L_clw (the unit of L_clw is mm) at each frequency (i.e., 1.4 GHz, 6.93 GHz, 10.65 GHz, 18.7 GHz, 23.8 GHz, 36.5 GHz and 89.0 GHz).

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A_{L} / L_{c l w} = a_{L} \times T_{c l w} + b_{L}

A_L can thus be written as:

A_{L} = L_{c l w} \times (a_{L} \times T_{c l w} + b_{L})

The coefficients (α_L and b_L) were determined by linear regression using Data set 1. Moreover, the estimated values of A_L and τ_L can be retrieved from Eqs. (9) and (4) using the coefficients (α_L and b_L). The coefficients and the RMSEs of A_L and τ_L at each frequency are shown in Table 3 and indicate that τ_L was obtained accurately with RMSEs from 0 to 0.0057.

Table 3. Coefficients and RMSEs for the relationships between T_clw and A_L/L_clw at each frequency (i.e., 1.4 GHz, 6.93 GHz, 10.65 GHz, 18.7 GHz, 23.8 GHz, 36.5 GHz and 89.0 GHz).

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Hence, the total transmittance τ can be rewritten from Eqs. (4), (5), (6) and (9) as:

τ = e^{- (a_{V} \times L_{w v} + b_{O} + L_{c l w} \times (a_{L} \times T_{c l w} + b_{L}))}

Using the coefficients given in Tables 1-3, the estimated value of τ can be computed from Eq. (10). Figure 12 shows the errors of τ, which have RMSEs ranging from 0.0018 to 0.0134. It also shows that the RMSEs of τ increase with increasing frequency.

Fig. 12 RMSEs of τ estimated from Eq. (10) using the coefficients given in Tables 1-3 at each frequency (i.e., 1.4 GHz, 6.93 GHz, 10.65 GHz, 18.7 GHz, 23.8 GHz, 36.5 GHz and 89.0 GHz).

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4.2 Parameterization of T_ba↑

Theoretically, T_ba↑ is a function of τ and the atmospheric effective temperature T_a [19] and can be approximated as:

T_{b a ↑} = (1 - τ) T_{a}

T_a depends on the vertical distributions of the atmospheric temperature, WV and CLW. Since T_a is difficult to obtain, but is related to WV, to tackle this problem, we focus on the relationship between L_wv and T_a. Fig. 13 shows scatter plots of L_wv and T_a at each frequency. The plots indicate that the relationship between L_wv and T_a can be approximated by a quadratic relationship as follows:

Fig. 13 Scatter plots of L_wv vs T_a at each frequency (i.e., 1.4 GHz, 6.93 GHz, 10.65 GHz, 18.7 GHz, 23.8 GHz, 36.5 GHz and 89.0 GHz).

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T_{a} = a_{T} L_{w v}^{2} + b_{T} L_{w v} + c_{T}

The coefficients (α_T, b_T and c_T) are determined by polynomial fitting using Data set 1. Moreover, the estimated value of T_a can be calculated using these coefficients. The coefficients (α_T, b_T and c_T) and the RMSEs of T_a are shown in Table 4. Although the RMSEs of T_a are greater than 3.5 K at all frequencies, this accuracy of T_a is acceptable. The main reason is that the error of T_a only slightly influences T_ba↑ because 1-τ is small.

Table 4. Coefficients and RMSEs for the relationships between L_wv and T_a at each frequency (i.e., 1.4 GHz, 6.93 GHz, 10.65 GHz, 18.7 GHz, 23.8 GHz, 36.5 GHz and 89.0 GHz).

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Thus, T_ba↑ can be rewritten from Eqs. (10), (11) and (12) as follows:

T_{b a ↑} = (1 - e^{- (a_{V} \times L_{w v} + b_{O} + L_{c l w} \times (a_{L} \times T_{c l w} + b_{L}))}) \times (a_{T} L_{w v}^{2} + b_{T} L_{w v} + c_{T})

Using the coefficients given in Tables 1-4, the estimated values of T_ba↑ were calculated from Eq. (13). Figure 14 shows the RMSEs of T_ba↑, which range from 0.44 K to 4.15 K, and demonstrates that the RMSEs of T_ba↑ increase with increasing frequency.

Fig. 14 RMSEs of T_ba↑ estimated with Eq. (13) using the coefficients given in Tables 1-4 at each frequency (i.e., 1.4 GHz, 6.93 GHz, 10.65 GHz, 18.7 GHz, 23.8 GHz, 36.5 GHz and 89.0 GHz).

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4.3 Generalized atmospheric correction method

Based on the parameterization of τ and T_ba↑, the generalized atmospheric correction method can be expressed as:

T_{g} = (T_{b} - T_{b a ↑}) / τ

where T_ba↑ and τ can be estimated from Eqs. (13) and (10).

Using the coefficients shown in Tables 1-4, T_g was estimated using Eq. (14). Table 5 shows the RMSEs of T_g, which range from 0.10 K to 5.74 K. High accuracies of T_g with RMSEs less than 1 K are achieved at 1.4 GHz, 6.93 GHz, 10.65 GHz and 18.7 GHz. RMSEs less than 2 K are obtained at 23.8 GHz and 36.5 GHz, and the low accuracy is achieved at 89.0 GHz with an RMSE of 5.74 K.

Table 5. RMSEs of T_g estimated from Eq. (14) using the coefficients shown in Tables 1-4 at each frequency (i.e., 1.4 GHz, 6.93 GHz, 10.65 GHz, 18.7 GHz, 23.8 GHz, 36.5 GHz and 89.0 GHz).

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To improve the accuracy of the generalized atmospheric correction method [Eq. (14)], all of the coefficients in Eq. (14) are re-determined by nonlinear fitting with Data set 1. Using the new coefficients, the estimated T_g can be calculated using Eq. (14). Table 6 shows the new coefficients and the RMSEs of T_g at each frequency. The results show that if the input variables (i.e., L_wv, L_clw and T_clw) can be obtained accurately, favorable accuracy of T_g can be achieved with RMSEs less than 1 K at 1.4 GHz, 6.93 GHz, 10.65 GHz, 18.7 GHz and 36.5 GHz. The RMSE is relatively large at 23.8 GHz due to the high sensitivity to WV. Because the accuracies of the parameterizations of τ and T_ba↑ are relatively low at 89.0 GHz, the worst accuracy of T_g is obtained at 89.0 GHz with an RMSE = 4.35 K. The value of T_g at 89.0 GHz is severely affected by the atmosphere; thus, it is not suitable for the retrieval of land surface parameters. Histograms of the differences between the actual and estimated values of T_g at each frequency are shown in Fig. 15 for comparison. The most prominent feature in Fig. 15 is that the distributions of the differences between the actual and estimated values of T_g at each frequency are approximately Gaussian. However, tails are observed on the left sides of the histograms for 18.7 GHz and 23.8 GHz; they are caused by an atmospheric profile with a WV of 4.31 cm. Although the WV of this atmospheric profile is not large, more than 90 percent of the WV is concentrated at height of less than 2 km; and the lower atmosphere affects T_g much more than the upper atmosphere. Thus, large errors are caused under this atmospheric condition.

Table 6. New coefficients of Eq. (14) determined by nonlinear fitting with Data set 1 and the RMSEs of T_g estimated from Eq. (14) using the new coefficients at each frequency (i.e., 1.4 GHz, 6.93 GHz, 10.65 GHz, 18.7 GHz, 23.8 GHz, 36.5 GHz and 89.0 GHz).

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Fig. 15 Histograms of the differences between actual and estimated values of T_g, which were obtained from Eq. (14) using the coefficients in Table 6 at each frequency (i.e., 1.4 GHz, 6.93 GHz, 10.65 GHz, 18.7 GHz, 23.8 GHz, 36.5 GHz and 89.0 GHz).

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4.4 Accuracy assessment

Data set 2, which has the same variations of CLW and emissivities as Data set 1 but different atmospheric conditions, was used to evaluate the performance of the generalized atmospheric correction method. The estimated T_g was retrieved from Eq. (14) using the coefficients given in Table 6. Figure 16 shows the biases and RMSEs of the differences between the actual and estimated values of T_g at each frequency. Better accuracies with biases less than 0.1 K and RMSEs less than 1 K are obtained for all frequencies except 23.8 GHz and 89.0 GHz. An RMSE of 1.18 K is obtained for 23.8 GHz due to the strong absorption of WV, and an RMSE of 3.97 K is obtained for 89.0 GHz due to the low transmittance.

Fig. 16 Biases and RMSEs of T_g retrieved from Eq. (14) using the coefficients given in Table 6 with Data set 2 at each frequency (i.e., 1.4 GHz, 6.93 GHz, 10.65 GHz, 18.7 GHz, 23.8 GHz, 36.5 GHz and 89.0 GHz).

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5. Discussion

The simulated data that were used in this study are more representative than those in previous studies of atmospheric impacts and corrections [9, 28–30, 37], because they cover both clear sky and cloudy conditions, include worldwide atmospheric situations and consider the emissivities of numerous types of land surfaces. These characteristics make the proposed atmospheric correction method much more applicable.

To ensure the reliability of the generalized atmospheric correction method, an accuracy assessment was performed using Data set 2 with different atmospheric conditions compared to Data set 1. Nevertheless, a shortcoming of this study is that the generalized atmospheric correction method was only evaluated using simulated data, but not field measurements due to the lack of in situ data.

The accuracy assessment that was conducted in Section 4.4 indicates that the generalized atmospheric correction method performs well for retrieving T_g. However, the uncertainties of the input variables (L_wv, L_clw and T_clw) also influence the performance of the generalized atmospheric correction method. Thus, the uncertainty of the generalized atmospheric correction method is evaluated with a sensitivity analysis. The uncertainty of T_g (δT_g) is estimated as the combination of the algorithm’s uncertainty (δalg) and the uncertainties associated with the errors of the input variables (δT_b, δL_wv, δL_clw and δT_clw):

δ T_{g} = \sqrt{δ a l g^{2} + δ T_{b}^{2} + δ L_{w v}^{2} + δ L_{c l w}^{2} + δ T_{c l w}^{2}}

with δ X = | \frac{\partial T_{g}}{\partial X} Δ X |

where X denotes the input variables (i.e., T_b, L_wv, L_clw and T_clw), and ΔX denotes the errors of the input variables.

As shown in Table 6, the errors of the algorithm range from 0.05 K to 4.35 K at frequencies from 1.4 GHz to 89.0 GHz. The errors of T_b are mainly from the noise equivalent temperature difference (NEΔT), which have typical values of 0.3, 0.3, 0.5, 0.5, 0.5, 0.5, 0.8 at 1.4 GHz, 6.93 GHz, 10.65 GHz, 18.7 GHz, 23.8 GHz, 36.5 GHz and 89.0 GHz, respectively [38]. These values are obtained from commonly-used microwave radiometers. The accuracy of the MODIS Total Precipitable Water product ranges from 5% to 15% [39, 40]. Thus, an error of ± 15% × L_wv cm was considered for the sensitivity analysis of L_wv. Based on Ji and Shi (2012) [41], an uncertainty of ± 0.14 mm was considered for the sensitivity analysis of L_clw. Because T_clw is difficult to obtain, the cloud top temperature product of MODIS was used to replace T_clw, and an error of ± 10 K was considered.

Figure 17 shows the sensitivity of the input variables X (i.e., T_b, L_wv, L_clw and T_clw) to T_g at each frequency. The results illustrate that the sensitivity of X to T_g varies at different frequencies. The sensitivities of T_b and L_wv to T_g increase slightly with increasing frequency except for 23.8 GHz. Due to the strong absorption of WV at 23.8 GHz, the sensitivities of T_b and L_wv to T_g at 23.8 GHz are greater than those at 36.5 GHz. The ∂T_g /∂T_b ranges from 1.02 to 1.89, and ∂T_g /∂L_wv varies from −0.03 to −3.89 with increasing frequency. The sensitivity of T_clw to T_g is much less than that of the other input variables, with ∂T_g /∂T_clw varying between −0.11 and 0.01. The sensitivity of L_clw to T_g increases significantly as the frequency increases, with ∂T_g /∂L_wv ranging from −0.03 to −23.58. Additionally, the sensitivity to T_g varies with the different input variables at a specific frequency. At frequencies less than 18.7 GHz, T_g is more sensitive to the error of T_b than the errors of the other input variables, while at frequencies higher than 18.7 GHz, T_g is more sensitive to the error of L_clw than the errors of the other input variables. At 18.7 GHz, the sensitivities of T_b and L_clw to T_g are nearly the same. The sensitivity of L_wv to T_g is relatively small compared to those of T_b and L_clw.

Fig. 17 Sensitivities of input variables X (i.e., T_b, L_wv, L_clw and T_clw) to T_g at each frequency (i.e., 1.4 GHz, 6.93 GHz, 10.65 GHz, 18.7 GHz, 23.8 GHz, 36.5 GHz and 89.0 GHz).

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Figure 18 shows the variation of δT_g with different values of L_wv at each frequency. There is no significant variation of δT_g as L_wv increases at 1.4 GHz, 6.93 GHz and 10.65 GHz because τ_V is large at these frequencies. This is consistent with the results shown in Fig. 6. The values of δT_g at 1.4 GHz, 6.93 GHz and 10.65 GHz are approximately 0.31 K, 0.33 K and 0.58 K, respectively, for all L_wv. Additionally, the values of δT_g at 18.7 GHz and 36.5 GHz increase slightly as L_wv increases. The values of δT_g at 23.8 GHz and 89.0 GHz increase rapidly as L_wv increases due to the strong absorption of water vapor at 23.8 GHz and the low transmittance at 89.0 GHz. Figure 18 also shows that δT_g increases as the frequency increases except at 23.8 GHz. Overall, the best accuracy of δT_g = 0.31 K is obtained for L_wv = 1.0 cm at 1.4 GHz, whereas the worst accuracy of δT_g = 8.21 K is achieved for L_wv = 10.0 cm at 89.0 GHz.

Fig. 18 Variation of δT_g caused by the errors of the algorithm, T_b, L_wv, L_clw and T_clw, with increasing L_wv at each frequency (i.e., 1.4 GHz, 6.93 GHz, 10.65 GHz, 18.7 GHz, 23.8 GHz, 36.5 GHz and 89.0 GHz).

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Based on this analysis, the generalized atmospheric correction method provides favorable accuracy, but the input variables restrict its application when the input variables are difficult to obtain or have low levels of accuracy. Thus, a simplification of the generalized atmospheric method is necessary to improve its applicability. The T_ba↑ and τ are treated as constants at each frequency because the ranges of T_ba↑ and τ are relatively small at the lower frequencies [Fig. 3]. The values of T_ba↑ and τ were determined from Eq. (1) by linear fitting using Data set 1. Table 7 shows the values of T_ba↑ and τ for the simplified atmospheric correction method at each frequency. Using the values of T_ba↑ and τ given in Table 7, T_g can be estimated using Eq. (1). Figure 19 shows the RMSEs of the differences between the actual and estimated values of T_g. Better accuracies with RMSEs of T_g less than 1 K are obtained at 1.4 GHz, 6.93 GHz and 10.65 GHz. However, the simplified atmospheric correction method performs significantly worse than the generalized atmospheric correction method at higher frequencies. The RMSEs of T_g are 3.20 K, 7.06 K, 3.93 K and 8.44 K at 18.7 GHz, 23.8 GHz, 36.5 GHz and 89.0 GHz, respectively.

Table 7. Values of T_ba↑ and τ of the simplified atmospheric correction method at each frequency (i.e., 1.4 GHz, 6.93 GHz, 10.65 GHz, 18.7 GHz, 23.8 GHz, 36.5 GHz and 89.0 GHz).

View Table | View all tables in this article

Fig. 19 RMSEs of the differences between the actual and estimated values of T_g, which were obtained from Eq. (1) using the values of T_ba↑ and τ given in Table 7, at each frequency (i.e., 1.4 GHz, 6.93 GHz, 10.65 GHz, 18.7 GHz, 23.8 GHz, 36.5 GHz and 89.0 GHz).

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Additionally, as was explained in Section 3, the means of ΔT (namely, the biases of the differences between T_g and T_b) vary with different emissivities and frequencies. Figure 20 shows the relationship between ε and the biases at each frequency. Linear relationships can accurately describe the relationships between $ε$ and biases. Thus, the bias can be written as:

Fig. 20 Relationships between ε and the biases of the differences between the actual and estimated values of T_g.

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b i a s = m \times ε + n

Table 8 shows the coefficients m and n that are determined by linear regression using Data set 1. Consequently, for studies with known emissivities, an emissivity-based atmospheric correction can be performed with Eq. (17). Using the coefficients given in Table 8, T_g was estimated from Eqs. (16) and (17). The RMSEs of the differences between the actual and estimated values of T_g for each emissivity at each frequency are shown in Fig. 21. The results show that better accuracies with RMSEs less than 1 K are achieved for all emissivities at 1.4 GHz, 6.93 GHz and 10.65 GHz, for emissivities of 0.9 and 1.0 at 18.7 GHz, and for emissivity of 0.9 at 23.8 GHz and 36.5 GHz.

Table 8. Coefficients m and n determined by linear regression using Data set 1 for Eq. (16) at each frequency (i.e., 1.4 GHz, 6.93 GHz, 10.65 GHz, 18.7 GHz, 23.8 GHz, 36.5 GHz and 89.0 GHz).

View Table | View all tables in this article

Fig. 21 RMSEs of the differences between the actual and estimated values of T_g, which were obtained from Eqs. (16) and (17) using the coefficients given in Table 8, for each emissivity (i.e., 0.6, 0.7, 0.8, 0.9 and 1.0) at each frequency (i.e., 1.4 GHz, 6.93 GHz, 10.65 GHz, 18.7 GHz, 23.8 GHz, 36.5 GHz and 89.0 GHz).

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T_{g} = T_{b} + b i a s

6. Conclusion

Analyses of the atmospheric impacts were performed for many types of land surfaces, and an appropriate atmospheric correction method was developed for each frequency (i.e., 1.4 GHz, 6.93 GHz, 10.65 GHz, 18.7 GHz, 23.8 GHz, 36.5 GHz and 89.0 GHz).

To analyze the atmospheric impact on T_g, a representative data set (Data set 1) that was simulated with MonoRTM was used to compare T_g and T_b for different emissivities at different frequencies. The results show that the atmospheric effect varies with the different emissivities and frequencies. The RMSEs of ΔT decrease when the emissivity ranges from 0.6 to 0.9, whereas the variation of the RMSEs of ΔT depends on the frequency when the emissivity ranges from 0.9 to 1.0. The RMSEs of ΔT increase as the frequency increases except for 23.8 GHz. The minimum RMSEs of ΔT, which are equal to the standard deviations of ΔT, are obtained when the emissivities are 0.9066, 0.9430, 0.9569, 0.9599, 0.9381, 0.9518, and 0.9146 at frequencies of 1.4 GHz, 6.93 GHz, 10.65 GHz, 18.7 GHz, 23.8 GHz, 36.5 GHz and 89.0 GHz, respectively. Consequently, T_g can be approximately described using T_b with a relatively favorable accuracy at these emissivities and frequencies.

To remove the atmospheric effect, the generalized atmospheric correction method was developed based on the microwave radiative transfer equation. The parameters T_ba↑ and τ were expressed as functions of three atmospheric parameters (L_wv, L_clw and T_clw). The results indicate that fitting errors with RMSEs of T_g less than 1 K are achieved at all frequencies except for 23.8 GHz and 89.0 GHz. The RMSEs of T_g for 23.8 GHz and 89.0 GHz are 1.34 K and 4.35 K, respectively. The behavior of the generalized atmospheric correction method was evaluated using Data set 2, which was simulated independently using the same variations of CLW and emissivity as in Data set 1 but different atmospheric conditions. Better accuracies with RMSEs of T_g less than 1 K are obtained at all frequencies except for 23.8 GHz and 89.0 GHz, and RMSEs of 1.18 K and 3.97 K are achieved at 23.8 GHz and 89.0 GHz, respectively.

To meet the demands of studies that lack accurate input parameters (L_wv, L_clw and T_clw) to perform the generalized atmospheric correction, the generalized atmospheric correction method was simplified. The parameters T_ba↑ and τ were treated as constants in the simplified method. The results show that high accuracies with RMSEs of T_g less than 1 K are obtained at 1.40 GHz, 6.93 GHz and 10.65 GHz. The RMSEs of T_g are 3.20 K, 7.06 K, 3.93 K and 8.44 K at 18.7 GHz, 23.8 GHz, 36.5 GHz and 89.0 GHz, respectively.

Additionally, an emissivity-based atmospheric correction method was developed from Eq. (2) for studies with known emissivities. The biases of the differences between T_g and T_b were expressed as a function of ε at each frequency. The results show that high accuracies with RMSEs of T_g less than 1 K are achieved for all emissivities at 1.4 GHz, 6.93 GHz and 10.65 GHz, for emissivities of 0.9 and 1.0 at 18.7 GHz, and for emissivity of 0.9 at 23.8 GHz and 36.5 GHz.

In general, the atmospheric effect on T_g is negligible at 1.4 GHz for land surfaces with emissivities greater than 0.7, at 6.93 GHz for land surfaces with emissivities greater than 0.8, and at 10.65 GHz for land surfaces with emissivities greater than 0.9 if an accuracy less than 1 K is desired. Otherwise, errors are caused by atmospheric effects [Fig. 4(b)]. Thus, the atmospheric correction is necessary under these conditions. If an accuracy less than 1 K is desired, the simplified atmospheric correction is needed at 1.4 GHz, 6.93 GHz and 10.65 GHz, and the generalized atmospheric correction is needed at 18.7 GHz and 36.5 GHz. The best accuracy at 23.8 GHz is RMSE = 1.34 K, whereas that at 89.0 GHz is RMSE = 4.35 K. The appropriate atmospheric correction method can be used based on the available data, frequency and accuracy requirements.

This study also provides a method to estimate τ and T_ba↑ at different frequencies using the atmospheric parameters (L_wv, L_clw and T_clw), which is important for simultaneously determining the land surface parameters using multi-frequency passive microwave satellite data. Additionally, this work will help scientists to rely more or better on higher microwave frequencies to improve spatial resolution [42].

Funding

National Natural Science Foundation of China (NSFC) (41231170).

Acknowledgments

We thank two reviewers for their valuable and detailed comments that have greatly improved the paper. We also thank the Laboratory of Dynamic Meteorology (LMD) and Atmospheric & Environmental Research (AER) Radiative Transfer Working Group for providing the atmospheric profiles and MonoRTM, separately.

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Freq. (GHz)	1.4	6.93	10.65	18.7	23.8	36.5	89.0
α_V (cm⁻¹)	0.00003	0.00070	0.00187	0.01639	0.05262	0.01814	0.08561
RMSE (A_V)	0.0000	0.0002	0.0004	0.0021	0.0020	0.0045	0.0232
RMSE (τ_V)	0.0000	0.0002	0.0004	0.0019	0.0014	0.0040	0.0141

Freq. (GHz)	1.4	6.93	10.65	18.7	23.8	36.5	89.0
b_O	0.0152	0.0172	0.0185	0.0236	0.0296	0.0697	0.0872
RMSE (A_O)	0.0018	0.0021	0.0022	0.0028	0.0034	0.0079	0.0142
RMSE (τ_O)	0.0018	0.0020	0.0022	0.0027	0.0033	0.0073	0.0128

Freq. (GHz)	1.4	6.93	10.65	18.7	23.8	36.5	89.0
α_L (K⁻¹·mm⁻¹)	−0.00001	−0.00029	−0.00066	−0.00175	−0.00252	−0.00429	−0.00698
b_L (mm⁻¹)	0.004	0.091	0.207	0.554	0.806	1.419	2.855
RMSE (A_L)	0.0000	0.0004	0.0009	0.0018	0.0021	0.0017	0.0078
RMSE (τ_L)	0.0000	0.0004	0.0009	0.0018	0.0020	0.0016	0.0057

Freq. (GHz)	1.4	6.93	10.65	18.7	23.8	36.5	89.0
α_T (K·cm⁻²)	−0.4997	−0.5758	−0.6328	−0.6972	−0.6908	−0.6840	−0.7180
b_T (K·cm⁻¹)	7.825	9.202	9.924	10.496	10.174	10.475	10.545
c_T (K)	241.16	247.63	250.24	252.32	251.86	250.37	251.08
RMSE (K)	3.54	3.84	4.12	4.38	4.34	4.59	5.09

Freq. (GHz)	1.40	6.93	10.65	18.70	23.80	36.50	89.00
RMSE (K)	0.10	0.13	0.17	0.56	1.60	1.11	5.74

Atmospheric correction for retrieving ground brightness temperature at commonly-used passive microwave frequencies

Abstract

1. Introduction

2. Simulated data

2.1 Data simulation

2.2 Atmospheric profiles

3. Atmospheric impact on the passive microwave ground brightness temperature

4. Atmospheric correction method and results

4.1 Parameterization of τ

4.2 Parameterization of T_ba↑

4.3 Generalized atmospheric correction method

4.4 Accuracy assessment

5. Discussion

6. Conclusion

Funding

Acknowledgments

References and links

Cited By

Figures (21)

Tables (8)

Equations (19)

Optics Express

Freq. (GHz)	1.40	6.93	10.65	18.70	23.80	36.50	89.00
α_V (cm⁻¹)	−0.000379	0.000169	0.001203	0.014909	0.049189	0.014281	0.053736
b_O	0.01704	0.01938	0.02125	0.03007	0.04019	0.08509	0.17127
α_L (K⁻¹·mm⁻¹)	−0.000057	−0.000241	−0.000452	−0.000760	−0.000219	−0.001117	0.013326
b_L (mm⁻¹)	0.0168	0.0779	0.1490	0.2744	0.1606	0.5212	−2.9704
α_T (K·cm⁻²)	−0.3196	−0.3020	−0.3158	−0.3758	−0.3457	−0.3731	−0.3572
b_T (K·cm⁻¹)	5.889	6.332	6.579	6.842	6.007	6.955	6.019
c_T (K)	245.05	253.02	256.52	260.54	262.62	258.29	264.55
RMSE	0.05	0.08	0.12	0.47	1.34	0.89	4.35

Freq. (GHz)	1.40	6.93	10.65	18.70	23.80	36.50	89.00
T_ba↑ (K)	3.85	5.82	8.28	23.56	36.58	41.63	63.58
τ	0.98546	0.97898	0.97075	0.92057	0.88208	0.85663	0.78701

Freq. (GHz)	1.40	6.93	10.65	18.70	23.80	36.50	89.00
m (K)	4.3	6.4	9.0	28.7	53.7	44.5	69.4
n (K)	−3.92	−5.99	−8.64	−27.52	−50.34	−42.33	−63.43

Abstract

1. Introduction

2. Simulated data

2.1 Data simulation

2.2 Atmospheric profiles

3. Atmospheric impact on the passive microwave ground brightness temperature

4. Atmospheric correction method and results

4.1 Parameterization of τ

4.2 Parameterization of Tba↑

4.3 Generalized atmospheric correction method

4.4 Accuracy assessment

5. Discussion

6. Conclusion

Funding

Acknowledgments

References and links

Cited By

Figures (21)

Tables (8)

Equations (19)

Optics Express

4.2 Parameterization of T_ba↑